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

Percentage Accurate: 95.9% → 99.4%

Time: 10.4s

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: 95.9% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := -\frac{\frac{x}{t}}{z}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+253}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+205}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 t) z))))
   (if (<= (* z t) -2e+253)
     t_1
     (if (<= (* z t) 1e+205) (/ x (- y (* z t))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = -((x / t) / z);
	double tmp;
	if ((z * t) <= -2e+253) {
		tmp = t_1;
	} else if ((z * t) <= 1e+205) {
		tmp = x / (y - (z * t));
	} 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 / t) / z)
    if ((z * t) <= (-2d+253)) then
        tmp = t_1
    else if ((z * t) <= 1d+205) then
        tmp = x / (y - (z * t))
    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 / t) / z);
	double tmp;
	if ((z * t) <= -2e+253) {
		tmp = t_1;
	} else if ((z * t) <= 1e+205) {
		tmp = x / (y - (z * t));
	} 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 / t) / z)
	tmp = 0
	if (z * t) <= -2e+253:
		tmp = t_1
	elif (z * t) <= 1e+205:
		tmp = x / (y - (z * t))
	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(-Float64(Float64(x / t) / z))
	tmp = 0.0
	if (Float64(z * t) <= -2e+253)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+205)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	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 / t) / z);
	tmp = 0.0;
	if ((z * t) <= -2e+253)
		tmp = t_1;
	elseif ((z * t) <= 1e+205)
		tmp = x / (y - (z * t));
	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[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+253], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+205], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.9999999999999999e253 or 1.00000000000000002e205 < (*.f64 z t)

   1. Initial program 83.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. lower-*.f64 N/A

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

      9. clear-num N/A

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

      10. flip3-- N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 83.1

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 83.1

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

   7. Applied rewrites 83.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. un-div-inv N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if -1.9999999999999999e253 < (*.f64 z t) < 1.00000000000000002e205

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+253}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 10^{+205}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+202}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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+202) t_1 (if (<= (* z t) 5e+88) (/ 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+202) {
		tmp = t_1;
	} else if ((z * t) <= 5e+88) {
		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+202)) then
        tmp = t_1
    else if ((z * t) <= 5d+88) 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+202) {
		tmp = t_1;
	} else if ((z * t) <= 5e+88) {
		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+202:
		tmp = t_1
	elif (z * t) <= 5e+88:
		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+202)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e+88)
		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+202)
		tmp = t_1;
	elseif ((z * t) <= 5e+88)
		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+202], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e+88], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.999999999999999e201 or 4.99999999999999997e88 < (*.f64 z t)

   1. Initial program 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. lower-*.f64 N/A

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

      9. clear-num N/A

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

      10. flip3-- N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 87.5

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

   4. Applied rewrites 87.5%

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

   6. Applied rewrites 57.9%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 56.7

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

   9. Applied rewrites 56.7%

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

   if -9.999999999999999e201 < (*.f64 z t) < 4.99999999999999997e88

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 72.1

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

   5. Applied rewrites 72.1%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(z, t, y\right)}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (fma z t y))))
   (if (<= y -3.5e-137) t_1 (if (<= y 150000000.0) (/ x (* z (- t))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / fma(z, t, y);
	double tmp;
	if (y <= -3.5e-137) {
		tmp = t_1;
	} else if (y <= 150000000.0) {
		tmp = x / (z * -t);
	} 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 / fma(z, t, y))
	tmp = 0.0
	if (y <= -3.5e-137)
		tmp = t_1;
	elseif (y <= 150000000.0)
		tmp = Float64(x / Float64(z * Float64(-t)));
	else
		tmp = t_1;
	end
	return 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 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.5e-137], t$95$1, If[LessEqual[y, 150000000.0], N[(x / N[(z * (-t)), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5000000000000001e-137 or 1.5e8 < y

   1. Initial program 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. lower-*.f64 N/A

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

      9. clear-num N/A

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

      10. flip3-- N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   6. Applied rewrites 83.6%

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

   if -3.5000000000000001e-137 < y < 1.5e8

   1. Initial program 95.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 75.1

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

   5. Applied rewrites 75.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, t, y\right)}\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, t, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% 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 96.1%

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

Alternative 5: 64.9% 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 96.1%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lift--.f64 N/A

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

   6. flip3-- N/A

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

   7. clear-num N/A

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

   8. lower-*.f64 N/A

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

   9. clear-num N/A

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

   10. flip3-- N/A

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

   11. lift--.f64 N/A

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

   12. lower-/.f64 95.9

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

4. Applied rewrites 95.9%

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

6. Applied rewrites 69.0%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(z, t, y\right)}} \]
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 96.1%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 55.3

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

5. Applied rewrites 55.3%

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

Developer Target 1: 96.7% 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 2024216 
(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))))