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

Percentage Accurate: 95.5% → 95.5%

Time: 6.7s

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: 95.5% 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: 95.5% 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 96.9%

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

Alternative 2: 62.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+171} \lor \neg \left(z \cdot t \leq 4 \cdot 10^{+112}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -2e+171) (not (<= (* z t) 4e+112)))
   (/ x (* z t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+171) || !((z * t) <= 4e+112)) {
		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) <= (-2d+171)) .or. (.not. ((z * t) <= 4d+112))) 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) <= -2e+171) || !((z * t) <= 4e+112)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -2e+171) or not ((z * t) <= 4e+112):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+171) || !(Float64(z * t) <= 4e+112))
		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) <= -2e+171) || ~(((z * t) <= 4e+112)))
		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], -2e+171], N[Not[LessEqual[N[(z * t), $MachinePrecision], 4e+112]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999991e171 or 3.9999999999999997e112 < (*.f64 z t)

   1. Initial program 90.0%

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

      1. clear-num 89.0%

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

      2. inv-pow 89.0%

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

   4. Applied egg-rr 89.0%

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

   5. Taylor expanded in y around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. associate-/l* 95.1%

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

      3. distribute-rgt-neg-in 95.1%

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

      4. distribute-neg-frac2 95.1%

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

   7. Simplified 95.1%

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

      1. unpow-1 95.1%

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

      2. associate-*r/ 86.5%

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

      3. *-commutative 86.5%

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

      4. frac-2neg 86.5%

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

      5. distribute-lft-neg-in 86.5%

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

      6. remove-double-neg 86.5%

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

      7. clear-num 87.4%

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

      8. add-sqr-sqrt 37.1%

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

      9. sqrt-unprod 65.2%

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

      10. sqr-neg 65.2%

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

      11. sqrt-unprod 33.0%

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

      12. add-sqr-sqrt 62.6%

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

   9. Applied egg-rr 62.6%

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

   if -1.99999999999999991e171 < (*.f64 z t) < 3.9999999999999997e112

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 65.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+171} \lor \neg \left(z \cdot t \leq 4 \cdot 10^{+112}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15.5 \lor \neg \left(y \leq 1.72 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -15.5) (not (<= y 1.72e-9))) (/ x y) (/ x (* z (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -15.5) || !(y <= 1.72e-9)) {
		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 <= (-15.5d0)) .or. (.not. (y <= 1.72d-9))) 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 <= -15.5) || !(y <= 1.72e-9)) {
		tmp = x / y;
	} else {
		tmp = x / (z * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -15.5) or not (y <= 1.72e-9):
		tmp = x / y
	else:
		tmp = x / (z * -t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -15.5) || !(y <= 1.72e-9))
		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 <= -15.5) || ~((y <= 1.72e-9)))
		tmp = x / y;
	else
		tmp = x / (z * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -15.5], N[Not[LessEqual[y, 1.72e-9]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(z * (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -15.5 or 1.72000000000000006e-9 < y

   1. Initial program 97.6%

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

   3. Taylor expanded in y around inf 85.2%

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

   if -15.5 < y < 1.72000000000000006e-9

   1. Initial program 96.4%

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

   3. Taylor expanded in y around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. neg-mul-1 77.4%

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

   5. Simplified 77.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15.5 \lor \neg \left(y \leq 1.72 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.6% 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 96.9%

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

3. Taylor expanded in y around inf 52.8%

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

Developer target: 96.8% 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 2024108 
(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))))