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

Percentage Accurate: 99.2% → 98.7%

Time: 11.3s

Alternatives: 11

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: 99.2% 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: 98.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. distribute-lft-in N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. --lowering--.f64 99.5

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 85.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)}\\ t_2 := \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+59}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ x (* (- y t) (- z y))))) (t_2 (/ x (* y (- z y)))))
   (if (<= t_1 -2e+59)
     (- 1.0 (/ x (* t z)))
     (if (<= t_1 -1000000.0) t_2 (if (<= t_1 50000000.0) 1.0 t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / ((y - t) * (z - y)));
	double t_2 = x / (y * (z - y));
	double tmp;
	if (t_1 <= -2e+59) {
		tmp = 1.0 - (x / (t * z));
	} else if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 + (x / ((y - t) * (z - y)))
    t_2 = x / (y * (z - y))
    if (t_1 <= (-2d+59)) then
        tmp = 1.0d0 - (x / (t * z))
    else if (t_1 <= (-1000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 50000000.0d0) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / ((y - t) * (z - y)));
	double t_2 = x / (y * (z - y));
	double tmp;
	if (t_1 <= -2e+59) {
		tmp = 1.0 - (x / (t * z));
	} else if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 + (x / ((y - t) * (z - y)))
	t_2 = x / (y * (z - y))
	tmp = 0
	if t_1 <= -2e+59:
		tmp = 1.0 - (x / (t * z))
	elif t_1 <= -1000000.0:
		tmp = t_2
	elif t_1 <= 50000000.0:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(x / Float64(Float64(y - t) * Float64(z - y))))
	t_2 = Float64(x / Float64(y * Float64(z - y)))
	tmp = 0.0
	if (t_1 <= -2e+59)
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	elseif (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 50000000.0)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + (x / ((y - t) * (z - y)));
	t_2 = x / (y * (z - y));
	tmp = 0.0;
	if (t_1 <= -2e+59)
		tmp = 1.0 - (x / (t * z));
	elseif (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 50000000.0)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(x / N[(N[(y - t), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+59], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, 50000000.0], 1.0, t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -1.99999999999999994e59

   1. Initial program 96.4%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 43.1

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

   5. Simplified 43.1%

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

   if -1.99999999999999994e59 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -1e6 or 5e7 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 99.5%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. --lowering--.f64 96.1

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

   5. Simplified 96.1%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 60.6%

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

   if -1e6 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 5e7

   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. Simplified 98.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq -2 \cdot 10^{+59}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq -1000000:\\ \;\;\;\;\frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - t\right) \cdot \left(z - y\right)}\\ t_2 := 1 + t\_1\\ \mathbf{if}\;t\_2 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y t) (- z y)))) (t_2 (+ 1.0 t_1)))
   (if (<= t_2 -1000000.0) t_1 (if (<= t_2 2.0) 1.0 t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - t) * (z - y));
	double t_2 = 1.0 + t_1;
	double tmp;
	if (t_2 <= -1000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - t) * (z - y))
    t_2 = 1.0d0 + t_1
    if (t_2 <= (-1000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - t) * (z - y));
	double t_2 = 1.0 + t_1;
	double tmp;
	if (t_2 <= -1000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - t) * (z - y))
	t_2 = 1.0 + t_1
	tmp = 0
	if t_2 <= -1000000.0:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - t) * Float64(z - y)))
	t_2 = Float64(1.0 + t_1)
	tmp = 0.0
	if (t_2 <= -1000000.0)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - t) * (z - y));
	t_2 = 1.0 + t_1;
	tmp = 0.0;
	if (t_2 <= -1000000.0)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - t), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000.0], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -1e6 or 2 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 98.2%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. --lowering--.f64 94.7

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

   5. Simplified 94.7%

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

   if -1e6 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 2

   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. Simplified 99.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq -1000000:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot \left(z - y\right)}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - t\right) \cdot z}\\ t_2 := 1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)}\\ \mathbf{if}\;t\_2 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y t) z))) (t_2 (+ 1.0 (/ x (* (- y t) (- z y))))))
   (if (<= t_2 -1000000.0) t_1 (if (<= t_2 2.0) 1.0 t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - t) * z);
	double t_2 = 1.0 + (x / ((y - t) * (z - y)));
	double tmp;
	if (t_2 <= -1000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - t) * z)
    t_2 = 1.0d0 + (x / ((y - t) * (z - y)))
    if (t_2 <= (-1000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - t) * z);
	double t_2 = 1.0 + (x / ((y - t) * (z - y)));
	double tmp;
	if (t_2 <= -1000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - t) * z)
	t_2 = 1.0 + (x / ((y - t) * (z - y)))
	tmp = 0
	if t_2 <= -1000000.0:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - t) * z))
	t_2 = Float64(1.0 + Float64(x / Float64(Float64(y - t) * Float64(z - y))))
	tmp = 0.0
	if (t_2 <= -1000000.0)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - t) * z);
	t_2 = 1.0 + (x / ((y - t) * (z - y)));
	tmp = 0.0;
	if (t_2 <= -1000000.0)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x / N[(N[(y - t), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000.0], t$95$1, If[LessEqual[t$95$2, 2.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -1e6 or 2 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 98.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

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

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

      6. distribute-frac-neg2 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 98.1

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 58.7

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

   7. Simplified 58.7%

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

   8. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 56.0

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

   10. Simplified 56.0%

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

   if -1e6 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 2

   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. Simplified 99.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq -1000000:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 + \frac{x}{y \cdot z}\\ t_2 := 1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)}\\ \mathbf{if}\;t\_2 \leq 0.9999998861946734:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ x (* y z)))) (t_2 (+ 1.0 (/ x (* (- y t) (- z y))))))
   (if (<= t_2 0.9999998861946734) t_1 (if (<= t_2 50000000.0) 1.0 t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / (y * z));
	double t_2 = 1.0 + (x / ((y - t) * (z - y)));
	double tmp;
	if (t_2 <= 0.9999998861946734) {
		tmp = t_1;
	} else if (t_2 <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 + (x / (y * z))
    t_2 = 1.0d0 + (x / ((y - t) * (z - y)))
    if (t_2 <= 0.9999998861946734d0) then
        tmp = t_1
    else if (t_2 <= 50000000.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / (y * z));
	double t_2 = 1.0 + (x / ((y - t) * (z - y)));
	double tmp;
	if (t_2 <= 0.9999998861946734) {
		tmp = t_1;
	} else if (t_2 <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 + (x / (y * z))
	t_2 = 1.0 + (x / ((y - t) * (z - y)))
	tmp = 0
	if t_2 <= 0.9999998861946734:
		tmp = t_1
	elif t_2 <= 50000000.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(x / Float64(y * z)))
	t_2 = Float64(1.0 + Float64(x / Float64(Float64(y - t) * Float64(z - y))))
	tmp = 0.0
	if (t_2 <= 0.9999998861946734)
		tmp = t_1;
	elseif (t_2 <= 50000000.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + (x / (y * z));
	t_2 = 1.0 + (x / ((y - t) * (z - y)));
	tmp = 0.0;
	if (t_2 <= 0.9999998861946734)
		tmp = t_1;
	elseif (t_2 <= 50000000.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x / N[(N[(y - t), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.9999998861946734], t$95$1, If[LessEqual[t$95$2, 50000000.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 0.999999886194673393 or 5e7 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 31.1

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

   8. Simplified 31.1%

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

   if 0.999999886194673393 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 5e7

   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. Simplified 98.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq 0.9999998861946734:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{y \cdot z}\\ t_2 := 1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)}\\ \mathbf{if}\;t\_2 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* y z))) (t_2 (+ 1.0 (/ x (* (- y t) (- z y))))))
   (if (<= t_2 -1000000.0) t_1 (if (<= t_2 50000000.0) 1.0 t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / (y * z);
	double t_2 = 1.0 + (x / ((y - t) * (z - y)));
	double tmp;
	if (t_2 <= -1000000.0) {
		tmp = t_1;
	} else if (t_2 <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / (y * z)
    t_2 = 1.0d0 + (x / ((y - t) * (z - y)))
    if (t_2 <= (-1000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 50000000.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / (y * z);
	double t_2 = 1.0 + (x / ((y - t) * (z - y)));
	double tmp;
	if (t_2 <= -1000000.0) {
		tmp = t_1;
	} else if (t_2 <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / (y * z)
	t_2 = 1.0 + (x / ((y - t) * (z - y)))
	tmp = 0
	if t_2 <= -1000000.0:
		tmp = t_1
	elif t_2 <= 50000000.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(y * z))
	t_2 = Float64(1.0 + Float64(x / Float64(Float64(y - t) * Float64(z - y))))
	tmp = 0.0
	if (t_2 <= -1000000.0)
		tmp = t_1;
	elseif (t_2 <= 50000000.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / (y * z);
	t_2 = 1.0 + (x / ((y - t) * (z - y)));
	tmp = 0.0;
	if (t_2 <= -1000000.0)
		tmp = t_1;
	elseif (t_2 <= 50000000.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x / N[(N[(y - t), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000.0], t$95$1, If[LessEqual[t$95$2, 50000000.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -1e6 or 5e7 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 57.5

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

   5. Simplified 57.5%

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

   6. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 30.1

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

   8. Simplified 30.1%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 28.9

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

   11. Simplified 28.9%

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

   if -1e6 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 5e7

   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. Simplified 98.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq -1000000:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - t\right) \cdot \left(z - y\right)} \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - t\right) \cdot \left(y - z\right)}\\ t_2 := 1 - \frac{x}{t \cdot z}\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y t) (- y z)))) (t_2 (- 1.0 (/ x (* t z)))))
   (if (<= t_1 -500.0) t_2 (if (<= t_1 2e-7) 1.0 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - t) * (y - z));
	double t_2 = 1.0 - (x / (t * z));
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-7) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - t) * (y - z))
    t_2 = 1.0d0 - (x / (t * z))
    if (t_1 <= (-500.0d0)) then
        tmp = t_2
    else if (t_1 <= 2d-7) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - t) * (y - z));
	double t_2 = 1.0 - (x / (t * z));
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-7) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - t) * (y - z))
	t_2 = 1.0 - (x / (t * z))
	tmp = 0
	if t_1 <= -500.0:
		tmp = t_2
	elif t_1 <= 2e-7:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - t) * Float64(y - z)))
	t_2 = Float64(1.0 - Float64(x / Float64(t * z)))
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 2e-7)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - t) * (y - z));
	t_2 = 1.0 - (x / (t * z));
	tmp = 0.0;
	if (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 2e-7)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 2e-7], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -500 or 1.9999999999999999e-7 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 98.2%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 38.7

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

   5. Simplified 38.7%

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

   if -500 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 1.9999999999999999e-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. Simplified 99.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - t\right) \cdot \left(y - z\right)} \leq -500:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{elif}\;\frac{x}{\left(y - t\right) \cdot \left(y - z\right)} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 + \frac{x}{\left(y - t\right) \cdot z}\\ \mathbf{if}\;z \leq -9.4 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-150}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ x (* (- y t) z)))))
   (if (<= z -9.4e-60)
     t_1
     (if (<= z 9e-150) (+ 1.0 (/ x (* y (- t y)))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / ((y - t) * z));
	double tmp;
	if (z <= -9.4e-60) {
		tmp = t_1;
	} else if (z <= 9e-150) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (x / ((y - t) * z))
    if (z <= (-9.4d-60)) then
        tmp = t_1
    else if (z <= 9d-150) then
        tmp = 1.0d0 + (x / (y * (t - y)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / ((y - t) * z));
	double tmp;
	if (z <= -9.4e-60) {
		tmp = t_1;
	} else if (z <= 9e-150) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 + (x / ((y - t) * z))
	tmp = 0
	if z <= -9.4e-60:
		tmp = t_1
	elif z <= 9e-150:
		tmp = 1.0 + (x / (y * (t - y)))
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(x / Float64(Float64(y - t) * z)))
	tmp = 0.0
	if (z <= -9.4e-60)
		tmp = t_1;
	elseif (z <= 9e-150)
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(t - y))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + (x / ((y - t) * z));
	tmp = 0.0;
	if (z <= -9.4e-60)
		tmp = t_1;
	elseif (z <= 9e-150)
		tmp = 1.0 + (x / (y * (t - y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.4e-60], t$95$1, If[LessEqual[z, 9e-150], N[(1.0 + N[(x / N[(y * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4e-60 or 9.0000000000000005e-150 < z

   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

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. --lowering--.f64 91.1

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

   5. Simplified 91.1%

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

   if -9.4e-60 < z < 9.0000000000000005e-150

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 91.9

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

   5. Simplified 91.9%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.4 \cdot 10^{-60}:\\ \;\;\;\;1 + \frac{x}{\left(y - t\right) \cdot z}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-150}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{if}\;y \leq -2.12 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-115}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ x (* y (- t y))))))
   (if (<= y -2.12e-106) t_1 (if (<= y 6.8e-115) (- 1.0 (/ x (* t z))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / (y * (t - y)));
	double tmp;
	if (y <= -2.12e-106) {
		tmp = t_1;
	} else if (y <= 6.8e-115) {
		tmp = 1.0 - (x / (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (x / (y * (t - y)))
    if (y <= (-2.12d-106)) then
        tmp = t_1
    else if (y <= 6.8d-115) then
        tmp = 1.0d0 - (x / (t * z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / (y * (t - y)));
	double tmp;
	if (y <= -2.12e-106) {
		tmp = t_1;
	} else if (y <= 6.8e-115) {
		tmp = 1.0 - (x / (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 + (x / (y * (t - y)))
	tmp = 0
	if y <= -2.12e-106:
		tmp = t_1
	elif y <= 6.8e-115:
		tmp = 1.0 - (x / (t * z))
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(x / Float64(y * Float64(t - y))))
	tmp = 0.0
	if (y <= -2.12e-106)
		tmp = t_1;
	elseif (y <= 6.8e-115)
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + (x / (y * (t - y)));
	tmp = 0.0;
	if (y <= -2.12e-106)
		tmp = t_1;
	elseif (y <= 6.8e-115)
		tmp = 1.0 - (x / (t * z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(x / N[(y * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.12e-106], t$95$1, If[LessEqual[y, 6.8e-115], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.12e-106 or 6.7999999999999996e-115 < y

   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 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 85.5

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

   5. Simplified 85.5%

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

   if -2.12e-106 < y < 6.7999999999999996e-115

   1. Initial program 98.7%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 82.2

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

   5. Simplified 82.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.12 \cdot 10^{-106}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-115}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + (x / ((y - t) * (z - y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 11: 74.4% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

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

5. Simplified 73.9%

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

Reproduce

herbie shell --seed 2024196 
(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)))))