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

Percentage Accurate: 95.5% → 97.2%

Time: 10.1s

Alternatives: 6

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: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

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 (<= (* z t) 2e+204) (/ x (- y (* z t))) (/ (/ x t) (- z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 2e+204) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	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 ((z * t) <= 2d+204) then
        tmp = x / (y - (z * t))
    else
        tmp = (x / t) / -z
    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 ((z * t) <= 2e+204) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= 2e+204:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / 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(z * t) <= 2e+204)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / t) / Float64(-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 ((z * t) <= 2e+204)
		tmp = x / (y - (z * t));
	else
		tmp = (x / 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[(z * t), $MachinePrecision], 2e+204], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1.99999999999999998e204

   1. Initial program 99.1%

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

   if 1.99999999999999998e204 < (*.f64 z t)

   1. Initial program 84.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.8%

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

Alternative 2: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{-z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (- (* z t)))))
   (if (<= (* z t) -5e-92) t_1 (if (<= (* z t) 1e+77) (/ x y) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / -(z * t);
	double tmp;
	if ((z * t) <= -5e-92) {
		tmp = t_1;
	} else if ((z * t) <= 1e+77) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x / -(z * t)
    if ((z * t) <= (-5d-92)) then
        tmp = t_1
    else if ((z * t) <= 1d+77) then
        tmp = x / y
    else
        tmp = t_1
    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 t_1 = x / -(z * t);
	double tmp;
	if ((z * t) <= -5e-92) {
		tmp = t_1;
	} else if ((z * t) <= 1e+77) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / -(z * t)
	tmp = 0
	if (z * t) <= -5e-92:
		tmp = t_1
	elif (z * t) <= 1e+77:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(-Float64(z * t)))
	tmp = 0.0
	if (Float64(z * t) <= -5e-92)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+77)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / -(z * t);
	tmp = 0.0;
	if ((z * t) <= -5e-92)
		tmp = t_1;
	elseif ((z * t) <= 1e+77)
		tmp = x / y;
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x / (-N[(z * t), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e-92], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+77], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.00000000000000011e-92 or 9.99999999999999983e76 < (*.f64 z t)

   1. Initial program 94.2%

      \[\frac{x}{y - z \cdot t} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   if -5.00000000000000011e-92 < (*.f64 z t) < 9.99999999999999983e76

   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 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* z t))))
   (if (<= (* z t) -1e+126) t_1 (if (<= (* z t) 5e+132) (/ x y) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / (z * t);
	double tmp;
	if ((z * t) <= -1e+126) {
		tmp = t_1;
	} else if ((z * t) <= 5e+132) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x / (z * t)
    if ((z * t) <= (-1d+126)) then
        tmp = t_1
    else if ((z * t) <= 5d+132) then
        tmp = x / y
    else
        tmp = t_1
    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 t_1 = x / (z * t);
	double tmp;
	if ((z * t) <= -1e+126) {
		tmp = t_1;
	} else if ((z * t) <= 5e+132) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / (z * t)
	tmp = 0
	if (z * t) <= -1e+126:
		tmp = t_1
	elif (z * t) <= 5e+132:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -1e+126)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e+132)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / (z * t);
	tmp = 0.0;
	if ((z * t) <= -1e+126)
		tmp = t_1;
	elseif ((z * t) <= 5e+132)
		tmp = x / y;
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+126], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e+132], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999925e125 or 5.0000000000000001e132 < (*.f64 z t)

   1. Initial program 91.2%

      \[\frac{x}{y - z \cdot t} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 91.2

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. neg-sub0 N/A

         \[\leadsto \frac{x}{y + \color{blue}{\left(0 - z \cdot t\right)}} \]

      8. flip3-- N/A

         \[\leadsto \frac{x}{y + \color{blue}{\frac{{0}^{3} - {\left(z \cdot t\right)}^{3}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{x}{y + \frac{\color{blue}{0} - {\left(z \cdot t\right)}^{3}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}} \]

      10. neg-sub0 N/A

          \[\leadsto \frac{x}{y + \frac{\color{blue}{\mathsf{neg}\left({\left(z \cdot t\right)}^{3}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}} \]

      11. distribute-neg-frac N/A

          \[\leadsto \frac{x}{y + \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(z \cdot t\right)}^{3}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)}} \]

      12. sqr-pow N/A

          \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(z \cdot t\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(z \cdot t\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

      13. unpow-prod-down N/A

          \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\left(z \cdot t\right) \cdot \left(z \cdot t\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

      14. sqr-neg N/A

          \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(z \cdot t\right)\right) \cdot \left(\mathsf{neg}\left(z \cdot t\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot t\right)\right)} \cdot \left(\mathsf{neg}\left(z \cdot t\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

      16. lift-neg.f64 N/A

          \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{{\left(\left(\mathsf{neg}\left(z \cdot t\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot t\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

      17. unpow-prod-down N/A

          \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(z \cdot t\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z \cdot t\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

      18. sqr-pow N/A

          \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(z \cdot t\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(\mathsf{neg}\left(z \cdot t\right)\right)}}^{3}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

      20. cube-neg N/A

          \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({\left(z \cdot t\right)}^{3}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

   6. Applied rewrites 57.7%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 55.5

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

   9. Applied rewrites 55.5%

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

   if -9.99999999999999925e125 < (*.f64 z t) < 5.0000000000000001e132

   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 76.3

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

   5. Applied rewrites 76.3%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 1.0× speedup

Math

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

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 (* z t))))

C

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

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 - (z * t))
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 - (z * t));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.1%

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

Alternative 5: 65.4% accurate, 1.1× speedup

Math

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

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 (fma z t y)))

C

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x / fma(z, t, 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 / N[(z * t + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.1%

   \[\frac{x}{y - z \cdot t} \]
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. lower-/.f64 N/A

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

   4. lower-/.f64 96.6

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

4. Applied rewrites 96.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 97.1

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. neg-sub0 N/A

      \[\leadsto \frac{x}{y + \color{blue}{\left(0 - z \cdot t\right)}} \]

   8. flip3-- N/A

      \[\leadsto \frac{x}{y + \color{blue}{\frac{{0}^{3} - {\left(z \cdot t\right)}^{3}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{x}{y + \frac{\color{blue}{0} - {\left(z \cdot t\right)}^{3}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}} \]

   10. neg-sub0 N/A

       \[\leadsto \frac{x}{y + \frac{\color{blue}{\mathsf{neg}\left({\left(z \cdot t\right)}^{3}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}} \]

   11. distribute-neg-frac N/A

       \[\leadsto \frac{x}{y + \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(z \cdot t\right)}^{3}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)}} \]

   12. sqr-pow N/A

       \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(z \cdot t\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(z \cdot t\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

   13. unpow-prod-down N/A

       \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\left(z \cdot t\right) \cdot \left(z \cdot t\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

   14. sqr-neg N/A

       \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(z \cdot t\right)\right) \cdot \left(\mathsf{neg}\left(z \cdot t\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

   15. lift-neg.f64 N/A

       \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot t\right)\right)} \cdot \left(\mathsf{neg}\left(z \cdot t\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

   16. lift-neg.f64 N/A

       \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{{\left(\left(\mathsf{neg}\left(z \cdot t\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot t\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

   17. unpow-prod-down N/A

       \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(z \cdot t\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z \cdot t\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

   18. sqr-pow N/A

       \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(z \cdot t\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

   19. lift-neg.f64 N/A

       \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(\mathsf{neg}\left(z \cdot t\right)\right)}}^{3}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

   20. cube-neg N/A

       \[\leadsto \frac{x}{y + \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({\left(z \cdot t\right)}^{3}\right)}}{0 \cdot 0 + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + 0 \cdot \left(z \cdot t\right)\right)}\right)\right)} \]

6. Applied rewrites 69.7%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(z, t, y\right)}} \]
7. Add Preprocessing

Alternative 6: 54.3% 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 97.1%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 57.3

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

5. Applied rewrites 57.3%

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

Developer Target 1: 96.2% 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 2024223 
(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))))