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

Percentage Accurate: 99.0% → 98.5%

Time: 12.1s

Alternatives: 12

Speedup: N/A×

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: 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. neg-mul-1 96.9%

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

   3. *-commutative 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-/r* 98.5%

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

   6. associate-*r/ 98.5%

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

   7. metadata-eval 98.5%

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

   8. times-frac 98.5%

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

   9. *-lft-identity 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. sub-neg 98.5%

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

   12. +-commutative 98.5%

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

   13. distribute-neg-out 98.5%

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

   14. remove-double-neg 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 2: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 - \frac{x}{t \cdot z}\\ \mathbf{if}\;t \leq -8 \cdot 10^{-120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-82}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y}\\ \end{array} \end{array} \]

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 (- 1.0 (/ x (* t z)))))
   (if (<= t -8e-120)
     t_1
     (if (<= t 1.5e-82)
       (+ 1.0 (/ x (* y z)))
       (if (<= t 1.95e-7)
         (+ 1.0 (/ x (* y t)))
         (if (<= t 3.3e+97) t_1 (+ 1.0 (/ (/ x t) y))))))))

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 (t <= -8e-120) {
		tmp = t_1;
	} else if (t <= 1.5e-82) {
		tmp = 1.0 + (x / (y * z));
	} else if (t <= 1.95e-7) {
		tmp = 1.0 + (x / (y * t));
	} else if (t <= 3.3e+97) {
		tmp = t_1;
	} else {
		tmp = 1.0 + ((x / t) / 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 = 1.0d0 - (x / (t * z))
    if (t <= (-8d-120)) then
        tmp = t_1
    else if (t <= 1.5d-82) then
        tmp = 1.0d0 + (x / (y * z))
    else if (t <= 1.95d-7) then
        tmp = 1.0d0 + (x / (y * t))
    else if (t <= 3.3d+97) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((x / t) / 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 = 1.0 - (x / (t * z));
	double tmp;
	if (t <= -8e-120) {
		tmp = t_1;
	} else if (t <= 1.5e-82) {
		tmp = 1.0 + (x / (y * z));
	} else if (t <= 1.95e-7) {
		tmp = 1.0 + (x / (y * t));
	} else if (t <= 3.3e+97) {
		tmp = t_1;
	} else {
		tmp = 1.0 + ((x / t) / y);
	}
	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 t <= -8e-120:
		tmp = t_1
	elif t <= 1.5e-82:
		tmp = 1.0 + (x / (y * z))
	elif t <= 1.95e-7:
		tmp = 1.0 + (x / (y * t))
	elif t <= 3.3e+97:
		tmp = t_1
	else:
		tmp = 1.0 + ((x / t) / y)
	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 (t <= -8e-120)
		tmp = t_1;
	elseif (t <= 1.5e-82)
		tmp = Float64(1.0 + Float64(x / Float64(y * z)));
	elseif (t <= 1.95e-7)
		tmp = Float64(1.0 + Float64(x / Float64(y * t)));
	elseif (t <= 3.3e+97)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / 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 = 1.0 - (x / (t * z));
	tmp = 0.0;
	if (t <= -8e-120)
		tmp = t_1;
	elseif (t <= 1.5e-82)
		tmp = 1.0 + (x / (y * z));
	elseif (t <= 1.95e-7)
		tmp = 1.0 + (x / (y * t));
	elseif (t <= 3.3e+97)
		tmp = t_1;
	else
		tmp = 1.0 + ((x / t) / 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[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8e-120], t$95$1, If[LessEqual[t, 1.5e-82], N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.95e-7], N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e+97], t$95$1, N[(1.0 + N[(N[(x / t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.99999999999999983e-120 or 1.95000000000000012e-7 < t < 3.3000000000000001e97

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0 79.3%

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

   if -7.99999999999999983e-120 < t < 1.4999999999999999e-82

   1. Initial program 91.0%

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

      1. sub-neg 91.0%

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

      2. neg-mul-1 91.0%

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

      3. *-commutative 91.0%

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

      4. *-commutative 91.0%

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

      5. associate-/r* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. times-frac 99.9%

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

      9. *-lft-identity 99.9%

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

      10. neg-mul-1 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-out 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 70.6%

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

      1. associate-/r* 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in y around inf 63.3%

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

      1. *-commutative 63.3%

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

   10. Simplified 63.3%

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

   if 1.4999999999999999e-82 < t < 1.95000000000000012e-7

   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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. times-frac 99.9%

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

      9. *-lft-identity 99.9%

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

      10. neg-mul-1 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-out 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 54.4%

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

   if 3.3000000000000001e97 < 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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. times-frac 99.9%

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

      9. *-lft-identity 99.9%

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

      10. neg-mul-1 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-out 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around 0 89.6%

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

      1. associate-/r* 89.6%

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

      2. div-inv 89.6%

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

   10. Applied egg-rr 89.6%

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

       1. un-div-inv 89.6%

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

   12. Applied egg-rr 89.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-120}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-82}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+97}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-82}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+96}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y}\\ \end{array} \end{array} \]

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 -1.06e-119)
   (- 1.0 (/ x (* t z)))
   (if (<= t 9.2e-82)
     (+ 1.0 (/ x (* y z)))
     (if (<= t 2.1e-7)
       (+ 1.0 (/ x (* y t)))
       (if (<= t 6.2e+96) (- 1.0 (/ (/ x t) z)) (+ 1.0 (/ (/ x t) y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.06e-119) {
		tmp = 1.0 - (x / (t * z));
	} else if (t <= 9.2e-82) {
		tmp = 1.0 + (x / (y * z));
	} else if (t <= 2.1e-7) {
		tmp = 1.0 + (x / (y * t));
	} else if (t <= 6.2e+96) {
		tmp = 1.0 - ((x / t) / z);
	} else {
		tmp = 1.0 + ((x / t) / 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 <= (-1.06d-119)) then
        tmp = 1.0d0 - (x / (t * z))
    else if (t <= 9.2d-82) then
        tmp = 1.0d0 + (x / (y * z))
    else if (t <= 2.1d-7) then
        tmp = 1.0d0 + (x / (y * t))
    else if (t <= 6.2d+96) then
        tmp = 1.0d0 - ((x / t) / z)
    else
        tmp = 1.0d0 + ((x / t) / 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 <= -1.06e-119) {
		tmp = 1.0 - (x / (t * z));
	} else if (t <= 9.2e-82) {
		tmp = 1.0 + (x / (y * z));
	} else if (t <= 2.1e-7) {
		tmp = 1.0 + (x / (y * t));
	} else if (t <= 6.2e+96) {
		tmp = 1.0 - ((x / t) / z);
	} else {
		tmp = 1.0 + ((x / t) / y);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -1.06e-119:
		tmp = 1.0 - (x / (t * z))
	elif t <= 9.2e-82:
		tmp = 1.0 + (x / (y * z))
	elif t <= 2.1e-7:
		tmp = 1.0 + (x / (y * t))
	elif t <= 6.2e+96:
		tmp = 1.0 - ((x / t) / z)
	else:
		tmp = 1.0 + ((x / t) / y)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.06e-119)
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	elseif (t <= 9.2e-82)
		tmp = Float64(1.0 + Float64(x / Float64(y * z)));
	elseif (t <= 2.1e-7)
		tmp = Float64(1.0 + Float64(x / Float64(y * t)));
	elseif (t <= 6.2e+96)
		tmp = Float64(1.0 - Float64(Float64(x / t) / z));
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / 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 <= -1.06e-119)
		tmp = 1.0 - (x / (t * z));
	elseif (t <= 9.2e-82)
		tmp = 1.0 + (x / (y * z));
	elseif (t <= 2.1e-7)
		tmp = 1.0 + (x / (y * t));
	elseif (t <= 6.2e+96)
		tmp = 1.0 - ((x / t) / z);
	else
		tmp = 1.0 + ((x / t) / 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, -1.06e-119], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e-82], N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e-7], N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.2e+96], N[(1.0 - N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.05999999999999999e-119

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0 79.3%

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

   if -1.05999999999999999e-119 < t < 9.19999999999999988e-82

   1. Initial program 91.0%

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

      1. sub-neg 91.0%

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

      2. neg-mul-1 91.0%

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

      3. *-commutative 91.0%

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

      4. *-commutative 91.0%

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

      5. associate-/r* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. times-frac 99.9%

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

      9. *-lft-identity 99.9%

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

      10. neg-mul-1 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-out 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 70.6%

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

      1. associate-/r* 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in y around inf 63.3%

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

      1. *-commutative 63.3%

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

   10. Simplified 63.3%

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

   if 9.19999999999999988e-82 < t < 2.1e-7

   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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. times-frac 99.9%

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

      9. *-lft-identity 99.9%

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

      10. neg-mul-1 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-out 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 54.4%

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

   if 2.1e-7 < t < 6.1999999999999996e96

   1. Initial program 99.9%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r* 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 79.0%

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

      1. associate-/r* 79.0%

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

   7. Simplified 79.0%

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

   if 6.1999999999999996e96 < 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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. times-frac 99.9%

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

      9. *-lft-identity 99.9%

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

      10. neg-mul-1 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-out 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around 0 89.6%

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

      1. associate-/r* 89.6%

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

      2. div-inv 89.6%

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

   10. Applied egg-rr 89.6%

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

       1. un-div-inv 89.6%

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

   12. Applied egg-rr 89.6%

       \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t}}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-82}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+96}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-123}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-84}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+97}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y}\\ \end{array} \end{array} \]

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 -3.9e-123)
   (- 1.0 (/ x (* t z)))
   (if (<= t 1.62e-84)
     (+ 1.0 (/ (/ x z) y))
     (if (<= t 2.05e-7)
       (+ 1.0 (/ x (* y t)))
       (if (<= t 4.6e+97) (- 1.0 (/ (/ x t) z)) (+ 1.0 (/ (/ x t) y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.9e-123) {
		tmp = 1.0 - (x / (t * z));
	} else if (t <= 1.62e-84) {
		tmp = 1.0 + ((x / z) / y);
	} else if (t <= 2.05e-7) {
		tmp = 1.0 + (x / (y * t));
	} else if (t <= 4.6e+97) {
		tmp = 1.0 - ((x / t) / z);
	} else {
		tmp = 1.0 + ((x / t) / 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 <= (-3.9d-123)) then
        tmp = 1.0d0 - (x / (t * z))
    else if (t <= 1.62d-84) then
        tmp = 1.0d0 + ((x / z) / y)
    else if (t <= 2.05d-7) then
        tmp = 1.0d0 + (x / (y * t))
    else if (t <= 4.6d+97) then
        tmp = 1.0d0 - ((x / t) / z)
    else
        tmp = 1.0d0 + ((x / t) / 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 <= -3.9e-123) {
		tmp = 1.0 - (x / (t * z));
	} else if (t <= 1.62e-84) {
		tmp = 1.0 + ((x / z) / y);
	} else if (t <= 2.05e-7) {
		tmp = 1.0 + (x / (y * t));
	} else if (t <= 4.6e+97) {
		tmp = 1.0 - ((x / t) / z);
	} else {
		tmp = 1.0 + ((x / t) / y);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -3.9e-123:
		tmp = 1.0 - (x / (t * z))
	elif t <= 1.62e-84:
		tmp = 1.0 + ((x / z) / y)
	elif t <= 2.05e-7:
		tmp = 1.0 + (x / (y * t))
	elif t <= 4.6e+97:
		tmp = 1.0 - ((x / t) / z)
	else:
		tmp = 1.0 + ((x / t) / y)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.9e-123)
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	elseif (t <= 1.62e-84)
		tmp = Float64(1.0 + Float64(Float64(x / z) / y));
	elseif (t <= 2.05e-7)
		tmp = Float64(1.0 + Float64(x / Float64(y * t)));
	elseif (t <= 4.6e+97)
		tmp = Float64(1.0 - Float64(Float64(x / t) / z));
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / 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 <= -3.9e-123)
		tmp = 1.0 - (x / (t * z));
	elseif (t <= 1.62e-84)
		tmp = 1.0 + ((x / z) / y);
	elseif (t <= 2.05e-7)
		tmp = 1.0 + (x / (y * t));
	elseif (t <= 4.6e+97)
		tmp = 1.0 - ((x / t) / z);
	else
		tmp = 1.0 + ((x / t) / 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, -3.9e-123], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.62e-84], N[(1.0 + N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05e-7], N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e+97], N[(1.0 - N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -3.89999999999999976e-123

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0 79.3%

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

   if -3.89999999999999976e-123 < t < 1.62000000000000008e-84

   1. Initial program 91.0%

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

   3. Taylor expanded in t around 0 83.6%

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

   4. Taylor expanded in y around 0 63.3%

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

      1. associate-*r/ 63.3%

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

      2. *-commutative 63.3%

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

      3. associate-/r* 65.5%

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

      4. associate-*r/ 65.5%

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

      5. mul-1-neg 65.5%

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

   6. Simplified 65.5%

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

   if 1.62000000000000008e-84 < t < 2.05e-7

   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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. times-frac 99.9%

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

      9. *-lft-identity 99.9%

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

      10. neg-mul-1 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-out 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 54.4%

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

   if 2.05e-7 < t < 4.60000000000000011e97

   1. Initial program 99.9%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r* 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 79.0%

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

      1. associate-/r* 79.0%

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

   7. Simplified 79.0%

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

   if 4.60000000000000011e97 < 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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. times-frac 99.9%

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

      9. *-lft-identity 99.9%

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

      10. neg-mul-1 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-out 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around 0 89.6%

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

      1. associate-/r* 89.6%

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

      2. div-inv 89.6%

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

   10. Applied egg-rr 89.6%

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

       1. un-div-inv 89.6%

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

   12. Applied egg-rr 89.6%

       \[\leadsto 1 + \color{blue}{\frac{\frac{x}{t}}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-123}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-84}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+97}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-80} \lor \neg \left(y \leq 4.2 \cdot 10^{-35}\right):\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \end{array} \end{array} \]

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 -4.6e-80) (not (<= y 4.2e-35)))
   (+ 1.0 (/ (/ x y) (- z 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 <= -4.6e-80) || !(y <= 4.2e-35)) {
		tmp = 1.0 + ((x / y) / (z - 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 <= (-4.6d-80)) .or. (.not. (y <= 4.2d-35))) then
        tmp = 1.0d0 + ((x / y) / (z - 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 <= -4.6e-80) || !(y <= 4.2e-35)) {
		tmp = 1.0 + ((x / y) / (z - 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 <= -4.6e-80) or not (y <= 4.2e-35):
		tmp = 1.0 + ((x / y) / (z - 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 <= -4.6e-80) || !(y <= 4.2e-35))
		tmp = Float64(1.0 + Float64(Float64(x / y) / Float64(z - 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 <= -4.6e-80) || ~((y <= 4.2e-35)))
		tmp = 1.0 + ((x / y) / (z - 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, -4.6e-80], N[Not[LessEqual[y, 4.2e-35]], $MachinePrecision]], N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5999999999999996e-80 or 4.2e-35 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/r* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

      8. times-frac 100.0%

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

      9. *-lft-identity 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. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-out 100.0%

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

      14. remove-double-neg 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 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around inf 95.3%

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

   10. Taylor expanded in x around 0 95.3%

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

       1. associate-/r* 94.8%

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

   12. Simplified 94.8%

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

   if -4.5999999999999996e-80 < y < 4.2e-35

   1. Initial program 91.9%

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

      1. sub-neg 91.9%

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

      2. neg-mul-1 91.9%

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

      3. *-commutative 91.9%

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

      4. *-commutative 91.9%

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

      5. associate-/r* 96.0%

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

      6. associate-*r/ 96.0%

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

      7. metadata-eval 96.0%

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

      8. times-frac 96.0%

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

      9. *-lft-identity 96.0%

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

      10. neg-mul-1 96.0%

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

      11. sub-neg 96.0%

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

      12. +-commutative 96.0%

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

      13. distribute-neg-out 96.0%

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

      14. remove-double-neg 96.0%

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

      15. sub-neg 96.0%

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

   3. Simplified 96.0%

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

      1. clear-num 96.0%

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

      2. inv-pow 96.0%

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

      3. div-inv 96.0%

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

      4. clear-num 97.3%

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

   6. Applied egg-rr 97.3%

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

      1. unpow-1 97.3%

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

      2. associate-/r* 97.3%

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

   8. Simplified 97.3%

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

   9. Taylor expanded in y around 0 75.5%

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

       1. mul-1-neg 75.5%

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

       2. associate-/l/ 73.9%

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

       3. distribute-neg-frac 73.9%

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

       4. distribute-neg-frac 73.9%

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

   11. Simplified 73.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-80} \lor \neg \left(y \leq 4.2 \cdot 10^{-35}\right):\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-80} \lor \neg \left(y \leq 1.65 \cdot 10^{-9}\right):\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \end{array} \end{array} \]

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 -4.8e-80) (not (<= y 1.65e-9)))
   (+ 1.0 (/ (/ x y) (- z 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 <= -4.8e-80) || !(y <= 1.65e-9)) {
		tmp = 1.0 + ((x / y) / (z - 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 <= (-4.8d-80)) .or. (.not. (y <= 1.65d-9))) then
        tmp = 1.0d0 + ((x / y) / (z - 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 <= -4.8e-80) || !(y <= 1.65e-9)) {
		tmp = 1.0 + ((x / y) / (z - 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 <= -4.8e-80) or not (y <= 1.65e-9):
		tmp = 1.0 + ((x / y) / (z - 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 <= -4.8e-80) || !(y <= 1.65e-9))
		tmp = Float64(1.0 + Float64(Float64(x / y) / Float64(z - 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 <= -4.8e-80) || ~((y <= 1.65e-9)))
		tmp = 1.0 + ((x / y) / (z - 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, -4.8e-80], N[Not[LessEqual[y, 1.65e-9]], $MachinePrecision]], N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(z - y), $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 < -4.7999999999999998e-80 or 1.65000000000000009e-9 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/r* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

      8. times-frac 100.0%

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

      9. *-lft-identity 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. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-out 100.0%

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

      14. remove-double-neg 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 100.0%

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

      2. inv-pow 100.0%

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

      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* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around inf 96.4%

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

   10. Taylor expanded in x around 0 96.4%

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

       1. associate-/r* 95.8%

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

   12. Simplified 95.8%

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

   if -4.7999999999999998e-80 < y < 1.65000000000000009e-9

   1. Initial program 92.5%

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

      1. sub-neg 92.5%

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

      2. neg-mul-1 92.5%

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

      3. *-commutative 92.5%

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

      4. *-commutative 92.5%

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

      5. associate-/r* 96.3%

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

      6. associate-*r/ 96.3%

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

      7. metadata-eval 96.3%

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

      8. times-frac 96.3%

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

      9. *-lft-identity 96.3%

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

      10. neg-mul-1 96.3%

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

      11. sub-neg 96.3%

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

      12. +-commutative 96.3%

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

      13. distribute-neg-out 96.3%

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

      14. remove-double-neg 96.3%

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

      15. sub-neg 96.3%

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

   3. Simplified 96.3%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{z - y}}{y - t}} \]
   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 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-80} \lor \neg \left(y \leq 1.65 \cdot 10^{-9}\right):\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-59}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.55e-59)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= z 4.4e-78) (- 1.0 (/ x (* y (- y t)))) (- 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 (z <= -1.55e-59) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (z <= 4.4e-78) {
		tmp = 1.0 - (x / (y * (y - t)));
	} 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 (z <= (-1.55d-59)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (z <= 4.4d-78) then
        tmp = 1.0d0 - (x / (y * (y - t)))
    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 (z <= -1.55e-59) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (z <= 4.4e-78) {
		tmp = 1.0 - (x / (y * (y - t)));
	} 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 z <= -1.55e-59:
		tmp = 1.0 + ((x / z) / (y - t))
	elif z <= 4.4e-78:
		tmp = 1.0 - (x / (y * (y - t)))
	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 (z <= -1.55e-59)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (z <= 4.4e-78)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - t))));
	else
		tmp = Float64(1.0 - Float64(Float64(x / 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 (z <= -1.55e-59)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (z <= 4.4e-78)
		tmp = 1.0 - (x / (y * (y - t)));
	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[LessEqual[z, -1.55e-59], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e-78], N[(1.0 - N[(x / N[(y * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55e-59

   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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 97.6%

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

      6. associate-*r/ 97.6%

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

      7. metadata-eval 97.6%

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

      8. times-frac 97.6%

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

      9. *-lft-identity 97.6%

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

      10. neg-mul-1 97.6%

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

      11. sub-neg 97.6%

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

      12. +-commutative 97.6%

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

      13. distribute-neg-out 97.6%

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

      14. remove-double-neg 97.6%

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

      15. sub-neg 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in z around inf 96.4%

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

      1. associate-/r* 94.0%

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

   7. Simplified 94.0%

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

   if -1.55e-59 < z < 4.3999999999999998e-78

   1. Initial program 91.5%

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

   3. Taylor expanded in z around 0 84.0%

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

   if 4.3999999999999998e-78 < z

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r* 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 85.0%

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

      1. associate-/r* 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-59}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-57}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-82}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.8e-57)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= z 7.8e-82) (- 1.0 (/ (/ x y) (- y t))) (- 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 (z <= -3.8e-57) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (z <= 7.8e-82) {
		tmp = 1.0 - ((x / y) / (y - t));
	} 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 (z <= (-3.8d-57)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (z <= 7.8d-82) then
        tmp = 1.0d0 - ((x / y) / (y - t))
    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 (z <= -3.8e-57) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (z <= 7.8e-82) {
		tmp = 1.0 - ((x / y) / (y - t));
	} 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 z <= -3.8e-57:
		tmp = 1.0 + ((x / z) / (y - t))
	elif z <= 7.8e-82:
		tmp = 1.0 - ((x / y) / (y - t))
	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 (z <= -3.8e-57)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (z <= 7.8e-82)
		tmp = Float64(1.0 - Float64(Float64(x / y) / Float64(y - t)));
	else
		tmp = Float64(1.0 - Float64(Float64(x / 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 (z <= -3.8e-57)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (z <= 7.8e-82)
		tmp = 1.0 - ((x / y) / (y - t));
	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[LessEqual[z, -3.8e-57], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e-82], N[(1.0 - N[(N[(x / y), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.7999999999999997e-57

   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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 97.6%

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

      6. associate-*r/ 97.6%

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

      7. metadata-eval 97.6%

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

      8. times-frac 97.6%

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

      9. *-lft-identity 97.6%

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

      10. neg-mul-1 97.6%

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

      11. sub-neg 97.6%

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

      12. +-commutative 97.6%

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

      13. distribute-neg-out 97.6%

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

      14. remove-double-neg 97.6%

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

      15. sub-neg 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in z around inf 96.4%

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

      1. associate-/r* 94.0%

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

   7. Simplified 94.0%

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

   if -3.7999999999999997e-57 < z < 7.79999999999999947e-82

   1. Initial program 91.4%

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

      1. clear-num 91.3%

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

      2. associate-/r/ 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-/r* 90.5%

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

   4. Applied egg-rr 90.5%

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

   5. Taylor expanded in z around 0 83.8%

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

      1. associate-/r* 87.0%

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

   7. Simplified 87.0%

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

   if 7.79999999999999947e-82 < z

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r* 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 84.0%

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

      1. associate-/r* 82.9%

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

   7. Simplified 82.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-57}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-82}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-274}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-118}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

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 -4.1e-274)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= t 1.9e-118)
     (+ 1.0 (/ (/ x y) (- z y)))
     (- 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 <= -4.1e-274) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 1.9e-118) {
		tmp = 1.0 + ((x / y) / (z - y));
	} 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 <= (-4.1d-274)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (t <= 1.9d-118) then
        tmp = 1.0d0 + ((x / y) / (z - y))
    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 <= -4.1e-274) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 1.9e-118) {
		tmp = 1.0 + ((x / y) / (z - y));
	} 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 <= -4.1e-274:
		tmp = 1.0 + ((x / z) / (y - t))
	elif t <= 1.9e-118:
		tmp = 1.0 + ((x / y) / (z - y))
	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 <= -4.1e-274)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (t <= 1.9e-118)
		tmp = Float64(1.0 + Float64(Float64(x / y) / Float64(z - y)));
	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 <= -4.1e-274)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (t <= 1.9e-118)
		tmp = 1.0 + ((x / y) / (z - y));
	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, -4.1e-274], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-118], N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(z - y), $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 < -4.09999999999999987e-274

   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. neg-mul-1 98.3%

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

      3. *-commutative 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-/r* 97.7%

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

      6. associate-*r/ 97.7%

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

      7. metadata-eval 97.7%

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

      8. times-frac 97.7%

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

      9. *-lft-identity 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. sub-neg 97.7%

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

      12. +-commutative 97.7%

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

      13. distribute-neg-out 97.7%

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

      14. remove-double-neg 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 -4.09999999999999987e-274 < t < 1.9e-118

   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. neg-mul-1 86.1%

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

      3. *-commutative 86.1%

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

      4. *-commutative 86.1%

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

      5. associate-/r* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

      8. times-frac 100.0%

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

      9. *-lft-identity 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. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-out 100.0%

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

      14. remove-double-neg 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%

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

   10. Taylor expanded in x around 0 83.9%

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

       1. associate-/r* 94.3%

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

   12. Simplified 94.3%

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

   if 1.9e-118 < 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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 98.9%

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

      6. associate-*r/ 98.9%

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

      7. metadata-eval 98.9%

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

      8. times-frac 98.9%

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

      9. *-lft-identity 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. sub-neg 98.9%

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

      12. +-commutative 98.9%

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

      13. distribute-neg-out 98.9%

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

      14. remove-double-neg 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 t around inf 94.4%

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

      1. associate-*r/ 94.4%

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

      2. neg-mul-1 94.4%

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

      3. associate-/r* 94.4%

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

   7. Simplified 94.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-274}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-118}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-82}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \end{array} \]

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 9e-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 <= 9e-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 <= 9d-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 <= 9e-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 <= 9e-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 <= 9e-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 <= 9e-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, 9e-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 < 8.9999999999999997e-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. neg-mul-1 95.5%

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

      3. *-commutative 95.5%

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

      4. *-commutative 95.5%

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

      5. associate-/r* 98.3%

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

      6. associate-*r/ 98.3%

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

      7. metadata-eval 98.3%

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

      8. times-frac 98.3%

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

      9. *-lft-identity 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. sub-neg 98.3%

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

      12. +-commutative 98.3%

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

      13. distribute-neg-out 98.3%

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

      14. remove-double-neg 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 8.9999999999999997e-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. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 98.7%

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

      6. associate-*r/ 98.7%

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

      7. metadata-eval 98.7%

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

      8. times-frac 98.7%

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

      9. *-lft-identity 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. sub-neg 98.7%

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

      12. +-commutative 98.7%

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

      13. distribute-neg-out 98.7%

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

      14. remove-double-neg 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. Taylor expanded in t around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

      3. associate-/r* 98.7%

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

   7. Simplified 98.7%

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

   8. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-82}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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. neg-mul-1 96.9%

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

   3. *-commutative 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-/r* 98.5%

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

   6. associate-*r/ 98.5%

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

   7. metadata-eval 98.5%

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

   8. times-frac 98.5%

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

   9. *-lft-identity 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. sub-neg 98.5%

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

   12. +-commutative 98.5%

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

   13. distribute-neg-out 98.5%

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

   14. remove-double-neg 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 12: 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. neg-mul-1 96.9%

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

   3. *-commutative 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-/r* 98.5%

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

   6. associate-*r/ 98.5%

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

   7. metadata-eval 98.5%

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

   8. times-frac 98.5%

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

   9. *-lft-identity 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. sub-neg 98.5%

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

   12. +-commutative 98.5%

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

   13. distribute-neg-out 98.5%

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

   14. remove-double-neg 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. Taylor expanded in t around inf 80.2%

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

   1. associate-*r/ 80.2%

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

   2. neg-mul-1 80.2%

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

   3. associate-/r* 80.2%

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

7. Simplified 80.2%

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

8. Taylor expanded in z around 0 57.6%

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

9. Final simplification 57.6%

   \[\leadsto 1 + \frac{x}{y \cdot t} \]
10. 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)))))