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

Percentage Accurate: 95.8% → 99.7%

Time: 9.4s

Alternatives: 7

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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+293}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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+228)
   (/ (/ x (- z)) t)
   (if (<= (* z t) 1e+293) (/ x (fma (- z) t y)) (- (/ (/ 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+228) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 1e+293) {
		tmp = x / fma(-z, t, y);
	} 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+228)
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif (Float64(z * t) <= 1e+293)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(-Float64(Float64(x / t) / z));
	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_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+228], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+293], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.9999999999999998e228

   1. Initial program 74.1%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. clear-num N/A

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

      7. flip-- N/A

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

      8. /-lowering-/.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

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

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. neg-lowering-neg.f64 74.3

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 74.3

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

   7. Simplified 74.3%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. neg-mul-1 N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. remove-double-neg N/A

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

      8. /-lowering-/.f64 N/A

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

      9. neg-lowering-neg.f64 97.4

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

   9. Applied egg-rr 97.4%

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

   if -1.9999999999999998e228 < (*.f64 z t) < 9.9999999999999992e292

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   if 9.9999999999999992e292 < (*.f64 z t)

   1. Initial program 64.3%

      \[\frac{x}{y - z \cdot t} \]
   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. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 99.9

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

   8. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+293}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := -\frac{\frac{x}{t}}{z}\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+293}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 t) z))))
   (if (<= (* z t) (- INFINITY))
     t_1
     (if (<= (* z t) 1e+293) (/ x (fma (- z) t 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 ((z * t) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((z * t) <= 1e+293) {
		tmp = x / fma(-z, t, 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(-Float64(Float64(x / t) / z))
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+293)
		tmp = Float64(x / fma(Float64(-z), t, y));
	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[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+293], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 9.9999999999999992e292 < (*.f64 z t)

   1. Initial program 63.8%

      \[\frac{x}{y - z \cdot t} \]
   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. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 99.9

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

   8. Simplified 99.9%

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

   if -inf.0 < (*.f64 z t) < 9.9999999999999992e292

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 10^{+293}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := -\frac{x}{z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -5000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+59}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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) -5000000.0) t_1 (if (<= (* z t) 1e+59) (/ 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) <= -5000000.0) {
		tmp = t_1;
	} else if ((z * t) <= 1e+59) {
		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) <= (-5000000.0d0)) then
        tmp = t_1
    else if ((z * t) <= 1d+59) 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) <= -5000000.0) {
		tmp = t_1;
	} else if ((z * t) <= 1e+59) {
		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) <= -5000000.0:
		tmp = t_1
	elif (z * t) <= 1e+59:
		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(-Float64(x / Float64(z * t)))
	tmp = 0.0
	if (Float64(z * t) <= -5000000.0)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+59)
		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) <= -5000000.0)
		tmp = t_1;
	elseif ((z * t) <= 1e+59)
		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], -5000000.0], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+59], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5e6 or 9.99999999999999972e58 < (*.f64 z t)

   1. Initial program 86.5%

      \[\frac{x}{y - z \cdot t} \]
   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. /-lowering-/.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. *-lowering-*.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. neg-lowering-neg.f64 77.8

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

   5. Simplified 77.8%

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

   if -5e6 < (*.f64 z t) < 9.99999999999999972e58

   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. /-lowering-/.f64 81.7

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

   5. Simplified 81.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5000000:\\ \;\;\;\;-\frac{x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+59}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+183}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 3 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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) -2e+183) t_1 (if (<= (* z t) 3e+101) (/ 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) <= -2e+183) {
		tmp = t_1;
	} else if ((z * t) <= 3e+101) {
		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) <= (-2d+183)) then
        tmp = t_1
    else if ((z * t) <= 3d+101) 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) <= -2e+183) {
		tmp = t_1;
	} else if ((z * t) <= 3e+101) {
		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) <= -2e+183:
		tmp = t_1
	elif (z * t) <= 3e+101:
		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) <= -2e+183)
		tmp = t_1;
	elseif (Float64(z * t) <= 3e+101)
		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) <= -2e+183)
		tmp = t_1;
	elseif ((z * t) <= 3e+101)
		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], -2e+183], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 3e+101], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999989e183 or 2.99999999999999993e101 < (*.f64 z t)

   1. Initial program 79.4%

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

      1. flip-- N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. accelerator-lowering-fma.f64 25.7

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

   4. Applied egg-rr 25.7%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 25.7

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

   7. Simplified 25.7%

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

   8. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 55.8

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

   10. Simplified 55.8%

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

   if -1.99999999999999989e183 < (*.f64 z t) < 2.99999999999999993e101

   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. /-lowering-/.f64 69.8

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

   5. Simplified 69.8%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+183}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 3 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% accurate, 1.0× 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(Float64(-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 93.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-lowering-neg.f64 93.8

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

4. Applied egg-rr 93.8%

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

Alternative 6: 95.8% 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 93.8%

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

Alternative 7: 54.2% 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 93.8%

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

3. Taylor expanded in y around inf

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

   1. /-lowering-/.f64 53.9

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

5. Simplified 53.9%

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

Developer Target 1: 96.1% 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 2024204 
(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))))