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

Percentage Accurate: 95.9% → 99.7%

Time: 8.0s

Alternatives: 7

Speedup: 0.3×

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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+231}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ x (- z)) t)
   (if (<= (* z t) 1e+231) (/ x (- y (* z t))) (/ (/ x t) (- z)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 1e+231) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) / -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 ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 1e+231) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (x / -z) / t
	elif (z * t) <= 1e+231:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / t) / -z
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif (Float64(z * t) <= 1e+231)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (x / -z) / t;
	elseif ((z * t) <= 1e+231)
		tmp = x / (y - (z * t));
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+231], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 51.1%

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

      1. clear-num 51.1%

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

      2. inv-pow 51.1%

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

   4. Applied egg-rr 51.1%

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

   5. Taylor expanded in y around 0 51.1%

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

      1. associate-/l/ 99.6%

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

      2. associate-*r/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. neg-mul-1 99.6%

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

   7. Simplified 99.6%

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

   if -inf.0 < (*.f64 z t) < 1.0000000000000001e231

   1. Initial program 99.8%

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

   if 1.0000000000000001e231 < (*.f64 z t)

   1. Initial program 68.3%

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

   3. Taylor expanded in y around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unsub-neg 64.2%

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

      3. associate-/r* 88.3%

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

      4. associate-*r/ 88.3%

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

      5. associate-/r* 88.3%

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

      6. unpow2 88.3%

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

      7. associate-/r* 88.4%

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

      8. associate-/r* 88.3%

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

      9. div-sub 88.3%

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

      10. unsub-neg 88.3%

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

      11. mul-1-neg 88.3%

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

      12. distribute-lft-out 88.3%

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

      13. mul-1-neg 88.3%

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

      14. distribute-neg-frac 88.3%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+231}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.01:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{-1}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -0.01)
   (/ (/ x (- z)) t)
   (if (<= (* z t) 5e-84)
     (/ x y)
     (if (<= (* z t) 2e+14) (* x (/ -1.0 (* z t))) (/ (/ x t) (- z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -0.01) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 5e-84) {
		tmp = x / y;
	} else if ((z * t) <= 2e+14) {
		tmp = x * (-1.0 / (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * t) <= (-0.01d0)) then
        tmp = (x / -z) / t
    else if ((z * t) <= 5d-84) then
        tmp = x / y
    else if ((z * t) <= 2d+14) then
        tmp = x * ((-1.0d0) / (z * t))
    else
        tmp = (x / t) / -z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -0.01) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 5e-84) {
		tmp = x / y;
	} else if ((z * t) <= 2e+14) {
		tmp = x * (-1.0 / (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -0.01:
		tmp = (x / -z) / t
	elif (z * t) <= 5e-84:
		tmp = x / y
	elif (z * t) <= 2e+14:
		tmp = x * (-1.0 / (z * t))
	else:
		tmp = (x / t) / -z
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -0.01)
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif (Float64(z * t) <= 5e-84)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 2e+14)
		tmp = Float64(x * Float64(-1.0 / Float64(z * t)));
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -0.01)
		tmp = (x / -z) / t;
	elseif ((z * t) <= 5e-84)
		tmp = x / y;
	elseif ((z * t) <= 2e+14)
		tmp = x * (-1.0 / (z * t));
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -0.01], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-84], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+14], N[(x * N[(-1.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -0.0100000000000000002

   1. Initial program 88.7%

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

      1. clear-num 86.1%

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

      2. inv-pow 86.1%

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in y around 0 73.0%

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

      1. associate-/l/ 81.1%

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

      2. associate-*r/ 81.1%

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

      3. associate-*r/ 81.1%

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

      4. neg-mul-1 81.1%

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

   7. Simplified 81.1%

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

   if -0.0100000000000000002 < (*.f64 z t) < 5.0000000000000002e-84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 86.4%

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

   if 5.0000000000000002e-84 < (*.f64 z t) < 2e14

   1. Initial program 99.7%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 82.2%

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

   if 2e14 < (*.f64 z t)

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 0 58.7%

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

      1. mul-1-neg 58.7%

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

      2. unsub-neg 58.7%

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

      3. associate-/r* 68.0%

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

      4. associate-*r/ 68.0%

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

      5. associate-/r* 67.9%

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

      6. unpow2 67.9%

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

      7. associate-/r* 71.4%

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

      8. associate-/r* 71.4%

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

      9. div-sub 71.4%

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

      10. unsub-neg 71.4%

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

      11. mul-1-neg 71.4%

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

      12. distribute-lft-out 71.4%

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

      13. mul-1-neg 71.4%

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

      14. distribute-neg-frac 71.4%

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

   5. Simplified 78.8%

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

   6. Taylor expanded in y around 0 80.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.01:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{-1}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x}{t}}{-z}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+231}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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) -1e+53)
     t_1
     (if (<= (* z t) 5e-84)
       (/ x y)
       (if (<= (* z t) 1e+231) (/ x (* 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 / t) / -z;
	double tmp;
	if ((z * t) <= -1e+53) {
		tmp = t_1;
	} else if ((z * t) <= 5e-84) {
		tmp = x / y;
	} else if ((z * t) <= 1e+231) {
		tmp = x / (t * -z);
	} 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) <= (-1d+53)) then
        tmp = t_1
    else if ((z * t) <= 5d-84) then
        tmp = x / y
    else if ((z * t) <= 1d+231) then
        tmp = x / (t * -z)
    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) <= -1e+53) {
		tmp = t_1;
	} else if ((z * t) <= 5e-84) {
		tmp = x / y;
	} else if ((z * t) <= 1e+231) {
		tmp = x / (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 / t) / -z
	tmp = 0
	if (z * t) <= -1e+53:
		tmp = t_1
	elif (z * t) <= 5e-84:
		tmp = x / y
	elif (z * t) <= 1e+231:
		tmp = x / (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 / t) / Float64(-z))
	tmp = 0.0
	if (Float64(z * t) <= -1e+53)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-84)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 1e+231)
		tmp = Float64(x / Float64(t * Float64(-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 / t) / -z;
	tmp = 0.0;
	if ((z * t) <= -1e+53)
		tmp = t_1;
	elseif ((z * t) <= 5e-84)
		tmp = x / y;
	elseif ((z * t) <= 1e+231)
		tmp = x / (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 / t), $MachinePrecision] / (-z)), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+53], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-84], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+231], N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999999e52 or 1.0000000000000001e231 < (*.f64 z t)

   1. Initial program 79.3%

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

   3. Taylor expanded in y around 0 68.5%

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

      1. mul-1-neg 68.5%

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

      2. unsub-neg 68.5%

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

      3. associate-/r* 81.5%

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

      4. associate-*r/ 81.5%

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

      5. associate-/r* 81.5%

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

      6. unpow2 81.5%

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

      7. associate-/r* 83.0%

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

      8. associate-/r* 82.9%

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

      9. div-sub 82.9%

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

      10. unsub-neg 82.9%

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

      11. mul-1-neg 82.9%

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

      12. distribute-lft-out 82.9%

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

      13. mul-1-neg 82.9%

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

      14. distribute-neg-frac 82.9%

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

   5. Simplified 92.3%

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

   6. Taylor expanded in y around 0 92.5%

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

   if -9.9999999999999999e52 < (*.f64 z t) < 5.0000000000000002e-84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 81.9%

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

   if 5.0000000000000002e-84 < (*.f64 z t) < 1.0000000000000001e231

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 75.6%

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

      1. associate-*r/ 75.6%

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

      2. neg-mul-1 75.6%

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

   5. Simplified 75.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+231}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.01:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+231}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

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) -0.01)
   (/ (/ x (- z)) t)
   (if (<= (* z t) 5e-84)
     (/ x y)
     (if (<= (* z t) 1e+231) (/ x (* t (- z))) (/ (/ 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) <= -0.01) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 5e-84) {
		tmp = x / y;
	} else if ((z * t) <= 1e+231) {
		tmp = x / (t * -z);
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * t) <= (-0.01d0)) then
        tmp = (x / -z) / t
    else if ((z * t) <= 5d-84) then
        tmp = x / y
    else if ((z * t) <= 1d+231) then
        tmp = x / (t * -z)
    else
        tmp = (x / t) / -z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -0.01) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 5e-84) {
		tmp = x / y;
	} else if ((z * t) <= 1e+231) {
		tmp = x / (t * -z);
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -0.01:
		tmp = (x / -z) / t
	elif (z * t) <= 5e-84:
		tmp = x / y
	elif (z * t) <= 1e+231:
		tmp = x / (t * -z)
	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) <= -0.01)
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif (Float64(z * t) <= 5e-84)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 1e+231)
		tmp = Float64(x / Float64(t * Float64(-z)));
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -0.01)
		tmp = (x / -z) / t;
	elseif ((z * t) <= 5e-84)
		tmp = x / y;
	elseif ((z * t) <= 1e+231)
		tmp = x / (t * -z);
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -0.01], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-84], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+231], N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -0.0100000000000000002

   1. Initial program 88.7%

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

      1. clear-num 86.1%

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

      2. inv-pow 86.1%

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in y around 0 73.0%

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

      1. associate-/l/ 81.1%

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

      2. associate-*r/ 81.1%

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

      3. associate-*r/ 81.1%

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

      4. neg-mul-1 81.1%

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

   7. Simplified 81.1%

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

   if -0.0100000000000000002 < (*.f64 z t) < 5.0000000000000002e-84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 86.4%

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

   if 5.0000000000000002e-84 < (*.f64 z t) < 1.0000000000000001e231

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 75.6%

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

      1. associate-*r/ 75.6%

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

      2. neg-mul-1 75.6%

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

   5. Simplified 75.6%

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

   if 1.0000000000000001e231 < (*.f64 z t)

   1. Initial program 68.3%

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

   3. Taylor expanded in y around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unsub-neg 64.2%

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

      3. associate-/r* 88.3%

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

      4. associate-*r/ 88.3%

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

      5. associate-/r* 88.3%

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

      6. unpow2 88.3%

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

      7. associate-/r* 88.4%

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

      8. associate-/r* 88.3%

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

      9. div-sub 88.3%

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

      10. unsub-neg 88.3%

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

      11. mul-1-neg 88.3%

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

      12. distribute-lft-out 88.3%

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

      13. mul-1-neg 88.3%

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

      14. distribute-neg-frac 88.3%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 99.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.01:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+231}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -6 \cdot 10^{-27} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (or (<= (* z t) -6e-27) (not (<= (* z t) 5e-84)))
   (/ x (* t (- z)))
   (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -6e-27) || !((z * t) <= 5e-84)) {
		tmp = x / (t * -z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-6d-27)) .or. (.not. ((z * t) <= 5d-84))) then
        tmp = x / (t * -z)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -6e-27) || !((z * t) <= 5e-84)) {
		tmp = x / (t * -z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -6e-27) or not ((z * t) <= 5e-84):
		tmp = x / (t * -z)
	else:
		tmp = x / y
	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) <= -6e-27) || !(Float64(z * t) <= 5e-84))
		tmp = Float64(x / Float64(t * Float64(-z)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -6e-27) || ~(((z * t) <= 5e-84)))
		tmp = x / (t * -z);
	else
		tmp = x / y;
	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[Or[LessEqual[N[(z * t), $MachinePrecision], -6e-27], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e-84]], $MachinePrecision]], N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -6.0000000000000002e-27 or 5.0000000000000002e-84 < (*.f64 z t)

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 0 72.3%

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

      1. associate-*r/ 72.3%

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

      2. neg-mul-1 72.3%

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

   5. Simplified 72.3%

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

   if -6.0000000000000002e-27 < (*.f64 z t) < 5.0000000000000002e-84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 89.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -6 \cdot 10^{-27} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+57} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (or (<= (* z t) -2e+57) (not (<= (* z t) 5e+141))) (/ x (* z t)) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+57) || !((z * t) <= 5e+141)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-2d+57)) .or. (.not. ((z * t) <= 5d+141))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+57) || !((z * t) <= 5e+141)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -2e+57) or not ((z * t) <= 5e+141):
		tmp = x / (z * t)
	else:
		tmp = x / y
	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+57) || !(Float64(z * t) <= 5e+141))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -2e+57) || ~(((z * t) <= 5e+141)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	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[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+57], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+141]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e57 or 5.00000000000000025e141 < (*.f64 z t)

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0 71.2%

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

      1. mul-1-neg 71.2%

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

      2. unsub-neg 71.2%

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

      3. associate-/r* 81.5%

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

      4. associate-*r/ 81.5%

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

      5. associate-/r* 81.5%

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

      6. unpow2 81.5%

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

      7. associate-/r* 82.6%

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

      8. associate-/r* 82.6%

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

      9. div-sub 82.6%

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

      10. unsub-neg 82.6%

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

      11. mul-1-neg 82.6%

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

      12. distribute-lft-out 82.6%

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

      13. mul-1-neg 82.6%

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

      14. distribute-neg-frac 82.6%

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

   5. Simplified 92.2%

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

   6. Taylor expanded in y around 0 92.4%

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

      1. add-sqr-sqrt 39.4%

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

      2. sqrt-unprod 47.9%

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

      3. sqr-neg 47.9%

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

      4. sqrt-unprod 18.2%

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

      5. add-sqr-sqrt 35.8%

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

      6. div-inv 35.8%

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

      7. frac-2neg 35.8%

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

      8. metadata-eval 35.8%

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

      9. add-sqr-sqrt 17.6%

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

      10. sqrt-unprod 51.8%

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

      11. sqr-neg 51.8%

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

      12. sqrt-unprod 52.7%

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

      13. add-sqr-sqrt 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. *-commutative 92.3%

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

      2. frac-times 77.0%

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

      3. neg-mul-1 77.0%

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

      4. add-sqr-sqrt 37.7%

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

      5. sqrt-unprod 45.4%

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

      6. sqr-neg 45.4%

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

      7. sqrt-unprod 16.0%

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

      8. add-sqr-sqrt 36.1%

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

   10. Applied egg-rr 36.1%

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

   if -2.0000000000000001e57 < (*.f64 z t) < 5.00000000000000025e141

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 70.4%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+57} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.5% accurate, 2.3× 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 49.4%

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

4. Final simplification 49.4%

   \[\leadsto \frac{x}{y} \]
5. Add Preprocessing

Developer target: 96.4% accurate, 0.3× 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 2024096 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (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))))