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

Percentage Accurate: 98.9% → 98.5%

Time: 8.5s

Alternatives: 15

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -200000000000 \lor \neg \left(t\_1 \leq 100\right):\\ \;\;\;\;\frac{x}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ x (* (- y z) (- y t))))))
   (if (or (<= t_1 -200000000000.0) (not (<= t_1 100.0))) (/ x (* t y)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (x / ((y - z) * (y - t)));
	double tmp;
	if ((t_1 <= -200000000000.0) || !(t_1 <= 100.0)) {
		tmp = x / (t * 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) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - (x / ((y - z) * (y - t)))
    if ((t_1 <= (-200000000000.0d0)) .or. (.not. (t_1 <= 100.0d0))) then
        tmp = x / (t * 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 t_1 = 1.0 - (x / ((y - z) * (y - t)));
	double tmp;
	if ((t_1 <= -200000000000.0) || !(t_1 <= 100.0)) {
		tmp = x / (t * y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - (x / ((y - z) * (y - t)))
	tmp = 0
	if (t_1 <= -200000000000.0) or not (t_1 <= 100.0):
		tmp = x / (t * y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
	tmp = 0.0
	if ((t_1 <= -200000000000.0) || !(t_1 <= 100.0))
		tmp = Float64(x / Float64(t * y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - (x / ((y - z) * (y - t)));
	tmp = 0.0;
	if ((t_1 <= -200000000000.0) || ~((t_1 <= 100.0)))
		tmp = x / (t * y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -200000000000.0], N[Not[LessEqual[t$95$1, 100.0]], $MachinePrecision]], N[(x / N[(t * y), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -2e11 or 100 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 91.8

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 29.3%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 42.3%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 31.6%

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

   if -2e11 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 100

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -200000000000 \lor \neg \left(1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 100\right):\\ \;\;\;\;\frac{x}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -50:\\ \;\;\;\;\frac{-x}{\left(z - y\right) \cdot t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{\left(t - y\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -50.0)
     (/ (- x) (* (- z y) t))
     (if (<= t_1 5e-7) 1.0 (- 1.0 (/ x (* (- t y) z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -50.0) {
		tmp = -x / ((z - y) * t);
	} else if (t_1 <= 5e-7) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - (x / ((t - y) * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-50.0d0)) then
        tmp = -x / ((z - y) * t)
    else if (t_1 <= 5d-7) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 - (x / ((t - y) * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -50.0) {
		tmp = -x / ((z - y) * t);
	} else if (t_1 <= 5e-7) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - (x / ((t - y) * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -50.0:
		tmp = -x / ((z - y) * t)
	elif t_1 <= 5e-7:
		tmp = 1.0
	else:
		tmp = 1.0 - (x / ((t - y) * z))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -50.0)
		tmp = Float64(Float64(-x) / Float64(Float64(z - y) * t));
	elseif (t_1 <= 5e-7)
		tmp = 1.0;
	else
		tmp = Float64(1.0 - Float64(x / Float64(Float64(t - y) * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -50.0)
		tmp = -x / ((z - y) * t);
	elseif (t_1 <= 5e-7)
		tmp = 1.0;
	else
		tmp = 1.0 - (x / ((t - y) * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50.0], N[((-x) / N[(N[(z - y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], 1.0, N[(1.0 - N[(x / N[(N[(t - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -50

   1. Initial program 86.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 58.1%

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

   if -50 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.6%

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

   if 4.99999999999999977e-7 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 74.4

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

   5. Applied rewrites 74.4%

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

Alternative 4: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -200000000000:\\ \;\;\;\;\frac{-x}{y \cdot y}\\ \mathbf{elif}\;t\_1 \leq 100:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ x (* (- y z) (- y t))))))
   (if (<= t_1 -200000000000.0)
     (/ (- x) (* y y))
     (if (<= t_1 100.0) 1.0 (/ x (* t y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (x / ((y - z) * (y - t)));
	double tmp;
	if (t_1 <= -200000000000.0) {
		tmp = -x / (y * y);
	} else if (t_1 <= 100.0) {
		tmp = 1.0;
	} else {
		tmp = x / (t * 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) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - (x / ((y - z) * (y - t)))
    if (t_1 <= (-200000000000.0d0)) then
        tmp = -x / (y * y)
    else if (t_1 <= 100.0d0) then
        tmp = 1.0d0
    else
        tmp = x / (t * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (x / ((y - z) * (y - t)));
	double tmp;
	if (t_1 <= -200000000000.0) {
		tmp = -x / (y * y);
	} else if (t_1 <= 100.0) {
		tmp = 1.0;
	} else {
		tmp = x / (t * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - (x / ((y - z) * (y - t)))
	tmp = 0
	if t_1 <= -200000000000.0:
		tmp = -x / (y * y)
	elif t_1 <= 100.0:
		tmp = 1.0
	else:
		tmp = x / (t * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
	tmp = 0.0
	if (t_1 <= -200000000000.0)
		tmp = Float64(Float64(-x) / Float64(y * y));
	elseif (t_1 <= 100.0)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(t * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - (x / ((y - z) * (y - t)));
	tmp = 0.0;
	if (t_1 <= -200000000000.0)
		tmp = -x / (y * y);
	elseif (t_1 <= 100.0)
		tmp = 1.0;
	else
		tmp = x / (t * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200000000000.0], N[((-x) / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 100.0], 1.0, N[(x / N[(t * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -2e11

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 28.5%

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

   if -2e11 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 100

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.2%

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

   if 100 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 86.4%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 30.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 49.5%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 39.4%

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

Alternative 5: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -50 \lor \neg \left(t\_1 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{-x}{\left(z - y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (or (<= t_1 -50.0) (not (<= t_1 5e-7))) (/ (- x) (* (- z y) t)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if ((t_1 <= -50.0) || !(t_1 <= 5e-7)) {
		tmp = -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) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if ((t_1 <= (-50.0d0)) .or. (.not. (t_1 <= 5d-7))) then
        tmp = -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 t_1 = x / ((y - z) * (y - t));
	double tmp;
	if ((t_1 <= -50.0) || !(t_1 <= 5e-7)) {
		tmp = -x / ((z - y) * t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if (t_1 <= -50.0) or not (t_1 <= 5e-7):
		tmp = -x / ((z - y) * t)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if ((t_1 <= -50.0) || !(t_1 <= 5e-7))
		tmp = Float64(Float64(-x) / Float64(Float64(z - y) * t));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if ((t_1 <= -50.0) || ~((t_1 <= 5e-7)))
		tmp = -x / ((z - y) * t);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -50.0], N[Not[LessEqual[t$95$1, 5e-7]], $MachinePrecision]], N[((-x) / N[(N[(z - y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -50 or 4.99999999999999977e-7 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 92.2%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 66.1%

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

   if -50 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -50 \lor \neg \left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{-x}{\left(z - y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -50 \lor \neg \left(t\_1 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (or (<= t_1 -50.0) (not (<= t_1 5e-7))) (/ x (* (- y t) z)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if ((t_1 <= -50.0) || !(t_1 <= 5e-7)) {
		tmp = x / ((y - t) * 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) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if ((t_1 <= (-50.0d0)) .or. (.not. (t_1 <= 5d-7))) then
        tmp = x / ((y - t) * 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 t_1 = x / ((y - z) * (y - t));
	double tmp;
	if ((t_1 <= -50.0) || !(t_1 <= 5e-7)) {
		tmp = x / ((y - t) * z);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if (t_1 <= -50.0) or not (t_1 <= 5e-7):
		tmp = x / ((y - t) * z)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if ((t_1 <= -50.0) || !(t_1 <= 5e-7))
		tmp = Float64(x / Float64(Float64(y - t) * z));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if ((t_1 <= -50.0) || ~((t_1 <= 5e-7)))
		tmp = x / ((y - t) * z);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -50.0], N[Not[LessEqual[t$95$1, 5e-7]], $MachinePrecision]], N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -50 or 4.99999999999999977e-7 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 92.2%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 63.5%

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

   if -50 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -50 \lor \neg \left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{-x}{\left(y - t\right) \cdot y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -4e+22)
     (/ (- x) (* (- y t) y))
     (if (<= t_1 5e-7) 1.0 (/ x (* (- y t) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -4e+22) {
		tmp = -x / ((y - t) * y);
	} else if (t_1 <= 5e-7) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - t) * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-4d+22)) then
        tmp = -x / ((y - t) * y)
    else if (t_1 <= 5d-7) then
        tmp = 1.0d0
    else
        tmp = x / ((y - t) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -4e+22) {
		tmp = -x / ((y - t) * y);
	} else if (t_1 <= 5e-7) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - t) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -4e+22:
		tmp = -x / ((y - t) * y)
	elif t_1 <= 5e-7:
		tmp = 1.0
	else:
		tmp = x / ((y - t) * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -4e+22)
		tmp = Float64(Float64(-x) / Float64(Float64(y - t) * y));
	elseif (t_1 <= 5e-7)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(Float64(y - t) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -4e+22)
		tmp = -x / ((y - t) * y);
	elseif (t_1 <= 5e-7)
		tmp = 1.0;
	else
		tmp = x / ((y - t) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+22], N[((-x) / N[(N[(y - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], 1.0, N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -4e22

   1. Initial program 86.4%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 30.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 49.5%

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

   if -4e22 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.2%

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

   if 4.99999999999999977e-7 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.7%

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

Alternative 8: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -0.004:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -0.004)
     (- 1.0 (/ x (* t z)))
     (if (<= t_1 5e-7) 1.0 (/ x (* (- y t) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -0.004) {
		tmp = 1.0 - (x / (t * z));
	} else if (t_1 <= 5e-7) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - t) * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-0.004d0)) then
        tmp = 1.0d0 - (x / (t * z))
    else if (t_1 <= 5d-7) then
        tmp = 1.0d0
    else
        tmp = x / ((y - t) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -0.004) {
		tmp = 1.0 - (x / (t * z));
	} else if (t_1 <= 5e-7) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - t) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -0.004:
		tmp = 1.0 - (x / (t * z))
	elif t_1 <= 5e-7:
		tmp = 1.0
	else:
		tmp = x / ((y - t) * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -0.004)
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	elseif (t_1 <= 5e-7)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(Float64(y - t) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -0.004)
		tmp = 1.0 - (x / (t * z));
	elseif (t_1 <= 5e-7)
		tmp = 1.0;
	else
		tmp = x / ((y - t) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.004], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], 1.0, N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -0.0040000000000000001

   1. Initial program 87.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 45.1

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

   5. Applied rewrites 45.1%

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

   if -0.0040000000000000001 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.4%

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

   if 4.99999999999999977e-7 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.7%

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

Alternative 9: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -50 \lor \neg \left(t\_1 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (or (<= t_1 -50.0) (not (<= t_1 5e-7))) (/ (- x) (* z t)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if ((t_1 <= -50.0) || !(t_1 <= 5e-7)) {
		tmp = -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) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if ((t_1 <= (-50.0d0)) .or. (.not. (t_1 <= 5d-7))) then
        tmp = -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 t_1 = x / ((y - z) * (y - t));
	double tmp;
	if ((t_1 <= -50.0) || !(t_1 <= 5e-7)) {
		tmp = -x / (z * t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if (t_1 <= -50.0) or not (t_1 <= 5e-7):
		tmp = -x / (z * t)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if ((t_1 <= -50.0) || !(t_1 <= 5e-7))
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if ((t_1 <= -50.0) || ~((t_1 <= 5e-7)))
		tmp = -x / (z * t);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -50.0], N[Not[LessEqual[t$95$1, 5e-7]], $MachinePrecision]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -50 or 4.99999999999999977e-7 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 92.2%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.6%

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

   10. Applied rewrites 51.1%

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

   if -50 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -50 \lor \neg \left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-101}:\\ \;\;\;\;1 - \frac{x}{\left(t - y\right) \cdot z}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-229}:\\ \;\;\;\;1 - \frac{x}{\left(y - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{\left(z - y\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.22e-101)
   (- 1.0 (/ x (* (- t y) z)))
   (if (<= z 5.6e-229)
     (- 1.0 (/ x (* (- y t) y)))
     (- 1.0 (/ x (* (- z y) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.22e-101) {
		tmp = 1.0 - (x / ((t - y) * z));
	} else if (z <= 5.6e-229) {
		tmp = 1.0 - (x / ((y - t) * y));
	} else {
		tmp = 1.0 - (x / ((z - 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 <= (-1.22d-101)) then
        tmp = 1.0d0 - (x / ((t - y) * z))
    else if (z <= 5.6d-229) then
        tmp = 1.0d0 - (x / ((y - t) * y))
    else
        tmp = 1.0d0 - (x / ((z - 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 <= -1.22e-101) {
		tmp = 1.0 - (x / ((t - y) * z));
	} else if (z <= 5.6e-229) {
		tmp = 1.0 - (x / ((y - t) * y));
	} else {
		tmp = 1.0 - (x / ((z - y) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.22e-101:
		tmp = 1.0 - (x / ((t - y) * z))
	elif z <= 5.6e-229:
		tmp = 1.0 - (x / ((y - t) * y))
	else:
		tmp = 1.0 - (x / ((z - y) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.22e-101)
		tmp = Float64(1.0 - Float64(x / Float64(Float64(t - y) * z)));
	elseif (z <= 5.6e-229)
		tmp = Float64(1.0 - Float64(x / Float64(Float64(y - t) * y)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(Float64(z - y) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.22e-101)
		tmp = 1.0 - (x / ((t - y) * z));
	elseif (z <= 5.6e-229)
		tmp = 1.0 - (x / ((y - t) * y));
	else
		tmp = 1.0 - (x / ((z - y) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.22e-101], N[(1.0 - N[(x / N[(N[(t - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.6e-229], N[(1.0 - N[(x / N[(N[(y - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(N[(z - y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2199999999999999e-101

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -1.2199999999999999e-101 < z < 5.5999999999999998e-229

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 93.3

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

   5. Applied rewrites 93.3%

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

   if 5.5999999999999998e-229 < z

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

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

Alternative 11: 98.4% accurate, 0.8× speedup

Math

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

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(Float64(x / 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[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 12: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-101}:\\ \;\;\;\;1 - \frac{x}{\left(t - y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{\left(y - t\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.22e-101)
   (- 1.0 (/ x (* (- t y) z)))
   (- 1.0 (/ x (* (- y t) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.22e-101) {
		tmp = 1.0 - (x / ((t - y) * z));
	} else {
		tmp = 1.0 - (x / ((y - t) * 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 (z <= (-1.22d-101)) then
        tmp = 1.0d0 - (x / ((t - y) * z))
    else
        tmp = 1.0d0 - (x / ((y - t) * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.22e-101) {
		tmp = 1.0 - (x / ((t - y) * z));
	} else {
		tmp = 1.0 - (x / ((y - t) * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.22e-101:
		tmp = 1.0 - (x / ((t - y) * z))
	else:
		tmp = 1.0 - (x / ((y - t) * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.22e-101)
		tmp = Float64(1.0 - Float64(x / Float64(Float64(t - y) * z)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(Float64(y - t) * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.22e-101)
		tmp = 1.0 - (x / ((t - y) * z));
	else
		tmp = 1.0 - (x / ((y - t) * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.22e-101], N[(1.0 - N[(x / N[(N[(t - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(N[(y - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2199999999999999e-101

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -1.2199999999999999e-101 < z

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

Alternative 13: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. distribute-lft-neg-in N/A

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

   8. distribute-frac-neg2 N/A

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

   9. lower-fma.f64 N/A

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

   10. distribute-frac-neg2 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 98.1

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 98.1

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

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\left(y - t\right) \cdot \left(y - z\right)}, x, 1\right)} \]
5. Add Preprocessing

Alternative 14: 98.9% 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 98.0%

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

Alternative 15: 74.8% accurate, 26.0× 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 98.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 74.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024324 
(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)))))