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

Percentage Accurate: 95.8% → 97.9%

Time: 7.5s

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: 95.8% 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, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* t z) (- INFINITY)) (/ (/ x z) (- t)) (/ x (- y (* t z)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * z) <= -((double) INFINITY)) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / (y - (t * z));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * z) <= -Double.POSITIVE_INFINITY) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / (y - (t * z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (t * z) <= -math.inf:
		tmp = (x / z) / -t
	else:
		tmp = x / (y - (t * z))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * z) <= Float64(-Inf))
		tmp = Float64(Float64(x / z) / Float64(-t));
	else
		tmp = Float64(x / Float64(y - Float64(t * z)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t * z) <= -Inf)
		tmp = (x / z) / -t;
	else
		tmp = x / (y - (t * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(t * z), $MachinePrecision], (-Infinity)], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], N[(x / N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 78.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{{y}^{3} - {\left(z \cdot t\right)}^{3}}{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}}}}{\frac{1}{x}} \]

      7. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}}{\frac{1}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}{\frac{1}{x}}} \]

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + \left(-1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}} + -1 \cdot \frac{x \cdot {y}^{2}}{{t}^{2} \cdot {z}^{3}}\right)}{t}} \]

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in t around inf

      \[\leadsto \frac{\frac{x}{z}}{\mathsf{neg}\left(\color{blue}{t}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

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

   if -inf.0 < (*.f64 z t)

   1. Initial program 98.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{\left(-t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.5e+23) (/ x y) (if (<= y 7.2e-78) (/ x (* (- t) z)) (/ x y))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.5e+23) {
		tmp = x / y;
	} else if (y <= 7.2e-78) {
		tmp = x / (-t * z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 <= (-9.5d+23)) then
        tmp = x / y
    else if (y <= 7.2d-78) then
        tmp = x / (-t * z)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.5e+23) {
		tmp = x / y;
	} else if (y <= 7.2e-78) {
		tmp = x / (-t * z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -9.5e+23:
		tmp = x / y
	elif y <= 7.2e-78:
		tmp = x / (-t * z)
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.5e+23)
		tmp = Float64(x / y);
	elseif (y <= 7.2e-78)
		tmp = Float64(x / Float64(Float64(-t) * z));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9.5e+23)
		tmp = x / y;
	elseif (y <= 7.2e-78)
		tmp = x / (-t * z);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -9.5e+23], N[(x / y), $MachinePrecision], If[LessEqual[y, 7.2e-78], N[(x / N[((-t) * z), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.50000000000000038e23 or 7.2000000000000005e-78 < y

   1. Initial program 98.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if -9.50000000000000038e23 < y < 7.2000000000000005e-78

   1. Initial program 94.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 78.3

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

   5. Applied rewrites 78.3%

      \[\leadsto \frac{x}{\color{blue}{\left(-t\right) \cdot z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \frac{x}{y - t \cdot z} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x (- y (* t z))))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 - (t * z))
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return x / (y - (t * z))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(t * z)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / (y - (t * z));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x / N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

3. Final simplification 96.7%

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

Alternative 4: 54.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x y))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return x / y

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x / y)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / y;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 53.7

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

5. Applied rewrites 53.7%

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

Developer Target 1: 96.4% accurate, 0.4× 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 2024240 
(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))))