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

Percentage Accurate: 95.5% → 96.4%

Time: 9.8s

Alternatives: 5

Speedup: 0.6×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{y - z \cdot t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / (y - (z * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Python

def code(x, y, z, t):
	return x / (y - (z * t))

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y - z \cdot t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / (y - (z * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Python

def code(x, y, z, t):
	return x / (y - (z * t))

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+220}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \frac{1}{x}, z \cdot \frac{-t}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.25e+220)
   (/ 1.0 (fma y (/ 1.0 x) (* z (/ (- t) x))))
   (/ x (- y (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.25e+220) {
		tmp = 1.0 / fma(y, (1.0 / x), (z * (-t / x)));
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.25e+220)
		tmp = Float64(1.0 / fma(y, Float64(1.0 / x), Float64(z * Float64(Float64(-t) / x))));
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, -2.25e+220], N[(1.0 / N[(y * N[(1.0 / x), $MachinePrecision] + N[(z * N[((-t) / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25000000000000005e220

   1. Initial program 73.9%

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

      1. clear-num 74.1%

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

      2. associate-/r/ 73.7%

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

   3. Applied egg-rr 73.7%

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

      1. associate-/r/ 74.1%

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

   5. Applied egg-rr 74.1%

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

      1. div-sub 68.5%

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

      2. div-inv 68.5%

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

      3. *-un-lft-identity 68.5%

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

      4. times-frac 88.4%

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

      5. prod-diff 71.8%

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

   7. Applied egg-rr 71.8%

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

   8. Taylor expanded in t around 0 66.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(y, \frac{1}{x}, -\frac{t}{x} \cdot \frac{z}{1}\right) + \color{blue}{t \cdot \left(-1 \cdot \frac{z}{x} + \frac{z}{x}\right)}} \]
   9. Step-by-step derivation

      1. distribute-lft-in 66.2%

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

      2. mul-1-neg 66.2%

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

      3. distribute-rgt-neg-out 66.2%

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

      4. associate-*r/ 40.9%

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

      5. associate-*l/ 66.2%

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

      6. distribute-lft-neg-in 66.2%

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

      7. distribute-frac-neg 66.2%

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

      8. associate-*r/ 46.2%

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

      9. associate-*l/ 71.8%

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

      10. rem-log-exp 14.4%

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

      11. rem-log-exp 11.1%

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

      12. log-prod 11.1%

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

      13. prod-exp 71.8%

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

      14. distribute-rgt-in 88.4%

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

      15. exp-prod 88.4%

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

      16. distribute-frac-neg 88.4%

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

      17. mul-1-neg 88.4%

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

      18. distribute-lft1-in 88.4%

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

      19. metadata-eval 88.4%

          \[\leadsto \frac{1}{\mathsf{fma}\left(y, \frac{1}{x}, -\frac{t}{x} \cdot \frac{z}{1}\right) + \log \left({\left(e^{z}\right)}^{\left(\color{blue}{0} \cdot \frac{t}{x}\right)}\right)} \]

      20. mul0-lft 94.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(y, \frac{1}{x}, -\frac{t}{x} \cdot \frac{z}{1}\right) + \log \left({\left(e^{z}\right)}^{\color{blue}{0}}\right)} \]

   10. Simplified 94.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(y, \frac{1}{x}, -\frac{t}{x} \cdot \frac{z}{1}\right) + \color{blue}{0}} \]

   if -2.25000000000000005e220 < z

   1. Initial program 98.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+220}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \frac{1}{x}, z \cdot \frac{-t}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 2: 97.6% accurate, 0.6× speedup

Math

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

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (/ (/ (- x) z) t) (/ x (- y (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (-x / z) / t
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (-x / z) / t;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.2%

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

      1. clear-num 72.2%

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

      2. associate-/r/ 72.2%

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

   3. Applied egg-rr 72.2%

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

      1. associate-/r/ 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. div-sub 58.8%

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

      2. div-inv 58.8%

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

      3. *-un-lft-identity 58.8%

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

      4. times-frac 83.9%

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

      5. prod-diff 26.5%

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

   7. Applied egg-rr 26.5%

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

   8. Taylor expanded in y around 0 72.2%

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

      1. associate-*r/ 72.2%

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

      2. associate-/l/ 99.9%

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

      3. mul-1-neg 99.9%

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

   10. Simplified 99.9%

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

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

   1. Initial program 98.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 3: 69.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+47} \lor \neg \left(z \leq 1.15 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.45e+47) (not (<= z 1.15e-83))) (/ (- x) (* z t)) (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.45e+47) || !(z <= 1.15e-83)) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.45d+47)) .or. (.not. (z <= 1.15d-83))) then
        tmp = -x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.45e+47) || !(z <= 1.15e-83)) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -1.45e+47) or not (z <= 1.15e-83):
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.45e+47) || !(z <= 1.15e-83))
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.45e+47) || ~((z <= 1.15e-83)))
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.45e+47], N[Not[LessEqual[z, 1.15e-83]], $MachinePrecision]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4499999999999999e47 or 1.14999999999999995e-83 < z

   1. Initial program 94.1%

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

   2. Taylor expanded in y around 0 68.9%

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

      1. associate-*r/ 68.9%

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

      2. neg-mul-1 68.9%

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

   4. Simplified 68.9%

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

   if -1.4499999999999999e47 < z < 1.14999999999999995e-83

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 80.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+47} \lor \neg \left(z \leq 1.15 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 73.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+42} \lor \neg \left(z \leq 1.15 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.1e+42) (not (<= z 1.15e-83))) (/ (/ (- x) z) t) (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.1e+42) || !(z <= 1.15e-83)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.1d+42)) .or. (.not. (z <= 1.15d-83))) then
        tmp = (-x / z) / t
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.1e+42) || !(z <= 1.15e-83)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -2.1e+42) or not (z <= 1.15e-83):
		tmp = (-x / z) / t
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.1e+42) || !(z <= 1.15e-83))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.1e+42) || ~((z <= 1.15e-83)))
		tmp = (-x / z) / t;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.1e+42], N[Not[LessEqual[z, 1.15e-83]], $MachinePrecision]], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.09999999999999995e42 or 1.14999999999999995e-83 < z

   1. Initial program 94.1%

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

      1. clear-num 93.5%

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

      2. associate-/r/ 93.9%

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

   3. Applied egg-rr 93.9%

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

      1. associate-/r/ 93.5%

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

   5. Applied egg-rr 93.5%

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

      1. div-sub 89.9%

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

      2. div-inv 89.8%

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

      3. *-un-lft-identity 89.8%

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

      4. times-frac 90.9%

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

      5. prod-diff 70.9%

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

   7. Applied egg-rr 70.9%

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

   8. Taylor expanded in y around 0 68.9%

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

      1. associate-*r/ 68.9%

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

      2. associate-/l/ 72.4%

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

      3. mul-1-neg 72.4%

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

   10. Simplified 72.4%

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

   if -2.09999999999999995e42 < z < 1.14999999999999995e-83

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 80.2%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+42} \lor \neg \left(z \leq 1.15 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 53.5% accurate, 2.3× speedup

Math

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

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x y))

C

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

Fortran

NOTE: 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 z < t;
public static double code(double x, double y, double z, double t) {
	return x / y;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: 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 96.8%

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

2. Taylor expanded in y around inf 55.3%

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

3. Final simplification 55.3%

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

Developer target: 96.4% accurate, 0.5× 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 2023195 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

  (/ x (- y (* z t))))