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

Percentage Accurate: 95.6% → 95.6%

Time: 7.4s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

Alternative 2: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-45} \lor \neg \left(y \leq 1.3 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4.9e-45) (not (<= y 1.3e-43))) (/ x y) (* x (/ (/ -1.0 t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.9e-45) || !(y <= 1.3e-43)) {
		tmp = x / y;
	} else {
		tmp = x * ((-1.0 / t) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-4.9d-45)) .or. (.not. (y <= 1.3d-43))) then
        tmp = x / y
    else
        tmp = x * (((-1.0d0) / t) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.9e-45) || !(y <= 1.3e-43)) {
		tmp = x / y;
	} else {
		tmp = x * ((-1.0 / t) / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -4.9e-45) or not (y <= 1.3e-43):
		tmp = x / y
	else:
		tmp = x * ((-1.0 / t) / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.9e-45) || !(y <= 1.3e-43))
		tmp = Float64(x / y);
	else
		tmp = Float64(x * Float64(Float64(-1.0 / t) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -4.9e-45) || ~((y <= 1.3e-43)))
		tmp = x / y;
	else
		tmp = x * ((-1.0 / t) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4.9e-45], N[Not[LessEqual[y, 1.3e-43]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(N[(-1.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8999999999999998e-45 or 1.3e-43 < y

   1. Initial program 97.4%

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

   3. Taylor expanded in y around inf 81.0%

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

   if -4.8999999999999998e-45 < y < 1.3e-43

   1. Initial program 96.1%

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

      1. clear-num 95.9%

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

      2. associate-/r/ 96.1%

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

   4. Applied egg-rr 96.1%

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

   5. Taylor expanded in y around 0 74.2%

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

      1. associate-/r* 74.3%

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

   7. Simplified 74.3%

      \[\leadsto \color{blue}{\frac{\frac{-1}{t}}{z}} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-45} \lor \neg \left(y \leq 1.3 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-43} \lor \neg \left(y \leq 1.35 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -6.5e-43) (not (<= y 1.35e-43))) (/ x y) (/ x (* z (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6.5e-43) || !(y <= 1.35e-43)) {
		tmp = x / y;
	} else {
		tmp = x / (z * -t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-6.5d-43)) .or. (.not. (y <= 1.35d-43))) then
        tmp = x / y
    else
        tmp = x / (z * -t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6.5e-43) || !(y <= 1.35e-43)) {
		tmp = x / y;
	} else {
		tmp = x / (z * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -6.5e-43) or not (y <= 1.35e-43):
		tmp = x / y
	else:
		tmp = x / (z * -t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -6.5e-43) || !(y <= 1.35e-43))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(z * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -6.5e-43) || ~((y <= 1.35e-43)))
		tmp = x / y;
	else
		tmp = x / (z * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -6.5e-43], N[Not[LessEqual[y, 1.35e-43]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(z * (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.50000000000000001e-43 or 1.34999999999999996e-43 < y

   1. Initial program 97.4%

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

   3. Taylor expanded in y around inf 81.0%

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

   if -6.50000000000000001e-43 < y < 1.34999999999999996e-43

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0 74.2%

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

      1. associate-*r/ 74.2%

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

      2. neg-mul-1 74.2%

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

   5. Simplified 74.2%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-43} \lor \neg \left(y \leq 1.35 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in y around inf 58.4%

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

Developer target: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) then
        tmp = x / (y - (z * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024103 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

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

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