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

Percentage Accurate: 96.0% → 99.9%

Time: 7.1s

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: 96.0% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{x}{y - t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \end{array} \end{array} \]

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))
   (if (<= (* t z) 2e+293) (/ x (- y (* t z))) (* (/ -1.0 t) (/ x 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 if ((t * z) <= 2e+293) {
		tmp = x / (y - (t * z));
	} else {
		tmp = (-1.0 / t) * (x / 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 if ((t * z) <= 2e+293) {
		tmp = x / (y - (t * z));
	} else {
		tmp = (-1.0 / t) * (x / 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
	elif (t * z) <= 2e+293:
		tmp = x / (y - (t * z))
	else:
		tmp = (-1.0 / t) * (x / 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));
	elseif (Float64(t * z) <= 2e+293)
		tmp = Float64(x / Float64(y - Float64(t * z)));
	else
		tmp = Float64(Float64(-1.0 / t) * Float64(x / 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;
	elseif ((t * z) <= 2e+293)
		tmp = x / (y - (t * z));
	else
		tmp = (-1.0 / t) * (x / 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], If[LessEqual[N[(t * z), $MachinePrecision], 2e+293], N[(x / N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 75.6%

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

   3. Taylor expanded in t around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 z t) < 1.9999999999999998e293

   1. Initial program 99.9%

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

   if 1.9999999999999998e293 < (*.f64 z t)

   1. Initial program 55.8%

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

   3. Taylor expanded in t around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.4%

       \[\leadsto \frac{-1}{t} \cdot \color{blue}{\frac{x}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{x}{y - t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x}{z}}{-t}\\ \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\frac{x}{y - t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (<= (* t z) (- INFINITY))
     t_1
     (if (<= (* t z) 5e+266) (/ x (- y (* t z))) 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 ((t * z) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((t * z) <= 5e+266) {
		tmp = x / (y - (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 ((t * z) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((t * z) <= 5e+266) {
		tmp = x / (y - (t * z));
	} 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 (t * z) <= -math.inf:
		tmp = t_1
	elif (t * z) <= 5e+266:
		tmp = x / (y - (t * z))
	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(x / z) / Float64(-t))
	tmp = 0.0
	if (Float64(t * z) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(t * z) <= 5e+266)
		tmp = Float64(x / Float64(y - Float64(t * z)));
	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 ((t * z) <= -Inf)
		tmp = t_1;
	elseif ((t * z) <= 5e+266)
		tmp = x / (y - (t * z));
	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 / z), $MachinePrecision] / (-t)), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 5e+266], N[(x / N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 4.9999999999999999e266 < (*.f64 z t)

   1. Initial program 68.6%

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

   3. Taylor expanded in t around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 z t) < 4.9999999999999999e266

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\frac{x}{y - t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(-t\right) \cdot z}\\ \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 1.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (- t) z))))
   (if (<= (* t z) -1e-42) t_1 (if (<= (* t z) 1.3e-22) (/ 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 / (-t * z);
	double tmp;
	if ((t * z) <= -1e-42) {
		tmp = t_1;
	} else if ((t * z) <= 1.3e-22) {
		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 / (-t * z)
    if ((t * z) <= (-1d-42)) then
        tmp = t_1
    else if ((t * z) <= 1.3d-22) 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 / (-t * z);
	double tmp;
	if ((t * z) <= -1e-42) {
		tmp = t_1;
	} else if ((t * z) <= 1.3e-22) {
		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 / (-t * z)
	tmp = 0
	if (t * z) <= -1e-42:
		tmp = t_1
	elif (t * z) <= 1.3e-22:
		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(-t) * z))
	tmp = 0.0
	if (Float64(t * z) <= -1e-42)
		tmp = t_1;
	elseif (Float64(t * z) <= 1.3e-22)
		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 / (-t * z);
	tmp = 0.0;
	if ((t * z) <= -1e-42)
		tmp = t_1;
	elseif ((t * z) <= 1.3e-22)
		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[((-t) * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1e-42], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1.3e-22], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000004e-42 or 1.3e-22 < (*.f64 z t)

   1. Initial program 90.8%

      \[\frac{x}{y - z \cdot t} \]
   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 74.5

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

   5. Applied rewrites 74.5%

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

   if -1.00000000000000004e-42 < (*.f64 z t) < 1.3e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 88.7

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

   5. Applied rewrites 88.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\left(-t\right) \cdot z}\\ \mathbf{elif}\;t \cdot z \leq 1.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(-t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% 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 94.9%

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

3. Final simplification 94.9%

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

Alternative 5: 55.1% 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 94.9%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 53.2

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

5. Applied rewrites 53.2%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
6. 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 2024257 
(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))))