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

Percentage Accurate: 95.8% → 97.9%

Time: 9.3s

Alternatives: 5

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+304}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{t \cdot \left(z \cdot t\right)}, \frac{x}{t}\right)}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 1e+304)
   (/ x (- y (* z t)))
   (/ (fma x (/ y (* t (* z t))) (/ x t)) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 1e+304) {
		tmp = x / (y - (z * t));
	} else {
		tmp = fma(x, (y / (t * (z * t))), (x / t)) / -z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 1e+304)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(fma(x, Float64(y / Float64(t * Float64(z * t))), Float64(x / t)) / Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 1e+304], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / N[(t * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 9.9999999999999994e303

   1. Initial program 98.6%

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

   if 9.9999999999999994e303 < (*.f64 z t)

   1. Initial program 66.4%

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

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

      9. clear-num N/A

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

      10. flip3-- N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{t} + -1 \cdot \frac{x \cdot y}{{t}^{2} \cdot z}}{z}} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{t} + \frac{x \cdot y}{{t}^{2} \cdot z}}{z}} \]

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{\frac{x}{t} + \frac{x \cdot y}{{t}^{2} \cdot z}}{\color{blue}{-1 \cdot z}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{t} + \frac{x \cdot y}{{t}^{2} \cdot z}}{-1 \cdot z}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{{t}^{2} \cdot z} + \frac{x}{t}}}{-1 \cdot z} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{y}{{t}^{2} \cdot z}} + \frac{x}{t}}{-1 \cdot z} \]

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 99.9

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

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \frac{y}{t \cdot \left(t \cdot z\right)}, \frac{x}{t}\right)}{-z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+304}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{t \cdot \left(z \cdot t\right)}, \frac{x}{t}\right)}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{-z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (- (* z t)))))
   (if (<= (* z t) -1e-31) t_1 (if (<= (* z t) 5e-50) (/ 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) <= -1e-31) {
		tmp = t_1;
	} else if ((z * t) <= 5e-50) {
		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) <= (-1d-31)) then
        tmp = t_1
    else if ((z * t) <= 5d-50) 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) <= -1e-31) {
		tmp = t_1;
	} else if ((z * t) <= 5e-50) {
		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) <= -1e-31:
		tmp = t_1
	elif (z * t) <= 5e-50:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(-Float64(z * t)))
	tmp = 0.0
	if (Float64(z * t) <= -1e-31)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-50)
		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) <= -1e-31)
		tmp = t_1;
	elseif ((z * t) <= 5e-50)
		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], -1e-31], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-50], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1e-31 or 4.99999999999999968e-50 < (*.f64 z t)

   1. Initial program 92.5%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -1e-31 < (*.f64 z t) < 4.99999999999999968e-50

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 96.6% 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 2024228 
(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))))