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

Percentage Accurate: 99.1% → 99.1%

Time: 10.0s

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, 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]

Derivation

1. Initial program 98.8%

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

3. Final simplification 98.8%

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

Alternative 2: 91.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-98} \lor \neg \left(z \leq 2.5 \cdot 10^{-97}\right):\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.6e-98) (not (<= z 2.5e-97)))
   (+ 1.0 (/ (/ x z) (- y t)))
   (- 1.0 (/ x (* y (- y t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.6e-98) || !(z <= 2.5e-97)) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else {
		tmp = 1.0 - (x / (y * (y - t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-5.6d-98)) .or. (.not. (z <= 2.5d-97))) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else
        tmp = 1.0d0 - (x / (y * (y - t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.6e-98) || !(z <= 2.5e-97)) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else {
		tmp = 1.0 - (x / (y * (y - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.6e-98) or not (z <= 2.5e-97):
		tmp = 1.0 + ((x / z) / (y - t))
	else:
		tmp = 1.0 - (x / (y * (y - t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.6e-98) || !(z <= 2.5e-97))
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.6e-98) || ~((z <= 2.5e-97)))
		tmp = 1.0 + ((x / z) / (y - t));
	else
		tmp = 1.0 - (x / (y * (y - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.6e-98], N[Not[LessEqual[z, 2.5e-97]], $MachinePrecision]], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(y * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5999999999999998e-98 or 2.4999999999999998e-97 < 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. Add Preprocessing

   5. Taylor expanded in z around inf 96.9%

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

      1. +-commutative 96.9%

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

      2. associate-/r* 96.4%

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

   7. Simplified 96.4%

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

   if -5.5999999999999998e-98 < z < 2.4999999999999998e-97

   1. Initial program 96.7%

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

      1. sub-neg 96.7%

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

      2. distribute-frac-neg 96.7%

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

      3. *-lft-identity 96.7%

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

      4. associate-/r* 96.6%

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

      5. associate-*r/ 96.6%

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

      6. metadata-eval 96.6%

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

      7. times-frac 96.6%

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

      8. neg-mul-1 96.6%

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

      9. remove-double-neg 96.6%

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

      10. neg-mul-1 96.6%

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

      11. sub-neg 96.6%

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

      12. distribute-neg-out 96.6%

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

      13. remove-double-neg 96.6%

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

      14. +-commutative 96.6%

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

      15. sub-neg 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in z around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. sub-neg 85.5%

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

   7. Simplified 85.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-98} \lor \neg \left(z \leq 2.5 \cdot 10^{-97}\right):\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-128}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.02e-37)
   1.0
   (if (<= y 7.5e-128) (+ 1.0 (/ (/ x z) (- y t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.02e-37) {
		tmp = 1.0;
	} else if (y <= 7.5e-128) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.02d-37)) then
        tmp = 1.0d0
    else if (y <= 7.5d-128) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.02e-37) {
		tmp = 1.0;
	} else if (y <= 7.5e-128) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.02e-37:
		tmp = 1.0
	elif y <= 7.5e-128:
		tmp = 1.0 + ((x / z) / (y - t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.02e-37)
		tmp = 1.0;
	elseif (y <= 7.5e-128)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.02e-37)
		tmp = 1.0;
	elseif (y <= 7.5e-128)
		tmp = 1.0 + ((x / z) / (y - t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.02e-37], 1.0, If[LessEqual[y, 7.5e-128], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02000000000000006e-37 or 7.50000000000000021e-128 < 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. distribute-frac-neg 100.0%

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

      3. *-lft-identity 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 89.4%

      \[\leadsto \color{blue}{1} \]

   if -1.02000000000000006e-37 < y < 7.50000000000000021e-128

   1. Initial program 97.0%

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

      1. sub-neg 97.0%

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

      2. distribute-frac-neg 97.0%

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

      3. *-lft-identity 97.0%

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

      4. associate-/r* 96.1%

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

      5. associate-*r/ 96.1%

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

      6. metadata-eval 96.1%

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

      7. times-frac 96.1%

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

      8. neg-mul-1 96.1%

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

      9. remove-double-neg 96.1%

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

      10. neg-mul-1 96.1%

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

      11. sub-neg 96.1%

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

      12. distribute-neg-out 96.1%

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

      13. remove-double-neg 96.1%

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

      14. +-commutative 96.1%

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

      15. sub-neg 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in z around inf 87.4%

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

      1. +-commutative 87.4%

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

      2. associate-/r* 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-128}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-178}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-64}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.3e-178)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= t 5.5e-64)
     (- 1.0 (/ x (* y (- y z))))
     (- 1.0 (/ (/ x t) (- z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.3e-178) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 5.5e-64) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.3d-178)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (t <= 5.5d-64) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 - ((x / t) / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.3e-178) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 5.5e-64) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.3e-178:
		tmp = 1.0 + ((x / z) / (y - t))
	elif t <= 5.5e-64:
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 - ((x / t) / (z - y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.3e-178)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (t <= 5.5e-64)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 - Float64(Float64(x / t) / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.3e-178)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (t <= 5.5e-64)
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 - ((x / t) / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.3e-178], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-64], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.29999999999999999e-178

   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* 99.0%

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

      5. associate-*r/ 99.0%

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

      6. metadata-eval 99.0%

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

      7. times-frac 99.0%

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

      8. neg-mul-1 99.0%

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

      9. remove-double-neg 99.0%

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

      10. neg-mul-1 99.0%

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

      11. sub-neg 99.0%

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

      12. distribute-neg-out 99.0%

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

      13. remove-double-neg 99.0%

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

      14. +-commutative 99.0%

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

      15. sub-neg 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in z around inf 81.8%

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

      1. +-commutative 81.8%

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

      2. associate-/r* 80.8%

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

   7. Simplified 80.8%

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

   if -1.29999999999999999e-178 < t < 5.4999999999999999e-64

   1. Initial program 97.4%

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

   3. Taylor expanded in t around 0 94.2%

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

   if 5.4999999999999999e-64 < t

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 96.5%

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

      5. associate-*r/ 96.5%

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

      6. metadata-eval 96.5%

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

      7. times-frac 96.5%

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

      8. neg-mul-1 96.5%

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

      9. remove-double-neg 96.5%

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

      10. neg-mul-1 96.5%

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

      11. sub-neg 96.5%

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

      12. distribute-neg-out 96.5%

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

      13. remove-double-neg 96.5%

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

      14. +-commutative 96.5%

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

      15. sub-neg 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in t around inf 97.5%

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

      1. mul-1-neg 97.5%

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

      2. unsub-neg 97.5%

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

      3. associate-/r* 96.4%

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

   7. Simplified 96.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-178}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-64}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-178}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 10^{-62}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.4e-178)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= t 1e-62) (- 1.0 (/ (/ x y) (- y z))) (- 1.0 (/ (/ x t) (- z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.4e-178) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 1e-62) {
		tmp = 1.0 - ((x / y) / (y - z));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.4d-178)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (t <= 1d-62) then
        tmp = 1.0d0 - ((x / y) / (y - z))
    else
        tmp = 1.0d0 - ((x / t) / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.4e-178) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 1e-62) {
		tmp = 1.0 - ((x / y) / (y - z));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.4e-178:
		tmp = 1.0 + ((x / z) / (y - t))
	elif t <= 1e-62:
		tmp = 1.0 - ((x / y) / (y - z))
	else:
		tmp = 1.0 - ((x / t) / (z - y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.4e-178)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (t <= 1e-62)
		tmp = Float64(1.0 - Float64(Float64(x / y) / Float64(y - z)));
	else
		tmp = Float64(1.0 - Float64(Float64(x / t) / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.4e-178)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (t <= 1e-62)
		tmp = 1.0 - ((x / y) / (y - z));
	else
		tmp = 1.0 - ((x / t) / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.4e-178], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-62], N[(1.0 - N[(N[(x / y), $MachinePrecision] / N[(y - z), $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 < -2.40000000000000005e-178

   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* 99.0%

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

      5. associate-*r/ 99.0%

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

      6. metadata-eval 99.0%

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

      7. times-frac 99.0%

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

      8. neg-mul-1 99.0%

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

      9. remove-double-neg 99.0%

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

      10. neg-mul-1 99.0%

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

      11. sub-neg 99.0%

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

      12. distribute-neg-out 99.0%

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

      13. remove-double-neg 99.0%

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

      14. +-commutative 99.0%

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

      15. sub-neg 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in z around inf 81.8%

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

      1. +-commutative 81.8%

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

      2. associate-/r* 80.8%

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

   7. Simplified 80.8%

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

   if -2.40000000000000005e-178 < t < 1e-62

   1. Initial program 97.4%

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

      1. clear-num 97.4%

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

      2. associate-/r/ 97.4%

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

      3. *-commutative 97.4%

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in t around 0 94.2%

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

      1. associate-/r* 95.3%

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

   7. Simplified 95.3%

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

   if 1e-62 < t

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 96.5%

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

      5. associate-*r/ 96.5%

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

      6. metadata-eval 96.5%

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

      7. times-frac 96.5%

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

      8. neg-mul-1 96.5%

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

      9. remove-double-neg 96.5%

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

      10. neg-mul-1 96.5%

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

      11. sub-neg 96.5%

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

      12. distribute-neg-out 96.5%

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

      13. remove-double-neg 96.5%

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

      14. +-commutative 96.5%

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

      15. sub-neg 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in t around inf 97.5%

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

      1. mul-1-neg 97.5%

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

      2. unsub-neg 97.5%

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

      3. associate-/r* 96.4%

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

   7. Simplified 96.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-178}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 10^{-62}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.16 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-125}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.16e-38) 1.0 (if (<= y 1.9e-125) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.16e-38) {
		tmp = 1.0;
	} else if (y <= 1.9e-125) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.16d-38)) then
        tmp = 1.0d0
    else if (y <= 1.9d-125) then
        tmp = 1.0d0 - (x / (z * t))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.16e-38) {
		tmp = 1.0;
	} else if (y <= 1.9e-125) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.16e-38:
		tmp = 1.0
	elif y <= 1.9e-125:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.16e-38)
		tmp = 1.0;
	elseif (y <= 1.9e-125)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.16e-38)
		tmp = 1.0;
	elseif (y <= 1.9e-125)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.16e-38], 1.0, If[LessEqual[y, 1.9e-125], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.16e-38 or 1.9000000000000001e-125 < 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. distribute-frac-neg 100.0%

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

      3. *-lft-identity 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 89.4%

      \[\leadsto \color{blue}{1} \]

   if -2.16e-38 < y < 1.9000000000000001e-125

   1. Initial program 97.0%

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

      1. sub-neg 97.0%

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

      2. distribute-frac-neg 97.0%

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

      3. *-lft-identity 97.0%

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

      4. associate-/r* 96.1%

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

      5. associate-*r/ 96.1%

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

      6. metadata-eval 96.1%

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

      7. times-frac 96.1%

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

      8. neg-mul-1 96.1%

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

      9. remove-double-neg 96.1%

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

      10. neg-mul-1 96.1%

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

      11. sub-neg 96.1%

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

      12. distribute-neg-out 96.1%

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

      13. remove-double-neg 96.1%

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

      14. +-commutative 96.1%

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

      15. sub-neg 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in y around 0 76.4%

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

      1. mul-1-neg 76.4%

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

      2. sub-neg 76.4%

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

   7. Simplified 76.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.16 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-125}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\frac{x}{z - y}}{y - t} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 98.8%

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

   1. sub-neg 98.8%

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

   2. distribute-frac-neg 98.8%

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

   3. *-lft-identity 98.8%

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

   4. associate-/r* 98.4%

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

   5. associate-*r/ 98.4%

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

   6. metadata-eval 98.4%

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

   7. times-frac 98.4%

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

   8. neg-mul-1 98.4%

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

   9. remove-double-neg 98.4%

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

   10. neg-mul-1 98.4%

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

   11. sub-neg 98.4%

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

   12. distribute-neg-out 98.4%

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

   13. remove-double-neg 98.4%

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

   14. +-commutative 98.4%

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

   15. sub-neg 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 8: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-224}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.3e-224) (+ 1.0 (/ x (* y z))) 1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.3e-224) {
		tmp = 1.0 + (x / (y * z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.3d-224) then
        tmp = 1.0d0 + (x / (y * z))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.3e-224) {
		tmp = 1.0 + (x / (y * z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 2.3e-224:
		tmp = 1.0 + (x / (y * z))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.3e-224)
		tmp = Float64(1.0 + Float64(x / Float64(y * z)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.3e-224)
		tmp = 1.0 + (x / (y * z));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2.3e-224], N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999988e-224

   1. Initial program 98.0%

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

      1. sub-neg 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. *-lft-identity 98.0%

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

      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. Add Preprocessing

   5. Taylor expanded in z around inf 82.2%

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

      1. +-commutative 82.2%

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

      2. associate-/r* 82.8%

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

   7. Simplified 82.8%

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

   8. Taylor expanded in y around inf 66.8%

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

   if 2.29999999999999988e-224 < t

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 97.4%

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

      5. associate-*r/ 97.4%

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

      6. metadata-eval 97.4%

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

      7. times-frac 97.4%

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

      8. neg-mul-1 97.4%

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

      9. remove-double-neg 97.4%

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

      10. neg-mul-1 97.4%

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

      11. sub-neg 97.4%

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

      12. distribute-neg-out 97.4%

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

      13. remove-double-neg 97.4%

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

      14. +-commutative 97.4%

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

      15. sub-neg 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 79.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-224}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-224}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 8.2e-224) (+ 1.0 (/ (/ x z) y)) 1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8.2e-224) {
		tmp = 1.0 + ((x / z) / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 8.2d-224) then
        tmp = 1.0d0 + ((x / z) / y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8.2e-224) {
		tmp = 1.0 + ((x / z) / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 8.2e-224:
		tmp = 1.0 + ((x / z) / y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8.2e-224)
		tmp = Float64(1.0 + Float64(Float64(x / z) / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 8.2e-224)
		tmp = 1.0 + ((x / z) / y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 8.2e-224], N[(1.0 + N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < 8.19999999999999972e-224

   1. Initial program 98.0%

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

      1. sub-neg 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. *-lft-identity 98.0%

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

      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. Add Preprocessing

   5. Taylor expanded in z around inf 82.2%

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

      1. +-commutative 82.2%

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

      2. associate-/r* 82.8%

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

   7. Simplified 82.8%

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

   8. Taylor expanded in y around inf 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-/r* 68.0%

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

   10. Simplified 68.0%

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

   if 8.19999999999999972e-224 < t

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-/r* 97.4%

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

      5. associate-*r/ 97.4%

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

      6. metadata-eval 97.4%

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

      7. times-frac 97.4%

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

      8. neg-mul-1 97.4%

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

      9. remove-double-neg 97.4%

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

      10. neg-mul-1 97.4%

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

      11. sub-neg 97.4%

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

      12. distribute-neg-out 97.4%

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

      13. remove-double-neg 97.4%

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

      14. +-commutative 97.4%

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

      15. sub-neg 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 79.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-224}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 1.0)

C

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

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
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 98.8%

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

   1. sub-neg 98.8%

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

   2. distribute-frac-neg 98.8%

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

   3. *-lft-identity 98.8%

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

   4. associate-/r* 98.4%

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

   5. associate-*r/ 98.4%

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

   6. metadata-eval 98.4%

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

   7. times-frac 98.4%

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

   8. neg-mul-1 98.4%

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

   9. remove-double-neg 98.4%

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

   10. neg-mul-1 98.4%

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

   11. sub-neg 98.4%

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

   12. distribute-neg-out 98.4%

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

   13. remove-double-neg 98.4%

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

   14. +-commutative 98.4%

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

   15. sub-neg 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in x around 0 74.0%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 74.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024013 
(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)))))