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

Percentage Accurate: 99.1% → 99.1%

Time: 8.0s

Alternatives: 10

Speedup: 1.0×

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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 - (x / ((y - z) * (y - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.65 \cdot 10^{-12}:\\ \;\;\;\;1 + \frac{\frac{x}{z - y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

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 2.65e-12)
   (+ 1.0 (/ (/ x (- z y)) (- y t)))
   (- 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 <= 2.65e-12) {
		tmp = 1.0 + ((x / (z - y)) / (y - t));
	} 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 <= 2.65d-12) then
        tmp = 1.0d0 + ((x / (z - y)) / (y - t))
    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 <= 2.65e-12) {
		tmp = 1.0 + ((x / (z - y)) / (y - t));
	} 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 <= 2.65e-12:
		tmp = 1.0 + ((x / (z - y)) / (y - t))
	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 <= 2.65e-12)
		tmp = Float64(1.0 + Float64(Float64(x / Float64(z - y)) / Float64(y - t)));
	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 <= 2.65e-12)
		tmp = 1.0 + ((x / (z - y)) / (y - t));
	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, 2.65e-12], N[(1.0 + N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.64999999999999982e-12

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. *-lft-identity 99.3%

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

      4. associate-/r* 98.9%

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

      5. associate-*r/ 98.9%

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

      6. metadata-eval 98.9%

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

      7. times-frac 98.9%

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

      8. neg-mul-1 98.9%

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

      9. remove-double-neg 98.9%

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

      10. neg-mul-1 98.9%

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

      11. sub-neg 98.9%

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

      12. distribute-neg-out 98.9%

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

      13. remove-double-neg 98.9%

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

      14. +-commutative 98.9%

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

      15. sub-neg 98.9%

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

   3. Simplified 98.9%

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

   if 2.64999999999999982e-12 < t

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      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* 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in t around inf 100.0%

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

      1. associate-/r* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r* 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. associate-*r/ 99.8%

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

      6. associate-*l/ 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. associate-/r* 100.0%

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

      11. neg-mul-1 100.0%

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

      12. *-lft-identity 100.0%

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

      13. associate-*l/ 99.8%

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

      14. associate-*r/ 100.0%

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

      15. *-rgt-identity 100.0%

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

      16. associate-/r/ 96.0%

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

      17. associate-/l* 99.9%

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

      18. neg-mul-1 99.9%

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

      19. associate-*l/ 99.9%

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

      20. *-lft-identity 99.9%

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

      21. associate-/r* 99.9%

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

   9. Simplified 99.9%

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

       1. frac-2neg 99.9%

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

       2. div-inv 99.9%

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

       3. distribute-neg-frac 99.9%

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

       4. frac-2neg 99.9%

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

       5. sub-neg 99.9%

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

       6. distribute-neg-in 99.9%

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

       7. add-sqr-sqrt 51.1%

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

       8. sqrt-unprod 95.2%

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

       9. sqr-neg 95.2%

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

       10. sqrt-unprod 42.7%

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

       11. add-sqr-sqrt 90.3%

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

       12. add-sqr-sqrt 47.6%

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

       13. sqrt-unprod 95.2%

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

       14. sqr-neg 95.2%

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

       15. sqrt-unprod 48.7%

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

       16. add-sqr-sqrt 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. associate-*r/ 99.9%

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

       2. *-rgt-identity 99.9%

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

       3. +-commutative 99.9%

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

       4. unsub-neg 99.9%

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

   13. Simplified 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.65 \cdot 10^{-12}:\\ \;\;\;\;1 + \frac{\frac{x}{z - y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]

Alternative 2: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-91} \lor \neg \left(z \leq 6.5 \cdot 10^{-100}\right):\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \end{array} \]

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 (or (<= z -6.8e-91) (not (<= z 6.5e-100)))
   (+ 1.0 (/ x (* z (- y t))))
   (+ 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 ((z <= -6.8e-91) || !(z <= 6.5e-100)) {
		tmp = 1.0 + (x / (z * (y - t)));
	} 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 ((z <= (-6.8d-91)) .or. (.not. (z <= 6.5d-100))) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    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 ((z <= -6.8e-91) || !(z <= 6.5e-100)) {
		tmp = 1.0 + (x / (z * (y - t)));
	} 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 (z <= -6.8e-91) or not (z <= 6.5e-100):
		tmp = 1.0 + (x / (z * (y - t)))
	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 ((z <= -6.8e-91) || !(z <= 6.5e-100))
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	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 ((z <= -6.8e-91) || ~((z <= 6.5e-100)))
		tmp = 1.0 + (x / (z * (y - t)));
	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[Or[LessEqual[z, -6.8e-91], N[Not[LessEqual[z, 6.5e-100]], $MachinePrecision]], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.80000000000000053e-91 or 6.50000000000000013e-100 < z

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 99.4%

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

      5. associate-*r/ 99.4%

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

      6. metadata-eval 99.4%

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

      7. times-frac 99.4%

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

      8. neg-mul-1 99.4%

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

      9. remove-double-neg 99.4%

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

      10. neg-mul-1 99.4%

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

      11. sub-neg 99.4%

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

      12. distribute-neg-out 99.4%

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

      13. remove-double-neg 99.4%

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

      14. +-commutative 99.4%

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

      15. sub-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in z around inf 92.8%

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

      1. *-commutative 92.8%

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

   6. Simplified 92.8%

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

   if -6.80000000000000053e-91 < z < 6.50000000000000013e-100

   1. Initial program 98.9%

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

      1. sub-neg 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. *-lft-identity 98.9%

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

      4. associate-/r* 95.1%

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

      5. associate-*r/ 95.1%

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

      6. metadata-eval 95.1%

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

      7. times-frac 95.1%

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

      8. neg-mul-1 95.1%

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

      9. remove-double-neg 95.1%

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

      10. neg-mul-1 95.1%

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

      11. sub-neg 95.1%

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

      12. distribute-neg-out 95.1%

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

      13. remove-double-neg 95.1%

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

      14. +-commutative 95.1%

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

      15. sub-neg 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in z around 0 91.7%

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

      1. mul-1-neg 91.7%

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

      2. distribute-frac-neg 91.7%

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

      3. associate-/r* 87.9%

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

   6. Simplified 87.9%

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

   7. Taylor expanded in y around 0 75.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-91} \lor \neg \left(z \leq 6.5 \cdot 10^{-100}\right):\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]

Alternative 3: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-91} \lor \neg \left(z \leq 3 \cdot 10^{-95}\right):\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot \frac{-1}{y \cdot t}\\ \end{array} \end{array} \]

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 (or (<= z -2.35e-91) (not (<= z 3e-95)))
   (+ 1.0 (/ x (* z (- y t))))
   (- 1.0 (* x (/ -1.0 (* y t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.35e-91) || !(z <= 3e-95)) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else {
		tmp = 1.0 - (x * (-1.0 / (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 ((z <= (-2.35d-91)) .or. (.not. (z <= 3d-95))) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    else
        tmp = 1.0d0 - (x * ((-1.0d0) / (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 ((z <= -2.35e-91) || !(z <= 3e-95)) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else {
		tmp = 1.0 - (x * (-1.0 / (y * t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -2.35e-91) or not (z <= 3e-95):
		tmp = 1.0 + (x / (z * (y - t)))
	else:
		tmp = 1.0 - (x * (-1.0 / (y * t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.35e-91) || !(z <= 3e-95))
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	else
		tmp = Float64(1.0 - Float64(x * Float64(-1.0 / 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 ((z <= -2.35e-91) || ~((z <= 3e-95)))
		tmp = 1.0 + (x / (z * (y - t)));
	else
		tmp = 1.0 - (x * (-1.0 / (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[z, -2.35e-91], N[Not[LessEqual[z, 3e-95]], $MachinePrecision]], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x * N[(-1.0 / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.35000000000000003e-91 or 3e-95 < z

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 99.4%

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

      5. associate-*r/ 99.4%

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

      6. metadata-eval 99.4%

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

      7. times-frac 99.4%

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

      8. neg-mul-1 99.4%

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

      9. remove-double-neg 99.4%

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

      10. neg-mul-1 99.4%

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

      11. sub-neg 99.4%

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

      12. distribute-neg-out 99.4%

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

      13. remove-double-neg 99.4%

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

      14. +-commutative 99.4%

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

      15. sub-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in z around inf 92.7%

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

      1. *-commutative 92.7%

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

   6. Simplified 92.7%

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

   if -2.35000000000000003e-91 < z < 3e-95

   1. Initial program 98.9%

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

      1. clear-num 98.9%

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

      2. associate-/r/ 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-/r* 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in t around inf 81.0%

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

      1. associate-/r* 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in y around inf 75.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-91} \lor \neg \left(z \leq 3 \cdot 10^{-95}\right):\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot \frac{-1}{y \cdot t}\\ \end{array} \]

Alternative 4: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-38} \lor \neg \left(z \leq 1.36 \cdot 10^{-95}\right):\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.5e-38) (not (<= z 1.36e-95)))
   (+ 1.0 (/ x (* z (- y t))))
   (- 1.0 (/ x (* y (- y t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.5e-38) || !(z <= 1.36e-95)) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else {
		tmp = 1.0 - (x / (y * (y - t)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-6.5d-38)) .or. (.not. (z <= 1.36d-95))) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    else
        tmp = 1.0d0 - (x / (y * (y - t)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.5e-38) || !(z <= 1.36e-95)) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else {
		tmp = 1.0 - (x / (y * (y - t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -6.5e-38) or not (z <= 1.36e-95):
		tmp = 1.0 + (x / (z * (y - t)))
	else:
		tmp = 1.0 - (x / (y * (y - t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.5e-38) || !(z <= 1.36e-95))
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	else
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.5e-38) || ~((z <= 1.36e-95)))
		tmp = 1.0 + (x / (z * (y - t)));
	else
		tmp = 1.0 - (x / (y * (y - t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.5e-38], N[Not[LessEqual[z, 1.36e-95]], $MachinePrecision]], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(y * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.49999999999999949e-38 or 1.36e-95 < z

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 99.3%

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

      5. associate-*r/ 99.3%

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

      6. metadata-eval 99.3%

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

      7. times-frac 99.3%

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

      8. neg-mul-1 99.3%

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

      9. remove-double-neg 99.3%

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

      10. neg-mul-1 99.3%

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

      11. sub-neg 99.3%

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

      12. distribute-neg-out 99.3%

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

      13. remove-double-neg 99.3%

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

      14. +-commutative 99.3%

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

      15. sub-neg 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around inf 95.5%

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

      1. *-commutative 95.5%

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

   6. Simplified 95.5%

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

   if -6.49999999999999949e-38 < z < 1.36e-95

   1. Initial program 99.0%

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

   2. Taylor expanded in z around 0 91.7%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot \left(y - t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-38} \lor \neg \left(z \leq 1.36 \cdot 10^{-95}\right):\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \end{array} \]

Alternative 5: 92.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-40}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-132}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.1e-40)
   (+ 1.0 (/ x (* z (- y t))))
   (if (<= z 8.5e-132)
     (- 1.0 (/ x (* y (- y t))))
     (- 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 (z <= -4.1e-40) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (z <= 8.5e-132) {
		tmp = 1.0 - (x / (y * (y - t)));
	} 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 (z <= (-4.1d-40)) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    else if (z <= 8.5d-132) then
        tmp = 1.0d0 - (x / (y * (y - t)))
    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 (z <= -4.1e-40) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (z <= 8.5e-132) {
		tmp = 1.0 - (x / (y * (y - t)));
	} 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 z <= -4.1e-40:
		tmp = 1.0 + (x / (z * (y - t)))
	elif z <= 8.5e-132:
		tmp = 1.0 - (x / (y * (y - t)))
	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 (z <= -4.1e-40)
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	elseif (z <= 8.5e-132)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - t))));
	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 (z <= -4.1e-40)
		tmp = 1.0 + (x / (z * (y - t)));
	elseif (z <= 8.5e-132)
		tmp = 1.0 - (x / (y * (y - t)));
	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[z, -4.1e-40], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e-132], N[(1.0 - N[(x / N[(y * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.09999999999999963e-40

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. neg-mul-1 99.9%

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

      9. remove-double-neg 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. distribute-neg-out 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 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. Taylor expanded in z around inf 97.9%

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

      1. *-commutative 97.9%

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

   6. Simplified 97.9%

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

   if -4.09999999999999963e-40 < z < 8.49999999999999988e-132

   1. Initial program 98.9%

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

   2. Taylor expanded in z around 0 93.7%

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

   if 8.49999999999999988e-132 < z

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r* 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in t around inf 73.5%

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

      1. associate-/r* 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in t around 0 73.5%

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

      1. associate-*r/ 73.5%

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

      2. associate-*l/ 73.5%

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

      3. associate-/r* 73.5%

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

      4. *-rgt-identity 73.5%

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

      5. associate-*r/ 73.5%

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

      6. associate-*l/ 73.5%

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

      7. associate-*r/ 73.5%

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

      8. metadata-eval 73.5%

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

      9. metadata-eval 73.5%

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

      10. associate-/r* 73.5%

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

      11. neg-mul-1 73.5%

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

      12. *-lft-identity 73.5%

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

      13. associate-*l/ 73.5%

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

      14. associate-*r/ 73.5%

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

      15. *-rgt-identity 73.5%

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

      16. associate-/r/ 73.0%

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

      17. associate-/l* 73.3%

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

      18. neg-mul-1 73.3%

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

      19. associate-*l/ 73.4%

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

      20. *-lft-identity 73.4%

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

      21. associate-/r* 73.4%

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

   9. Simplified 73.4%

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

       1. frac-2neg 73.4%

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

       2. div-inv 73.3%

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

       3. distribute-neg-frac 73.3%

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

       4. frac-2neg 73.3%

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

       5. sub-neg 73.3%

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

       6. distribute-neg-in 73.3%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 61.0%

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

       9. sqr-neg 61.0%

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

       10. sqrt-unprod 61.1%

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

       11. add-sqr-sqrt 61.1%

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

       12. add-sqr-sqrt 0.0%

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

       13. sqrt-unprod 73.2%

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

       14. sqr-neg 73.2%

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

       15. sqrt-unprod 73.3%

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

       16. add-sqr-sqrt 73.3%

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

   11. Applied egg-rr 73.3%

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

       1. associate-*r/ 73.4%

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

       2. *-rgt-identity 73.4%

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

       3. +-commutative 73.4%

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

       4. unsub-neg 73.4%

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

   13. Simplified 73.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-40}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-132}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]

Alternative 6: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -0.017:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-95}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \end{array} \]

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 (<= z -0.017)
   (+ 1.0 (/ (/ x z) y))
   (if (<= z 1.55e-95) (+ 1.0 (/ x (* y t))) (- 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 (z <= -0.017) {
		tmp = 1.0 + ((x / z) / y);
	} else if (z <= 1.55e-95) {
		tmp = 1.0 + (x / (y * t));
	} 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 (z <= (-0.017d0)) then
        tmp = 1.0d0 + ((x / z) / y)
    else if (z <= 1.55d-95) then
        tmp = 1.0d0 + (x / (y * t))
    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 (z <= -0.017) {
		tmp = 1.0 + ((x / z) / y);
	} else if (z <= 1.55e-95) {
		tmp = 1.0 + (x / (y * t));
	} 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 z <= -0.017:
		tmp = 1.0 + ((x / z) / y)
	elif z <= 1.55e-95:
		tmp = 1.0 + (x / (y * t))
	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 (z <= -0.017)
		tmp = Float64(1.0 + Float64(Float64(x / z) / y));
	elseif (z <= 1.55e-95)
		tmp = Float64(1.0 + Float64(x / Float64(y * t)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(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 (z <= -0.017)
		tmp = 1.0 + ((x / z) / y);
	elseif (z <= 1.55e-95)
		tmp = 1.0 + (x / (y * t));
	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[LessEqual[z, -0.017], N[(1.0 + N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55e-95], N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.017000000000000001

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. neg-mul-1 99.9%

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

      9. remove-double-neg 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. distribute-neg-out 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 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. Taylor expanded in z around inf 98.4%

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

      1. associate-/r* 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in y around inf 76.6%

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

      1. *-commutative 76.6%

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

   9. Simplified 76.6%

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

   10. Taylor expanded in x around 0 76.6%

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

       1. associate-/l/ 76.6%

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

   12. Simplified 76.6%

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

   if -0.017000000000000001 < z < 1.54999999999999996e-95

   1. Initial program 99.0%

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

      1. sub-neg 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. *-lft-identity 99.0%

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

      4. associate-/r* 95.8%

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

      5. associate-*r/ 95.8%

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

      6. metadata-eval 95.8%

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

      7. times-frac 95.8%

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

      8. neg-mul-1 95.8%

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

      9. remove-double-neg 95.8%

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

      10. neg-mul-1 95.8%

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

      11. sub-neg 95.8%

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

      12. distribute-neg-out 95.8%

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

      13. remove-double-neg 95.8%

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

      14. +-commutative 95.8%

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

      15. sub-neg 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in z around 0 90.4%

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

      1. mul-1-neg 90.4%

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

      2. distribute-frac-neg 90.4%

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

      3. associate-/r* 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in y around 0 74.0%

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

   if 1.54999999999999996e-95 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 68.8%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.017:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-95}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \]

Alternative 7: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 - (x / ((y - z) * (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(Float64(y - z) * 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 - z) * (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[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

2. Final simplification 99.5%

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

Alternative 8: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -0.37:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \end{array} \]

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 (<= z -0.37) (+ 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 (z <= -0.37) {
		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 (z <= (-0.37d0)) 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 (z <= -0.37) {
		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 z <= -0.37:
		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 (z <= -0.37)
		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 (z <= -0.37)
		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[z, -0.37], 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 z < -0.37

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. neg-mul-1 99.9%

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

      9. remove-double-neg 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. distribute-neg-out 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 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. Taylor expanded in z around inf 98.4%

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

      1. associate-/r* 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in y around inf 77.7%

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

      1. *-commutative 77.7%

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

   9. Simplified 77.7%

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

   if -0.37 < z

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. *-lft-identity 99.4%

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

      4. associate-/r* 97.0%

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

      5. associate-*r/ 97.0%

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

      6. metadata-eval 97.0%

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

      7. times-frac 97.0%

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

      8. neg-mul-1 97.0%

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

      9. remove-double-neg 97.0%

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

      10. neg-mul-1 97.0%

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

      11. sub-neg 97.0%

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

      12. distribute-neg-out 97.0%

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

      13. remove-double-neg 97.0%

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

      14. +-commutative 97.0%

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

      15. sub-neg 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in z around 0 77.8%

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

      1. mul-1-neg 77.8%

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

      2. distribute-frac-neg 77.8%

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

      3. associate-/r* 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in y around 0 62.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.37:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]

Alternative 9: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -0.3:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \end{array} \]

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 (<= z -0.3) (+ 1.0 (/ (/ x z) y)) (+ 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 (z <= -0.3) {
		tmp = 1.0 + ((x / z) / y);
	} 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 (z <= (-0.3d0)) then
        tmp = 1.0d0 + ((x / z) / y)
    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 (z <= -0.3) {
		tmp = 1.0 + ((x / z) / y);
	} 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 z <= -0.3:
		tmp = 1.0 + ((x / z) / y)
	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 (z <= -0.3)
		tmp = Float64(1.0 + Float64(Float64(x / z) / y));
	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 (z <= -0.3)
		tmp = 1.0 + ((x / z) / y);
	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[z, -0.3], N[(1.0 + N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.299999999999999989

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. neg-mul-1 99.9%

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

      9. remove-double-neg 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. distribute-neg-out 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 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. Taylor expanded in z around inf 98.4%

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

      1. associate-/r* 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in y around inf 77.7%

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

      1. *-commutative 77.7%

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

   9. Simplified 77.7%

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

   10. Taylor expanded in x around 0 77.7%

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

       1. associate-/l/ 77.7%

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

   12. Simplified 77.7%

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

   if -0.299999999999999989 < z

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. *-lft-identity 99.4%

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

      4. associate-/r* 97.0%

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

      5. associate-*r/ 97.0%

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

      6. metadata-eval 97.0%

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

      7. times-frac 97.0%

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

      8. neg-mul-1 97.0%

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

      9. remove-double-neg 97.0%

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

      10. neg-mul-1 97.0%

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

      11. sub-neg 97.0%

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

      12. distribute-neg-out 97.0%

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

      13. remove-double-neg 97.0%

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

      14. +-commutative 97.0%

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

      15. sub-neg 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in z around 0 77.8%

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

      1. mul-1-neg 77.8%

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

      2. distribute-frac-neg 77.8%

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

      3. associate-/r* 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in y around 0 62.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.3:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]

Alternative 10: 56.5% 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 99.5%

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

   1. sub-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. *-lft-identity 99.5%

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

   4. associate-/r* 97.7%

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

   5. associate-*r/ 97.7%

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

   6. metadata-eval 97.7%

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

   7. times-frac 97.7%

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

   8. neg-mul-1 97.7%

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

   9. remove-double-neg 97.7%

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

   10. neg-mul-1 97.7%

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

   11. sub-neg 97.7%

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

   12. distribute-neg-out 97.7%

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

   13. remove-double-neg 97.7%

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

   14. +-commutative 97.7%

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

   15. sub-neg 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in z around 0 75.3%

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

   1. mul-1-neg 75.3%

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

   2. distribute-frac-neg 75.3%

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

   3. associate-/r* 73.8%

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

6. Simplified 73.8%

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

7. Taylor expanded in y around 0 60.9%

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

8. Final simplification 60.9%

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

Reproduce

herbie shell --seed 2023322 
(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)))))