Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A

Percentage Accurate: 99.0% → 99.5%

Time: 12.1s

Alternatives: 14

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 1.0 (/ x (* (- y z) (- y t)))))

C

double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - 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 = 1.0d0 - (x / ((y - z) * (y - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Python

def code(x, y, z, t):
	return 1.0 - (x / ((y - z) * (y - t)))

Julia

function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 - (x / ((y - z) * (y - t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 1.0 (/ x (* (- y z) (- y t)))))

C

double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - 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 = 1.0d0 - (x / ((y - z) * (y - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

Python

def code(x, y, z, t):
	return 1.0 - (x / ((y - z) * (y - t)))

Julia

function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 - (x / ((y - z) * (y - t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+36}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z - y}\\ \end{array} \end{array} \]

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 (<= z -2e+36)
   (+ 1.0 (/ (/ x z) (- y t)))
   (+ 1.0 (/ (/ x (- y t)) (- z y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e+36) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else {
		tmp = 1.0 + ((x / (y - t)) / (z - 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 <= (-2d+36)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else
        tmp = 1.0d0 + ((x / (y - t)) / (z - 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 <= -2e+36) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else {
		tmp = 1.0 + ((x / (y - t)) / (z - y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= -2e+36:
		tmp = 1.0 + ((x / z) / (y - t))
	else:
		tmp = 1.0 + ((x / (y - t)) / (z - y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2e+36)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	else
		tmp = Float64(1.0 + Float64(Float64(x / Float64(y - t)) / Float64(z - 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 <= -2e+36)
		tmp = 1.0 + ((x / z) / (y - t));
	else
		tmp = 1.0 + ((x / (y - t)) / (z - 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[LessEqual[z, -2e+36], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000008e36

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. times-frac 100.0%

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

      8. neg-mul-1 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate-+l- 100.0%

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

      13. neg-sub0 100.0%

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

      14. +-commutative 100.0%

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

      15. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.9%

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

      1. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

   if -2.00000000000000008e36 < z

   1. Initial program 96.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 96.0%

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

      2. distribute-frac-neg 96.0%

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

      3. *-lft-identity 96.0%

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

      4. associate-/r* 98.0%

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

      5. associate-*r/ 98.0%

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

      6. metadata-eval 98.0%

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

      7. times-frac 98.0%

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

      8. neg-mul-1 98.0%

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

      9. remove-double-neg 98.0%

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

      10. neg-mul-1 98.0%

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

      11. neg-sub0 98.0%

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

      12. associate-+l- 98.0%

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

      13. neg-sub0 98.0%

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

      14. +-commutative 98.0%

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

      15. sub-neg 98.0%

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

   3. Simplified 98.0%

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

      1. clear-num 98.0%

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

      2. inv-pow 98.0%

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

      3. div-inv 98.0%

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

      4. clear-num 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. unpow-1 98.7%

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

      2. associate-/r* 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around 0 96.0%

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

       1. associate-/r* 99.4%

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

   11. Simplified 99.4%

       \[\leadsto 1 + \color{blue}{\frac{\frac{x}{y - t}}{z - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+36}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{y - t}\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{-22}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-88}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-125} \lor \neg \left(z \leq 7.5 \cdot 10^{-79}\right):\\ \;\;\;\;1 + \frac{t\_1}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{t\_1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (- y t))))
   (if (<= z -2.3e-22)
     (+ 1.0 (/ (/ x z) (- y t)))
     (if (<= z -1.02e-88)
       (- 1.0 (/ x (* y (- y z))))
       (if (or (<= z -1.95e-125) (not (<= z 7.5e-79)))
         (+ 1.0 (/ t_1 z))
         (- 1.0 (/ t_1 y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / (y - t);
	double tmp;
	if (z <= -2.3e-22) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (z <= -1.02e-88) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if ((z <= -1.95e-125) || !(z <= 7.5e-79)) {
		tmp = 1.0 + (t_1 / z);
	} else {
		tmp = 1.0 - (t_1 / 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) :: t_1
    real(8) :: tmp
    t_1 = x / (y - t)
    if (z <= (-2.3d-22)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (z <= (-1.02d-88)) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else if ((z <= (-1.95d-125)) .or. (.not. (z <= 7.5d-79))) then
        tmp = 1.0d0 + (t_1 / z)
    else
        tmp = 1.0d0 - (t_1 / 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 t_1 = x / (y - t);
	double tmp;
	if (z <= -2.3e-22) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (z <= -1.02e-88) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if ((z <= -1.95e-125) || !(z <= 7.5e-79)) {
		tmp = 1.0 + (t_1 / z);
	} else {
		tmp = 1.0 - (t_1 / y);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / (y - t)
	tmp = 0
	if z <= -2.3e-22:
		tmp = 1.0 + ((x / z) / (y - t))
	elif z <= -1.02e-88:
		tmp = 1.0 - (x / (y * (y - z)))
	elif (z <= -1.95e-125) or not (z <= 7.5e-79):
		tmp = 1.0 + (t_1 / z)
	else:
		tmp = 1.0 - (t_1 / y)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(y - t))
	tmp = 0.0
	if (z <= -2.3e-22)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (z <= -1.02e-88)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	elseif ((z <= -1.95e-125) || !(z <= 7.5e-79))
		tmp = Float64(1.0 + Float64(t_1 / z));
	else
		tmp = Float64(1.0 - Float64(t_1 / y));
	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 / (y - t);
	tmp = 0.0;
	if (z <= -2.3e-22)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (z <= -1.02e-88)
		tmp = 1.0 - (x / (y * (y - z)));
	elseif ((z <= -1.95e-125) || ~((z <= 7.5e-79)))
		tmp = 1.0 + (t_1 / z);
	else
		tmp = 1.0 - (t_1 / 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_] := Block[{t$95$1 = N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.3e-22], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.02e-88], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -1.95e-125], N[Not[LessEqual[z, 7.5e-79]], $MachinePrecision]], N[(1.0 + N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.2999999999999998e-22

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. times-frac 100.0%

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

      8. neg-mul-1 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate-+l- 100.0%

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

      13. neg-sub0 100.0%

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

      14. +-commutative 100.0%

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

      15. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 97.3%

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

      1. associate-/r* 97.4%

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

   7. Simplified 97.4%

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

   if -2.2999999999999998e-22 < z < -1.02000000000000001e-88

   1. Initial program 100.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 73.8%

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

   if -1.02000000000000001e-88 < z < -1.94999999999999991e-125 or 7.49999999999999969e-79 < z

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 98.9%

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

      5. associate-*r/ 98.9%

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

      6. metadata-eval 98.9%

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

      7. times-frac 98.9%

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

      8. neg-mul-1 98.9%

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

      9. remove-double-neg 98.9%

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

      10. neg-mul-1 98.9%

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

      11. neg-sub0 98.9%

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

      12. associate-+l- 98.9%

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

      13. neg-sub0 98.9%

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

      14. +-commutative 98.9%

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

      15. sub-neg 98.9%

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

   3. Simplified 98.9%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.1%

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

      1. associate-/r* 98.1%

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

   7. Simplified 98.1%

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

      1. associate-/l/ 99.1%

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

      2. div-inv 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. un-div-inv 99.1%

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

       2. associate-/r* 98.1%

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

   11. Applied egg-rr 98.1%

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

   if -1.94999999999999991e-125 < z < 7.49999999999999969e-79

   1. Initial program 90.8%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 85.0%

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

      1. div-inv 85.0%

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

      2. associate-/r* 85.0%

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

   5. Applied egg-rr 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-*l/ 88.4%

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

      3. associate-*r/ 88.4%

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

      4. associate-*l/ 88.4%

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

      5. *-lft-identity 88.4%

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

   7. Simplified 88.4%

      \[\leadsto 1 - \color{blue}{\frac{\frac{x}{y - t}}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-22}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-88}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-125} \lor \neg \left(z \leq 7.5 \cdot 10^{-79}\right):\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{y - t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 - \frac{x}{t \cdot z}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-80}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (- 1.0 (/ x (* t z)))))
   (if (<= z -9.5e+76)
     t_1
     (if (<= z -2.25e-88)
       (+ 1.0 (/ x (* y z)))
       (if (<= z 3.1e-80) (+ 1.0 (/ (/ x 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 = 1.0 - (x / (t * z));
	double tmp;
	if (z <= -9.5e+76) {
		tmp = t_1;
	} else if (z <= -2.25e-88) {
		tmp = 1.0 + (x / (y * z));
	} else if (z <= 3.1e-80) {
		tmp = 1.0 + ((x / t) / 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 = 1.0d0 - (x / (t * z))
    if (z <= (-9.5d+76)) then
        tmp = t_1
    else if (z <= (-2.25d-88)) then
        tmp = 1.0d0 + (x / (y * z))
    else if (z <= 3.1d-80) then
        tmp = 1.0d0 + ((x / t) / 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 = 1.0 - (x / (t * z));
	double tmp;
	if (z <= -9.5e+76) {
		tmp = t_1;
	} else if (z <= -2.25e-88) {
		tmp = 1.0 + (x / (y * z));
	} else if (z <= 3.1e-80) {
		tmp = 1.0 + ((x / t) / 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 = 1.0 - (x / (t * z))
	tmp = 0
	if z <= -9.5e+76:
		tmp = t_1
	elif z <= -2.25e-88:
		tmp = 1.0 + (x / (y * z))
	elif z <= 3.1e-80:
		tmp = 1.0 + ((x / 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(1.0 - Float64(x / Float64(t * z)))
	tmp = 0.0
	if (z <= -9.5e+76)
		tmp = t_1;
	elseif (z <= -2.25e-88)
		tmp = Float64(1.0 + Float64(x / Float64(y * z)));
	elseif (z <= 3.1e-80)
		tmp = Float64(1.0 + Float64(Float64(x / t) / 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 = 1.0 - (x / (t * z));
	tmp = 0.0;
	if (z <= -9.5e+76)
		tmp = t_1;
	elseif (z <= -2.25e-88)
		tmp = 1.0 + (x / (y * z));
	elseif (z <= 3.1e-80)
		tmp = 1.0 + ((x / t) / 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[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.5e+76], t$95$1, If[LessEqual[z, -2.25e-88], N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e-80], N[(1.0 + N[(N[(x / t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.5000000000000003e76 or 3.10000000000000016e-80 < z

   1. Initial program 100.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 86.2%

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

   if -9.5000000000000003e76 < z < -2.24999999999999996e-88

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 94.2%

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

      5. associate-*r/ 94.2%

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

      6. metadata-eval 94.2%

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

      7. times-frac 94.2%

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

      8. neg-mul-1 94.2%

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

      9. remove-double-neg 94.2%

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

      10. neg-mul-1 94.2%

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

      11. neg-sub0 94.2%

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

      12. associate-+l- 94.2%

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

      13. neg-sub0 94.2%

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

      14. +-commutative 94.2%

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

      15. sub-neg 94.2%

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

   3. Simplified 94.2%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 91.4%

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

      1. associate-/r* 85.7%

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

   7. Simplified 85.7%

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

   8. Taylor expanded in y around inf 59.8%

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

      1. *-commutative 59.8%

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

   10. Simplified 59.8%

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

   if -2.24999999999999996e-88 < z < 3.10000000000000016e-80

   1. Initial program 91.2%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 91.2%

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

      2. distribute-frac-neg 91.2%

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

      3. *-lft-identity 91.2%

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

      4. associate-/r* 97.8%

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

      5. associate-*r/ 97.8%

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

      6. metadata-eval 97.8%

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

      7. times-frac 97.8%

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

      8. neg-mul-1 97.8%

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

      9. remove-double-neg 97.8%

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

      10. neg-mul-1 97.8%

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

      11. neg-sub0 97.8%

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

      12. associate-+l- 97.8%

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

      13. neg-sub0 97.8%

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

      14. +-commutative 97.8%

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

      15. sub-neg 97.8%

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

   3. Simplified 97.8%

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

      1. clear-num 97.7%

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

      2. inv-pow 97.7%

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

      3. div-inv 97.7%

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

      4. clear-num 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. unpow-1 98.7%

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

      2. associate-/r* 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around 0 91.2%

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

       1. associate-/r* 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in t around inf 71.8%

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

       1. mul-1-neg 71.8%

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

       2. associate-/r* 75.1%

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

       3. distribute-neg-frac 75.1%

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

       4. distribute-neg-frac 75.1%

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

   14. Simplified 75.1%

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

   15. Taylor expanded in z around 0 64.9%

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

       1. associate-/r* 65.1%

          \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t}}{y}} \]

   17. Simplified 65.1%

       \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+76}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-80}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-275}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-120}:\\ \;\;\;\;1 + \frac{\frac{1}{y}}{\frac{z - y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

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 (<= t -1.25e-275)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= t 2.5e-120)
     (+ 1.0 (/ (/ 1.0 y) (/ (- z y) x)))
     (- 1.0 (/ (/ x t) (- z y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.25e-275) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 2.5e-120) {
		tmp = 1.0 + ((1.0 / y) / ((z - y) / x));
	} else {
		tmp = 1.0 - ((x / t) / (z - 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 (t <= (-1.25d-275)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (t <= 2.5d-120) then
        tmp = 1.0d0 + ((1.0d0 / y) / ((z - y) / x))
    else
        tmp = 1.0d0 - ((x / t) / (z - 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 (t <= -1.25e-275) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 2.5e-120) {
		tmp = 1.0 + ((1.0 / y) / ((z - y) / x));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -1.25e-275:
		tmp = 1.0 + ((x / z) / (y - t))
	elif t <= 2.5e-120:
		tmp = 1.0 + ((1.0 / y) / ((z - y) / x))
	else:
		tmp = 1.0 - ((x / t) / (z - y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.25e-275)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (t <= 2.5e-120)
		tmp = Float64(1.0 + Float64(Float64(1.0 / y) / Float64(Float64(z - y) / x)));
	else
		tmp = Float64(1.0 - Float64(Float64(x / t) / Float64(z - 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 (t <= -1.25e-275)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (t <= 2.5e-120)
		tmp = 1.0 + ((1.0 / y) / ((z - y) / x));
	else
		tmp = 1.0 - ((x / t) / (z - 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[LessEqual[t, -1.25e-275], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e-120], N[(1.0 + N[(N[(1.0 / y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.24999999999999996e-275

   1. Initial program 98.3%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. *-lft-identity 98.3%

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

      4. associate-/r* 97.7%

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

      5. associate-*r/ 97.7%

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

      6. metadata-eval 97.7%

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

      7. times-frac 97.7%

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

      8. neg-mul-1 97.7%

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

      9. remove-double-neg 97.7%

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

      10. neg-mul-1 97.7%

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

      11. neg-sub0 97.7%

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

      12. associate-+l- 97.7%

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

      13. neg-sub0 97.7%

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

      14. +-commutative 97.7%

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

      15. sub-neg 97.7%

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

   3. Simplified 97.7%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 80.6%

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

      1. associate-/r* 79.4%

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

   7. Simplified 79.4%

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

   if -1.24999999999999996e-275 < t < 2.50000000000000003e-120

   1. Initial program 86.1%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 86.1%

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

      2. distribute-frac-neg 86.1%

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

      3. *-lft-identity 86.1%

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

      4. associate-/r* 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. times-frac 100.0%

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

      8. neg-mul-1 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate-+l- 100.0%

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

      13. neg-sub0 100.0%

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

      14. +-commutative 100.0%

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

      15. sub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. div-inv 99.9%

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

      4. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around inf 94.3%

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

   if 2.50000000000000003e-120 < t

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 98.9%

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

      5. associate-*r/ 98.9%

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

      6. metadata-eval 98.9%

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

      7. times-frac 98.9%

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

      8. neg-mul-1 98.9%

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

      9. remove-double-neg 98.9%

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

      10. neg-mul-1 98.9%

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

      11. neg-sub0 98.9%

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

      12. associate-+l- 98.9%

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

      13. neg-sub0 98.9%

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

      14. +-commutative 98.9%

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

      15. sub-neg 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

      3. div-inv 98.8%

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

      4. clear-num 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around inf 94.4%

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

       1. mul-1-neg 94.4%

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

       2. associate-/r* 94.4%

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

   11. Simplified 94.4%

       \[\leadsto 1 + \color{blue}{\left(-\frac{\frac{x}{t}}{z - y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-275}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-120}:\\ \;\;\;\;1 + \frac{\frac{1}{y}}{\frac{z - y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+16} \lor \neg \left(y \leq 3.4 \cdot 10^{-8}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \end{array} \end{array} \]

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 (<= y -3.4e+16) (not (<= y 3.4e-8)))
   (- 1.0 (/ (/ x y) y))
   (+ 1.0 (/ (/ x z) (- y t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.4e+16) || !(y <= 3.4e-8)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 + ((x / z) / (y - t));
	}
	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 ((y <= (-3.4d+16)) .or. (.not. (y <= 3.4d-8))) then
        tmp = 1.0d0 - ((x / y) / y)
    else
        tmp = 1.0d0 + ((x / z) / (y - t))
    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 ((y <= -3.4e+16) || !(y <= 3.4e-8)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 + ((x / z) / (y - t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -3.4e+16) or not (y <= 3.4e-8):
		tmp = 1.0 - ((x / y) / y)
	else:
		tmp = 1.0 + ((x / z) / (y - t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.4e+16) || !(y <= 3.4e-8))
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	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 ((y <= -3.4e+16) || ~((y <= 3.4e-8)))
		tmp = 1.0 - ((x / y) / y);
	else
		tmp = 1.0 + ((x / z) / (y - t));
	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[y, -3.4e+16], N[Not[LessEqual[y, 3.4e-8]], $MachinePrecision]], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e16 or 3.4e-8 < y

   1. Initial program 100.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 98.2%

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

      1. div-inv 98.2%

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

      2. associate-/r* 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l/ 98.2%

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

      3. associate-*r/ 98.2%

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

      4. associate-*l/ 98.2%

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

      5. *-lft-identity 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in y around inf 94.6%

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

   if -3.4e16 < y < 3.4e-8

   1. Initial program 93.7%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 93.7%

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

      2. distribute-frac-neg 93.7%

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

      3. *-lft-identity 93.7%

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

      4. associate-/r* 96.9%

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

      5. associate-*r/ 96.9%

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

      6. metadata-eval 96.9%

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

      7. times-frac 96.9%

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

      8. neg-mul-1 96.9%

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

      9. remove-double-neg 96.9%

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

      10. neg-mul-1 96.9%

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

      11. neg-sub0 96.9%

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

      12. associate-+l- 96.9%

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

      13. neg-sub0 96.9%

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

      14. +-commutative 96.9%

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

      15. sub-neg 96.9%

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

   3. Simplified 96.9%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 81.6%

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

      1. associate-/r* 82.0%

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

   7. Simplified 82.0%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{z}}{y - t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+16} \lor \neg \left(y \leq 3.4 \cdot 10^{-8}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-28} \lor \neg \left(y \leq 1.6 \cdot 10^{-63}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \end{array} \end{array} \]

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 (<= y -3.5e-28) (not (<= y 1.6e-63)))
   (- 1.0 (/ x (* y (- y t))))
   (+ 1.0 (/ (/ x z) (- y t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.5e-28) || !(y <= 1.6e-63)) {
		tmp = 1.0 - (x / (y * (y - t)));
	} else {
		tmp = 1.0 + ((x / z) / (y - t));
	}
	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 ((y <= (-3.5d-28)) .or. (.not. (y <= 1.6d-63))) then
        tmp = 1.0d0 - (x / (y * (y - t)))
    else
        tmp = 1.0d0 + ((x / z) / (y - t))
    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 ((y <= -3.5e-28) || !(y <= 1.6e-63)) {
		tmp = 1.0 - (x / (y * (y - t)));
	} else {
		tmp = 1.0 + ((x / z) / (y - t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -3.5e-28) or not (y <= 1.6e-63):
		tmp = 1.0 - (x / (y * (y - t)))
	else:
		tmp = 1.0 + ((x / z) / (y - t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.5e-28) || !(y <= 1.6e-63))
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - t))));
	else
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	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 ((y <= -3.5e-28) || ~((y <= 1.6e-63)))
		tmp = 1.0 - (x / (y * (y - t)));
	else
		tmp = 1.0 + ((x / z) / (y - t));
	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[y, -3.5e-28], N[Not[LessEqual[y, 1.6e-63]], $MachinePrecision]], N[(1.0 - N[(x / N[(y * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5e-28 or 1.59999999999999994e-63 < y

   1. Initial program 100.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 94.9%

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

   if -3.5e-28 < y < 1.59999999999999994e-63

   1. Initial program 92.6%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 92.6%

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

      2. distribute-frac-neg 92.6%

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

      3. *-lft-identity 92.6%

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

      4. associate-/r* 96.4%

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

      5. associate-*r/ 96.4%

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

      6. metadata-eval 96.4%

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

      7. times-frac 96.4%

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

      8. neg-mul-1 96.4%

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

      9. remove-double-neg 96.4%

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

      10. neg-mul-1 96.4%

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

      11. neg-sub0 96.4%

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

      12. associate-+l- 96.4%

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

      13. neg-sub0 96.4%

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

      14. +-commutative 96.4%

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

      15. sub-neg 96.4%

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

   3. Simplified 96.4%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 83.6%

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

      1. associate-/r* 84.1%

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

   7. Simplified 84.1%

      \[\leadsto 1 + \color{blue}{\frac{\frac{x}{z}}{y - t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-28} \lor \neg \left(y \leq 1.6 \cdot 10^{-63}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-81}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-62}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \end{array} \end{array} \]

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 (<= y -6.8e-81)
   (- 1.0 (/ x (* y (- y z))))
   (if (<= y 2.7e-62)
     (+ 1.0 (/ (/ x z) (- y t)))
     (- 1.0 (/ x (* y (- y t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.8e-81) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 2.7e-62) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else {
		tmp = 1.0 - (x / (y * (y - t)));
	}
	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 (y <= (-6.8d-81)) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else if (y <= 2.7d-62) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else
        tmp = 1.0d0 - (x / (y * (y - t)))
    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 (y <= -6.8e-81) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 2.7e-62) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else {
		tmp = 1.0 - (x / (y * (y - t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -6.8e-81:
		tmp = 1.0 - (x / (y * (y - z)))
	elif y <= 2.7e-62:
		tmp = 1.0 + ((x / z) / (y - t))
	else:
		tmp = 1.0 - (x / (y * (y - t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.8e-81)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	elseif (y <= 2.7e-62)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - t))));
	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 (y <= -6.8e-81)
		tmp = 1.0 - (x / (y * (y - z)));
	elseif (y <= 2.7e-62)
		tmp = 1.0 + ((x / z) / (y - t));
	else
		tmp = 1.0 - (x / (y * (y - t)));
	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[y, -6.8e-81], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e-62], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(y * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.7999999999999997e-81

   1. Initial program 100.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 96.3%

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

   if -6.7999999999999997e-81 < y < 2.70000000000000019e-62

   1. Initial program 91.7%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 91.7%

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

      2. distribute-frac-neg 91.7%

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

      3. *-lft-identity 91.7%

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

      4. associate-/r* 95.9%

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

      5. associate-*r/ 95.9%

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

      6. metadata-eval 95.9%

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

      7. times-frac 95.9%

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

      8. neg-mul-1 95.9%

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

      9. remove-double-neg 95.9%

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

      10. neg-mul-1 95.9%

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

      11. neg-sub0 95.9%

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

      12. associate-+l- 95.9%

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

      13. neg-sub0 95.9%

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

      14. +-commutative 95.9%

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

      15. sub-neg 95.9%

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

   3. Simplified 95.9%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 84.6%

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

      1. associate-/r* 85.1%

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

   7. Simplified 85.1%

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

   if 2.70000000000000019e-62 < y

   1. Initial program 100.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 94.4%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-81}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-62}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-116}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

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 (<= t -1.35e-272)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= t 1.25e-116)
     (- 1.0 (/ x (* y (- y z))))
     (- 1.0 (/ (/ x t) (- z y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.35e-272) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 1.25e-116) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - ((x / t) / (z - 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 (t <= (-1.35d-272)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (t <= 1.25d-116) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 - ((x / t) / (z - 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 (t <= -1.35e-272) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 1.25e-116) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -1.35e-272:
		tmp = 1.0 + ((x / z) / (y - t))
	elif t <= 1.25e-116:
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 - ((x / t) / (z - y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.35e-272)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (t <= 1.25e-116)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 - Float64(Float64(x / t) / Float64(z - 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 (t <= -1.35e-272)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (t <= 1.25e-116)
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 - ((x / t) / (z - 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[LessEqual[t, -1.35e-272], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e-116], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.34999999999999996e-272

   1. Initial program 98.2%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 98.2%

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

      2. distribute-frac-neg 98.2%

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

      3. *-lft-identity 98.2%

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

      4. associate-/r* 97.7%

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

      5. associate-*r/ 97.7%

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

      6. metadata-eval 97.7%

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

      7. times-frac 97.7%

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

      8. neg-mul-1 97.7%

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

      9. remove-double-neg 97.7%

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

      10. neg-mul-1 97.7%

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

      11. neg-sub0 97.7%

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

      12. associate-+l- 97.7%

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

      13. neg-sub0 97.7%

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

      14. +-commutative 97.7%

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

      15. sub-neg 97.7%

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

   3. Simplified 97.7%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 81.2%

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

      1. associate-/r* 80.0%

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

   7. Simplified 80.0%

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

   if -1.34999999999999996e-272 < t < 1.2500000000000001e-116

   1. Initial program 86.5%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 84.3%

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

   if 1.2500000000000001e-116 < t

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 98.9%

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

      5. associate-*r/ 98.9%

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

      6. metadata-eval 98.9%

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

      7. times-frac 98.9%

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

      8. neg-mul-1 98.9%

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

      9. remove-double-neg 98.9%

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

      10. neg-mul-1 98.9%

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

      11. neg-sub0 98.9%

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

      12. associate-+l- 98.9%

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

      13. neg-sub0 98.9%

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

      14. +-commutative 98.9%

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

      15. sub-neg 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

      3. div-inv 98.8%

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

      4. clear-num 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around inf 94.4%

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

       1. mul-1-neg 94.4%

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

       2. associate-/r* 94.4%

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

   11. Simplified 94.4%

       \[\leadsto 1 + \color{blue}{\left(-\frac{\frac{x}{t}}{z - y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-116}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -11800 \lor \neg \left(y \leq 1.68 \cdot 10^{-9}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \end{array} \]

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 (or (<= y -11800.0) (not (<= y 1.68e-9)))
   (- 1.0 (/ (/ x y) y))
   (- 1.0 (/ x (* t z)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -11800.0) || !(y <= 1.68e-9)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - (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 ((y <= (-11800.0d0)) .or. (.not. (y <= 1.68d-9))) then
        tmp = 1.0d0 - ((x / y) / y)
    else
        tmp = 1.0d0 - (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 ((y <= -11800.0) || !(y <= 1.68e-9)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -11800.0) or not (y <= 1.68e-9):
		tmp = 1.0 - ((x / y) / y)
	else:
		tmp = 1.0 - (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 ((y <= -11800.0) || !(y <= 1.68e-9))
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * 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 ((y <= -11800.0) || ~((y <= 1.68e-9)))
		tmp = 1.0 - ((x / y) / y);
	else
		tmp = 1.0 - (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[Or[LessEqual[y, -11800.0], N[Not[LessEqual[y, 1.68e-9]], $MachinePrecision]], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -11800 or 1.68e-9 < y

   1. Initial program 100.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 97.9%

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

      1. div-inv 97.9%

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

      2. associate-/r* 97.9%

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

   5. Applied egg-rr 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*l/ 97.8%

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

      3. associate-*r/ 97.9%

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

      4. associate-*l/ 97.9%

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

      5. *-lft-identity 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in y around inf 94.4%

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

   if -11800 < y < 1.68e-9

   1. Initial program 93.6%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 72.0%

      \[\leadsto 1 - \color{blue}{\frac{x}{t \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11800 \lor \neg \left(y \leq 1.68 \cdot 10^{-9}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-7} \lor \neg \left(y \leq 9 \cdot 10^{-10}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \end{array} \end{array} \]

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 (<= y -2.7e-7) (not (<= y 9e-10)))
   (- 1.0 (/ (/ x y) y))
   (- 1.0 (/ (/ x z) t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.7e-7) || !(y <= 9e-10)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - ((x / z) / t);
	}
	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 ((y <= (-2.7d-7)) .or. (.not. (y <= 9d-10))) then
        tmp = 1.0d0 - ((x / y) / y)
    else
        tmp = 1.0d0 - ((x / z) / t)
    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 ((y <= -2.7e-7) || !(y <= 9e-10)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - ((x / z) / t);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -2.7e-7) or not (y <= 9e-10):
		tmp = 1.0 - ((x / y) / y)
	else:
		tmp = 1.0 - ((x / z) / t)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.7e-7) || !(y <= 9e-10))
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	else
		tmp = Float64(1.0 - Float64(Float64(x / z) / t));
	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 ((y <= -2.7e-7) || ~((y <= 9e-10)))
		tmp = 1.0 - ((x / y) / y);
	else
		tmp = 1.0 - ((x / z) / t);
	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[y, -2.7e-7], N[Not[LessEqual[y, 9e-10]], $MachinePrecision]], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.70000000000000009e-7 or 8.9999999999999999e-10 < y

   1. Initial program 100.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 97.2%

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

      1. div-inv 97.1%

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

      2. associate-/r* 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-*l/ 97.1%

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

      3. associate-*r/ 97.1%

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

      4. associate-*l/ 97.1%

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

      5. *-lft-identity 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in y around inf 93.7%

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

   if -2.70000000000000009e-7 < y < 8.9999999999999999e-10

   1. Initial program 93.5%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 93.5%

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

      2. distribute-frac-neg 93.5%

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

      3. *-lft-identity 93.5%

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

      4. associate-/r* 96.8%

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

      5. associate-*r/ 96.8%

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

      6. metadata-eval 96.8%

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

      7. times-frac 96.8%

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

      8. neg-mul-1 96.8%

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

      9. remove-double-neg 96.8%

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

      10. neg-mul-1 96.8%

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

      11. neg-sub0 96.8%

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

      12. associate-+l- 96.8%

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

      13. neg-sub0 96.8%

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

      14. +-commutative 96.8%

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

      15. sub-neg 96.8%

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

   3. Simplified 96.8%

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

      1. clear-num 96.8%

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

      2. inv-pow 96.8%

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

      3. div-inv 96.7%

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

      4. clear-num 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. unpow-1 97.8%

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

      2. associate-/r* 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in x around 0 93.5%

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

       1. associate-/r* 96.8%

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

   11. Simplified 96.8%

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

   12. Taylor expanded in t around inf 81.0%

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

       1. mul-1-neg 81.0%

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

       2. associate-/r* 81.1%

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

       3. distribute-neg-frac 81.1%

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

       4. distribute-neg-frac 81.1%

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

   14. Simplified 81.1%

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

   15. Taylor expanded in z around inf 72.6%

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

       1. associate-*r/ 72.6%

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

       2. *-commutative 72.6%

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

       3. associate-/r* 71.4%

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

       4. neg-mul-1 71.4%

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

   17. Simplified 71.4%

       \[\leadsto 1 + \color{blue}{\frac{\frac{-x}{z}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-7} \lor \neg \left(y \leq 9 \cdot 10^{-10}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \frac{\frac{1}{y - t}}{\frac{z - y}{x}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ 1.0 (/ (/ 1.0 (- y t)) (/ (- z y) x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + ((1.0 / (y - t)) / ((z - y) / x));
}

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 = 1.0d0 + ((1.0d0 / (y - t)) / ((z - y) / x))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 + ((1.0 / (y - t)) / ((z - y) / x));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + ((1.0 / (y - t)) / ((z - y) / x))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(Float64(1.0 / Float64(y - t)) / Float64(Float64(z - y) / x)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + ((1.0 / (y - t)) / ((z - y) / x));
end

Wolfram

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

Derivation

1. Initial program 96.9%

   \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
2. Step-by-step derivation

   1. sub-neg 96.9%

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

   2. distribute-frac-neg 96.9%

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

   3. *-lft-identity 96.9%

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

   4. associate-/r* 98.5%

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

   5. associate-*r/ 98.5%

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

   6. metadata-eval 98.5%

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

   7. times-frac 98.5%

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

   8. neg-mul-1 98.5%

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

   9. remove-double-neg 98.5%

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

   10. neg-mul-1 98.5%

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

   11. neg-sub0 98.5%

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

   12. associate-+l- 98.5%

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

   13. neg-sub0 98.5%

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

   14. +-commutative 98.5%

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

   15. sub-neg 98.5%

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

3. Simplified 98.5%

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

   1. clear-num 98.4%

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

   2. inv-pow 98.4%

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

   3. div-inv 98.4%

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

   4. clear-num 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

   2. associate-/r* 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

   \[\leadsto 1 + \frac{\frac{1}{y - t}}{\frac{z - y}{x}} \]
10. Add Preprocessing

Alternative 12: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-82}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 9.5e-82) (+ 1.0 (/ x (* y z))) (+ 1.0 (/ x (* y t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.5e-82) {
		tmp = 1.0 + (x / (y * z));
	} else {
		tmp = 1.0 + (x / (y * t));
	}
	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 (t <= 9.5d-82) then
        tmp = 1.0d0 + (x / (y * z))
    else
        tmp = 1.0d0 + (x / (y * t))
    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 (t <= 9.5e-82) {
		tmp = 1.0 + (x / (y * z));
	} else {
		tmp = 1.0 + (x / (y * t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 9.5e-82:
		tmp = 1.0 + (x / (y * z))
	else:
		tmp = 1.0 + (x / (y * t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 9.5e-82)
		tmp = Float64(1.0 + Float64(x / Float64(y * z)));
	else
		tmp = Float64(1.0 + Float64(x / Float64(y * t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 9.5e-82)
		tmp = 1.0 + (x / (y * z));
	else
		tmp = 1.0 + (x / (y * t));
	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[t, 9.5e-82], N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.4999999999999996e-82

   1. Initial program 95.5%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 95.5%

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

      2. distribute-frac-neg 95.5%

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

      3. *-lft-identity 95.5%

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

      4. associate-/r* 98.3%

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

      5. associate-*r/ 98.3%

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

      6. metadata-eval 98.3%

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

      7. times-frac 98.3%

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

      8. neg-mul-1 98.3%

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

      9. remove-double-neg 98.3%

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

      10. neg-mul-1 98.3%

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

      11. neg-sub0 98.3%

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

      12. associate-+l- 98.3%

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

      13. neg-sub0 98.3%

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

      14. +-commutative 98.3%

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

      15. sub-neg 98.3%

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

   3. Simplified 98.3%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 78.0%

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

      1. associate-/r* 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in y around inf 59.6%

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

      1. *-commutative 59.6%

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

   10. Simplified 59.6%

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

   if 9.4999999999999996e-82 < t

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 98.7%

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

      5. associate-*r/ 98.7%

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

      6. metadata-eval 98.7%

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

      7. times-frac 98.7%

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

      8. neg-mul-1 98.7%

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

      9. remove-double-neg 98.7%

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

      10. neg-mul-1 98.7%

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

      11. neg-sub0 98.7%

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

      12. associate-+l- 98.7%

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

      13. neg-sub0 98.7%

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

      14. +-commutative 98.7%

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

      15. sub-neg 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.7%

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

      2. inv-pow 98.7%

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

      3. div-inv 98.7%

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

      4. clear-num 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 99.9%

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

       1. associate-/r* 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in t around inf 98.7%

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

       1. mul-1-neg 98.7%

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

       2. associate-/r* 98.7%

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

       3. distribute-neg-frac 98.7%

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

       4. distribute-neg-frac 98.7%

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

   14. Simplified 98.7%

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

   15. Taylor expanded in z around 0 74.3%

       \[\leadsto 1 + \color{blue}{\frac{x}{t \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-82}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \frac{\frac{x}{z - y}}{y - t} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ 1.0 (/ (/ x (- z y)) (- y t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + ((x / (z - y)) / (y - 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 = 1.0d0 + ((x / (z - y)) / (y - t))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 + ((x / (z - y)) / (y - t));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + ((x / (z - y)) / (y - t))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(Float64(x / Float64(z - y)) / Float64(y - t)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + ((x / (z - y)) / (y - 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[(1.0 + N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

   \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
2. Step-by-step derivation

   1. sub-neg 96.9%

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

   2. distribute-frac-neg 96.9%

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

   3. *-lft-identity 96.9%

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

   4. associate-/r* 98.5%

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

   5. associate-*r/ 98.5%

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

   6. metadata-eval 98.5%

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

   7. times-frac 98.5%

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

   8. neg-mul-1 98.5%

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

   9. remove-double-neg 98.5%

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

   10. neg-mul-1 98.5%

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

   11. neg-sub0 98.5%

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

   12. associate-+l- 98.5%

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

   13. neg-sub0 98.5%

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

   14. +-commutative 98.5%

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

   15. sub-neg 98.5%

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

3. Simplified 98.5%

   \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
4. Add Preprocessing

5. Final simplification 98.5%

   \[\leadsto 1 + \frac{\frac{x}{z - y}}{y - t} \]
6. Add Preprocessing

Alternative 14: 56.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 + \frac{x}{y \cdot t} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ 1.0 (/ x (* y t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 + (x / (y * 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 = 1.0d0 + (x / (y * t))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 + (x / (y * t));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 + (x / (y * t))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 + Float64(x / Float64(y * t)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 + (x / (y * 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[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

   \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
2. Step-by-step derivation

   1. sub-neg 96.9%

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

   2. distribute-frac-neg 96.9%

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

   3. *-lft-identity 96.9%

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

   4. associate-/r* 98.5%

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

   5. associate-*r/ 98.5%

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

   6. metadata-eval 98.5%

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

   7. times-frac 98.5%

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

   8. neg-mul-1 98.5%

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

   9. remove-double-neg 98.5%

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

   10. neg-mul-1 98.5%

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

   11. neg-sub0 98.5%

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

   12. associate-+l- 98.5%

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

   13. neg-sub0 98.5%

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

   14. +-commutative 98.5%

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

   15. sub-neg 98.5%

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

3. Simplified 98.5%

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

   1. clear-num 98.4%

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

   2. inv-pow 98.4%

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

   3. div-inv 98.4%

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

   4. clear-num 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

   2. associate-/r* 98.9%

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

8. Simplified 98.9%

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

9. Taylor expanded in x around 0 96.9%

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

    1. associate-/r* 98.5%

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

11. Simplified 98.5%

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

12. Taylor expanded in t around inf 80.2%

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

    1. mul-1-neg 80.2%

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

    2. associate-/r* 80.2%

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

    3. distribute-neg-frac 80.2%

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

    4. distribute-neg-frac 80.2%

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

14. Simplified 80.2%

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

15. Taylor expanded in z around 0 57.6%

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

16. Final simplification 57.6%

    \[\leadsto 1 + \frac{x}{y \cdot t} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A"
  :precision binary64
  (- 1.0 (/ x (* (- y z) (- y t)))))