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

Percentage Accurate: 88.9% → 97.5%

Time: 9.0s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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 = x / ((y - z) * (t - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = x / ((y - z) * (t - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* (- t z) (- y z)))))
   (* x_s (if (<= t_1 5e-265) (/ (/ x_m (- t z)) (- y z)) t_1))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / ((t - z) * (y - z));
	double tmp;
	if (t_1 <= 5e-265) {
		tmp = (x_m / (t - z)) / (y - z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m / ((t - z) * (y - z))
    if (t_1 <= 5d-265) then
        tmp = (x_m / (t - z)) / (y - z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / ((t - z) * (y - z));
	double tmp;
	if (t_1 <= 5e-265) {
		tmp = (x_m / (t - z)) / (y - z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / ((t - z) * (y - z))
	tmp = 0
	if t_1 <= 5e-265:
		tmp = (x_m / (t - z)) / (y - z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(Float64(t - z) * Float64(y - z)))
	tmp = 0.0
	if (t_1 <= 5e-265)
		tmp = Float64(Float64(x_m / Float64(t - z)) / Float64(y - z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / ((t - z) * (y - z));
	tmp = 0.0;
	if (t_1 <= 5e-265)
		tmp = (x_m / (t - z)) / (y - z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 5e-265], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < 5.0000000000000001e-265

   1. Initial program 87.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 5.0000000000000001e-265 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)} \leq 5 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.95 \cdot 10^{-89}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z z))))
   (*
    x_s
    (if (<= z -3.5e+62)
      t_1
      (if (<= z -2.95e-89)
        (/ x_m (* t (- y z)))
        (if (<= z 4.5e+20) (/ x_m (* (- t z) y)) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -3.5e+62) {
		tmp = t_1;
	} else if (z <= -2.95e-89) {
		tmp = x_m / (t * (y - z));
	} else if (z <= 4.5e+20) {
		tmp = x_m / ((t - z) * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m / (z * z)
    if (z <= (-3.5d+62)) then
        tmp = t_1
    else if (z <= (-2.95d-89)) then
        tmp = x_m / (t * (y - z))
    else if (z <= 4.5d+20) then
        tmp = x_m / ((t - z) * y)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -3.5e+62) {
		tmp = t_1;
	} else if (z <= -2.95e-89) {
		tmp = x_m / (t * (y - z));
	} else if (z <= 4.5e+20) {
		tmp = x_m / ((t - z) * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -3.5e+62:
		tmp = t_1
	elif z <= -2.95e-89:
		tmp = x_m / (t * (y - z))
	elif z <= 4.5e+20:
		tmp = x_m / ((t - z) * y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -3.5e+62)
		tmp = t_1;
	elseif (z <= -2.95e-89)
		tmp = Float64(x_m / Float64(t * Float64(y - z)));
	elseif (z <= 4.5e+20)
		tmp = Float64(x_m / Float64(Float64(t - z) * y));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -3.5e+62)
		tmp = t_1;
	elseif (z <= -2.95e-89)
		tmp = x_m / (t * (y - z));
	elseif (z <= 4.5e+20)
		tmp = x_m / ((t - z) * y);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3.5e+62], t$95$1, If[LessEqual[z, -2.95e-89], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e+20], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -3.49999999999999984e62 or 4.5e20 < z

   1. Initial program 88.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if -3.49999999999999984e62 < z < -2.9500000000000001e-89

   1. Initial program 88.4%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -2.9500000000000001e-89 < z < 4.5e20

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 70.5

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

   5. Applied rewrites 70.5%

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

Alternative 3: 67.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{t \cdot \left(y - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-256}:\\ \;\;\;\;\frac{x\_m}{\left(-z\right) \cdot y}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* t (- y z)))))
   (*
    x_s
    (if (<= t -7.5e-119)
      t_1
      (if (<= t -4.6e-256)
        (/ x_m (* (- z) y))
        (if (<= t 2.4e-58) (/ x_m (* z z)) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (t * (y - z));
	double tmp;
	if (t <= -7.5e-119) {
		tmp = t_1;
	} else if (t <= -4.6e-256) {
		tmp = x_m / (-z * y);
	} else if (t <= 2.4e-58) {
		tmp = x_m / (z * z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m / (t * (y - z))
    if (t <= (-7.5d-119)) then
        tmp = t_1
    else if (t <= (-4.6d-256)) then
        tmp = x_m / (-z * y)
    else if (t <= 2.4d-58) then
        tmp = x_m / (z * z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (t * (y - z));
	double tmp;
	if (t <= -7.5e-119) {
		tmp = t_1;
	} else if (t <= -4.6e-256) {
		tmp = x_m / (-z * y);
	} else if (t <= 2.4e-58) {
		tmp = x_m / (z * z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (t * (y - z))
	tmp = 0
	if t <= -7.5e-119:
		tmp = t_1
	elif t <= -4.6e-256:
		tmp = x_m / (-z * y)
	elif t <= 2.4e-58:
		tmp = x_m / (z * z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(t * Float64(y - z)))
	tmp = 0.0
	if (t <= -7.5e-119)
		tmp = t_1;
	elseif (t <= -4.6e-256)
		tmp = Float64(x_m / Float64(Float64(-z) * y));
	elseif (t <= 2.4e-58)
		tmp = Float64(x_m / Float64(z * z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (t * (y - z));
	tmp = 0.0;
	if (t <= -7.5e-119)
		tmp = t_1;
	elseif (t <= -4.6e-256)
		tmp = x_m / (-z * y);
	elseif (t <= 2.4e-58)
		tmp = x_m / (z * z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -7.5e-119], t$95$1, If[LessEqual[t, -4.6e-256], N[(x$95$m / N[((-z) * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-58], N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < -7.50000000000000044e-119 or 2.4000000000000001e-58 < t

   1. Initial program 89.4%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if -7.50000000000000044e-119 < t < -4.6e-256

   1. Initial program 90.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.8%

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

   if -4.6e-256 < t < 2.4000000000000001e-58

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

Alternative 4: 90.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+197}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -9e-97)
    (/ (/ x_m y) (- t z))
    (if (<= t 5.8e+197) (/ x_m (* (- t z) (- y z))) (/ (/ x_m t) (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -9e-97) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 5.8e+197) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9d-97)) then
        tmp = (x_m / y) / (t - z)
    else if (t <= 5.8d+197) then
        tmp = x_m / ((t - z) * (y - z))
    else
        tmp = (x_m / t) / (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -9e-97) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 5.8e+197) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -9e-97:
		tmp = (x_m / y) / (t - z)
	elif t <= 5.8e+197:
		tmp = x_m / ((t - z) * (y - z))
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -9e-97)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (t <= 5.8e+197)
		tmp = Float64(x_m / Float64(Float64(t - z) * Float64(y - z)));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -9e-97)
		tmp = (x_m / y) / (t - z);
	elseif (t <= 5.8e+197)
		tmp = x_m / ((t - z) * (y - z));
	else
		tmp = (x_m / t) / (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -9e-97], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e+197], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -9.0000000000000002e-97

   1. Initial program 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 53.3

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

   7. Applied rewrites 53.3%

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

   if -9.0000000000000002e-97 < t < 5.80000000000000005e197

   1. Initial program 94.4%

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

   if 5.80000000000000005e197 < t

   1. Initial program 82.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 92.9

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

   7. Applied rewrites 92.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-89}:\\ \;\;\;\;\frac{x\_m}{\left(-t\right) \cdot z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z z))))
   (*
    x_s
    (if (<= z -3.5e+62)
      t_1
      (if (<= z -3e-89)
        (/ x_m (* (- t) z))
        (if (<= z 1.05e-50) (/ x_m (* t y)) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -3.5e+62) {
		tmp = t_1;
	} else if (z <= -3e-89) {
		tmp = x_m / (-t * z);
	} else if (z <= 1.05e-50) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m / (z * z)
    if (z <= (-3.5d+62)) then
        tmp = t_1
    else if (z <= (-3d-89)) then
        tmp = x_m / (-t * z)
    else if (z <= 1.05d-50) then
        tmp = x_m / (t * y)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -3.5e+62) {
		tmp = t_1;
	} else if (z <= -3e-89) {
		tmp = x_m / (-t * z);
	} else if (z <= 1.05e-50) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -3.5e+62:
		tmp = t_1
	elif z <= -3e-89:
		tmp = x_m / (-t * z)
	elif z <= 1.05e-50:
		tmp = x_m / (t * y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -3.5e+62)
		tmp = t_1;
	elseif (z <= -3e-89)
		tmp = Float64(x_m / Float64(Float64(-t) * z));
	elseif (z <= 1.05e-50)
		tmp = Float64(x_m / Float64(t * y));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -3.5e+62)
		tmp = t_1;
	elseif (z <= -3e-89)
		tmp = x_m / (-t * z);
	elseif (z <= 1.05e-50)
		tmp = x_m / (t * y);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3.5e+62], t$95$1, If[LessEqual[z, -3e-89], N[(x$95$m / N[((-t) * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e-50], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -3.49999999999999984e62 or 1.05e-50 < z

   1. Initial program 88.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   if -3.49999999999999984e62 < z < -2.9999999999999999e-89

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 45.8%

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

   if -2.9999999999999999e-89 < z < 1.05e-50

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

Alternative 6: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot y}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{x\_m}{\left(z - y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -5.4e-148)
    (/ x_m (* (- t z) y))
    (if (<= t 1.6e-36) (/ x_m (* (- z y) z)) (/ x_m (* t (- y z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -5.4e-148) {
		tmp = x_m / ((t - z) * y);
	} else if (t <= 1.6e-36) {
		tmp = x_m / ((z - y) * z);
	} else {
		tmp = x_m / (t * (y - z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.4d-148)) then
        tmp = x_m / ((t - z) * y)
    else if (t <= 1.6d-36) then
        tmp = x_m / ((z - y) * z)
    else
        tmp = x_m / (t * (y - z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -5.4e-148) {
		tmp = x_m / ((t - z) * y);
	} else if (t <= 1.6e-36) {
		tmp = x_m / ((z - y) * z);
	} else {
		tmp = x_m / (t * (y - z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -5.4e-148:
		tmp = x_m / ((t - z) * y)
	elif t <= 1.6e-36:
		tmp = x_m / ((z - y) * z)
	else:
		tmp = x_m / (t * (y - z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -5.4e-148)
		tmp = Float64(x_m / Float64(Float64(t - z) * y));
	elseif (t <= 1.6e-36)
		tmp = Float64(x_m / Float64(Float64(z - y) * z));
	else
		tmp = Float64(x_m / Float64(t * Float64(y - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -5.4e-148)
		tmp = x_m / ((t - z) * y);
	elseif (t <= 1.6e-36)
		tmp = x_m / ((z - y) * z);
	else
		tmp = x_m / (t * (y - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -5.4e-148], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e-36], N[(x$95$m / N[(N[(z - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -5.39999999999999976e-148

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 48.8

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

   5. Applied rewrites 48.8%

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

   if -5.39999999999999976e-148 < t < 1.60000000000000011e-36

   1. Initial program 95.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 79.9

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

   5. Applied rewrites 79.9%

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

   if 1.60000000000000011e-36 < t

   1. Initial program 89.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 80.7

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

   5. Applied rewrites 80.7%

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

Alternative 7: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-42}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot y}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{x\_m}{\left(z - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -9e-42)
    (/ x_m (* (- t z) y))
    (if (<= y 8.2e-240) (/ x_m (* (- z t) z)) (/ x_m (* t (- y z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -9e-42) {
		tmp = x_m / ((t - z) * y);
	} else if (y <= 8.2e-240) {
		tmp = x_m / ((z - t) * z);
	} else {
		tmp = x_m / (t * (y - z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-9d-42)) then
        tmp = x_m / ((t - z) * y)
    else if (y <= 8.2d-240) then
        tmp = x_m / ((z - t) * z)
    else
        tmp = x_m / (t * (y - z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -9e-42) {
		tmp = x_m / ((t - z) * y);
	} else if (y <= 8.2e-240) {
		tmp = x_m / ((z - t) * z);
	} else {
		tmp = x_m / (t * (y - z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -9e-42:
		tmp = x_m / ((t - z) * y)
	elif y <= 8.2e-240:
		tmp = x_m / ((z - t) * z)
	else:
		tmp = x_m / (t * (y - z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -9e-42)
		tmp = Float64(x_m / Float64(Float64(t - z) * y));
	elseif (y <= 8.2e-240)
		tmp = Float64(x_m / Float64(Float64(z - t) * z));
	else
		tmp = Float64(x_m / Float64(t * Float64(y - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -9e-42)
		tmp = x_m / ((t - z) * y);
	elseif (y <= 8.2e-240)
		tmp = x_m / ((z - t) * z);
	else
		tmp = x_m / (t * (y - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -9e-42], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-240], N[(x$95$m / N[(N[(z - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -9e-42

   1. Initial program 91.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -9e-42 < y < 8.2000000000000003e-240

   1. Initial program 93.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if 8.2000000000000003e-240 < y

   1. Initial program 89.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 59.5

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

   5. Applied rewrites 59.5%

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

Alternative 8: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{+197}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t 5.8e+197) (/ x_m (* (- t z) (- y z))) (/ (/ x_m t) (- y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 5.8e+197) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 5.8d+197) then
        tmp = x_m / ((t - z) * (y - z))
    else
        tmp = (x_m / t) / (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 5.8e+197) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 5.8e+197:
		tmp = x_m / ((t - z) * (y - z))
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 5.8e+197)
		tmp = Float64(x_m / Float64(Float64(t - z) * Float64(y - z)));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 5.8e+197)
		tmp = x_m / ((t - z) * (y - z));
	else
		tmp = (x_m / t) / (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 5.8e+197], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.80000000000000005e197

   1. Initial program 92.1%

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

   if 5.80000000000000005e197 < t

   1. Initial program 82.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 92.9

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

   7. Applied rewrites 92.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= z 6.2e+65) (/ x_m (* (- t z) (- y z))) (/ (/ x_m z) (- z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 6.2e+65) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = (x_m / z) / (z - y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 6.2d+65) then
        tmp = x_m / ((t - z) * (y - z))
    else
        tmp = (x_m / z) / (z - y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 6.2e+65) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = (x_m / z) / (z - y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 6.2e+65:
		tmp = x_m / ((t - z) * (y - z))
	else:
		tmp = (x_m / z) / (z - y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 6.2e+65)
		tmp = Float64(x_m / Float64(Float64(t - z) * Float64(y - z)));
	else
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 6.2e+65)
		tmp = x_m / ((t - z) * (y - z));
	else
		tmp = (x_m / z) / (z - y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 6.2e+65], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.19999999999999981e65

   1. Initial program 92.5%

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

   if 6.19999999999999981e65 < z

   1. Initial program 84.7%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. unsub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z z))))
   (* x_s (if (<= z -1.8e-11) t_1 (if (<= z 1.05e-50) (/ x_m (* t y)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -1.8e-11) {
		tmp = t_1;
	} else if (z <= 1.05e-50) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m / (z * z)
    if (z <= (-1.8d-11)) then
        tmp = t_1
    else if (z <= 1.05d-50) then
        tmp = x_m / (t * y)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -1.8e-11) {
		tmp = t_1;
	} else if (z <= 1.05e-50) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -1.8e-11:
		tmp = t_1
	elif z <= 1.05e-50:
		tmp = x_m / (t * y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -1.8e-11)
		tmp = t_1;
	elseif (z <= 1.05e-50)
		tmp = Float64(x_m / Float64(t * y));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -1.8e-11)
		tmp = t_1;
	elseif (z <= 1.05e-50)
		tmp = x_m / (t * y);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.8e-11], t$95$1, If[LessEqual[z, 1.05e-50], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.79999999999999992e-11 or 1.05e-50 < z

   1. Initial program 88.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if -1.79999999999999992e-11 < z < 1.05e-50

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 57.8

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

   5. Applied rewrites 57.8%

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

Alternative 11: 88.9% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (/ x_m (* (- t z) (- y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / ((t - z) * (y - z)));
}

Fortran

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

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / ((t - z) * (y - z)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	return x_s * (x_m / ((t - z) * (y - z)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / Float64(Float64(t - z) * Float64(y - z))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / ((t - z) * (y - z)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

3. Final simplification 91.0%

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

Alternative 12: 39.0% accurate, 1.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (/ x_m (* t y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / (t * y));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m / (t * y))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / (t * y));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	return x_s * (x_m / (t * y))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / Float64(t * y)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / (t * y));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in z around 0

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

   1. lower-*.f64 38.4

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

5. Applied rewrites 38.4%

   \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]
6. Add Preprocessing

Developer Target 1: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ \mathbf{if}\;\frac{x}{t\_1} < 0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (- t z))))
   (if (< (/ x t_1) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if ((x / t_1) < 0.0) {
		tmp = (x / (y - z)) / (t - z);
	} else {
		tmp = x * (1.0 / t_1);
	}
	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 = (y - z) * (t - z)
    if ((x / t_1) < 0.0d0) then
        tmp = (x / (y - z)) / (t - z)
    else
        tmp = x * (1.0d0 / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if ((x / t_1) < 0.0) {
		tmp = (x / (y - z)) / (t - z);
	} else {
		tmp = x * (1.0 / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - z) * (t - z)
	tmp = 0
	if (x / t_1) < 0.0:
		tmp = (x / (y - z)) / (t - z)
	else:
		tmp = x * (1.0 / t_1)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[Less[N[(x / t$95$1), $MachinePrecision], 0.0], N[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ x (* (- y z) (- t z))) 0) (/ (/ x (- y z)) (- t z)) (* x (/ 1 (* (- y z) (- t z))))))

  (/ x (* (- y z) (- t z))))