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

Percentage Accurate: 89.0% → 97.9%

Time: 12.0s

Alternatives: 14

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: 89.0% 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: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (- y z) (- t z)))))
   (* x_s (if (<= t_1 0.0) (/ (/ 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 / ((y - z) * (t - z));
	double tmp;
	if (t_1 <= 0.0) {
		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 / ((y - z) * (t - z))
    if (t_1 <= 0.0d0) 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 / ((y - z) * (t - z));
	double tmp;
	if (t_1 <= 0.0) {
		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 / ((y - z) * (t - z))
	tmp = 0
	if t_1 <= 0.0:
		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(y - z) * Float64(t - z)))
	tmp = 0.0
	if (t_1 <= 0.0)
		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 / ((y - z) * (t - z));
	tmp = 0.0;
	if (t_1 <= 0.0)
		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[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 0.0], 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))) < -0.0

   1. Initial program 82.4%

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

      1. associate-/l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{t - z}\right), \color{blue}{\left(y - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(t - z\right)\right), \left(\color{blue}{y} - z\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, z\right)\right), \left(y - z\right)\right) \]

      5. --lowering--.f64 98.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, z\right)\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 98.3%

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

   if -0.0 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 99.7%

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

Alternative 2: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t}\\ \end{array} \end{array} \]

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 -6e-28)
    (/ (/ x_m (- t z)) y)
    (if (<= t 5.5e+156) (/ x_m (* (- y z) (- t z))) (/ (/ x_m (- y z)) t)))))

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 <= -6e-28) {
		tmp = (x_m / (t - z)) / y;
	} else if (t <= 5.5e+156) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	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 <= (-6d-28)) then
        tmp = (x_m / (t - z)) / y
    else if (t <= 5.5d+156) then
        tmp = x_m / ((y - z) * (t - z))
    else
        tmp = (x_m / (y - z)) / t
    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 <= -6e-28) {
		tmp = (x_m / (t - z)) / y;
	} else if (t <= 5.5e+156) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	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 <= -6e-28:
		tmp = (x_m / (t - z)) / y
	elif t <= 5.5e+156:
		tmp = x_m / ((y - z) * (t - z))
	else:
		tmp = (x_m / (y - z)) / t
	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 <= -6e-28)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	elseif (t <= 5.5e+156)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / Float64(y - z)) / t);
	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 <= -6e-28)
		tmp = (x_m / (t - z)) / y;
	elseif (t <= 5.5e+156)
		tmp = x_m / ((y - z) * (t - z));
	else
		tmp = (x_m / (y - z)) / t;
	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, -6e-28], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t, 5.5e+156], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -6.00000000000000005e-28

   1. Initial program 80.2%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(t - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\color{blue}{t} - z\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - z\right)\right) \]

      5. --lowering--.f64 96.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{\_.f64}\left(t, z\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 48.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{\_.f64}\left(\color{blue}{t}, z\right)\right) \]

   7. Simplified 48.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{t - z}\right), \color{blue}{y}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(t - z\right)\right), y\right) \]

      5. --lowering--.f64 55.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, z\right)\right), y\right) \]

   9. Applied egg-rr 55.3%

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

   if -6.00000000000000005e-28 < t < 5.5000000000000003e156

   1. Initial program 92.4%

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

   if 5.5000000000000003e156 < t

   1. Initial program 75.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{t}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), t\right) \]

      5. --lowering--.f64 99.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), t\right) \]

   5. Simplified 99.6%

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

Alternative 3: 82.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;t \leq 165000000:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t}\\ \end{array} \end{array} \]

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 -1.8e-115)
    (/ (/ x_m y) (- t z))
    (if (<= t 165000000.0) (/ (/ x_m z) (- z y)) (/ (/ x_m (- y z)) t)))))

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 <= -1.8e-115) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 165000000.0) {
		tmp = (x_m / z) / (z - y);
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	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 <= (-1.8d-115)) then
        tmp = (x_m / y) / (t - z)
    else if (t <= 165000000.0d0) then
        tmp = (x_m / z) / (z - y)
    else
        tmp = (x_m / (y - z)) / t
    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 <= -1.8e-115) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 165000000.0) {
		tmp = (x_m / z) / (z - y);
	} else {
		tmp = (x_m / (y - z)) / t;
	}
	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 <= -1.8e-115:
		tmp = (x_m / y) / (t - z)
	elif t <= 165000000.0:
		tmp = (x_m / z) / (z - y)
	else:
		tmp = (x_m / (y - z)) / t
	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 <= -1.8e-115)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (t <= 165000000.0)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	else
		tmp = Float64(Float64(x_m / Float64(y - z)) / t);
	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 <= -1.8e-115)
		tmp = (x_m / y) / (t - z);
	elseif (t <= 165000000.0)
		tmp = (x_m / z) / (z - y);
	else
		tmp = (x_m / (y - z)) / t;
	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, -1.8e-115], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 165000000.0], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.80000000000000005e-115

   1. Initial program 82.8%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(t - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\color{blue}{t} - z\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - z\right)\right) \]

      5. --lowering--.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{\_.f64}\left(t, z\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 52.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{\_.f64}\left(\color{blue}{t}, z\right)\right) \]

   7. Simplified 52.4%

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

   if -1.80000000000000005e-115 < t < 1.65e8

   1. Initial program 92.7%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(t - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\color{blue}{t} - z\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - z\right)\right) \]

      5. --lowering--.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 97.6%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{x}{t - z}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)}\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\mathsf{neg}\left(\left(t - z\right)\right)}\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\mathsf{neg}\left(\left(t - z\right)\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(0 - \left(t - z\right)\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{z}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(0 - \left(t + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(0 - \left(\left(\mathsf{neg}\left(z\right)\right) + t\right)\right)\right), \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - t\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{z}\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - t\right)\right), \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(z - t\right)\right), \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{z}\right)\right)\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(0 - \color{blue}{\left(y - z\right)}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(0 - \left(y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(z\right)\right) + \color{blue}{y}\right)\right)\right) \]

      17. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - \color{blue}{y}\right)\right) \]

      18. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y\right)\right) \]

      19. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(z - y\right)\right) \]

      20. --lowering--.f64 96.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \mathsf{\_.f64}\left(z, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, \mathsf{\_.f64}\left(z, y\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 85.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{\_.f64}\left(\color{blue}{z}, y\right)\right) \]

   9. Simplified 85.2%

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

   if 1.65e8 < t

   1. Initial program 82.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{t}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), t\right) \]

      5. --lowering--.f64 94.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), t\right) \]

   5. Simplified 94.1%

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

Alternative 4: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 -2.8e-115)
    (/ (/ x_m y) (- t z))
    (if (<= t 3.5e-7) (/ (/ x_m z) (- z y)) (/ (/ 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 <= -2.8e-115) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 3.5e-7) {
		tmp = (x_m / z) / (z - y);
	} 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 <= (-2.8d-115)) then
        tmp = (x_m / y) / (t - z)
    else if (t <= 3.5d-7) then
        tmp = (x_m / z) / (z - y)
    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 <= -2.8e-115) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 3.5e-7) {
		tmp = (x_m / z) / (z - y);
	} 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 <= -2.8e-115:
		tmp = (x_m / y) / (t - z)
	elif t <= 3.5e-7:
		tmp = (x_m / z) / (z - y)
	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 <= -2.8e-115)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (t <= 3.5e-7)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	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 <= -2.8e-115)
		tmp = (x_m / y) / (t - z);
	elseif (t <= 3.5e-7)
		tmp = (x_m / z) / (z - y);
	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, -2.8e-115], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e-7], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $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 < -2.79999999999999987e-115

   1. Initial program 82.8%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(t - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\color{blue}{t} - z\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - z\right)\right) \]

      5. --lowering--.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{\_.f64}\left(t, z\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 52.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{\_.f64}\left(\color{blue}{t}, z\right)\right) \]

   7. Simplified 52.4%

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

   if -2.79999999999999987e-115 < t < 3.49999999999999984e-7

   1. Initial program 93.2%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(t - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\color{blue}{t} - z\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - z\right)\right) \]

      5. --lowering--.f64 97.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 97.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{x}{t - z}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)}\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\mathsf{neg}\left(\left(t - z\right)\right)}\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\mathsf{neg}\left(\left(t - z\right)\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(0 - \left(t - z\right)\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{z}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(0 - \left(t + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(0 - \left(\left(\mathsf{neg}\left(z\right)\right) + t\right)\right)\right), \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - t\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{z}\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - t\right)\right), \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(z - t\right)\right), \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{z}\right)\right)\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(0 - \color{blue}{\left(y - z\right)}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(0 - \left(y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(z\right)\right) + \color{blue}{y}\right)\right)\right) \]

      17. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - \color{blue}{y}\right)\right) \]

      18. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y\right)\right) \]

      19. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \left(z - y\right)\right) \]

      20. --lowering--.f64 96.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(z, t\right)\right), \mathsf{\_.f64}\left(z, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, \mathsf{\_.f64}\left(z, y\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 85.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{\_.f64}\left(\color{blue}{z}, y\right)\right) \]

   9. Simplified 85.5%

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

   if 3.49999999999999984e-7 < t

   1. Initial program 82.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 77.2%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{t}\right), \color{blue}{\left(y - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), \left(\color{blue}{y} - z\right)\right) \]

      5. --lowering--.f64 87.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 87.2%

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

Alternative 5: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 -1.6e-285)
    (/ (/ x_m y) (- t z))
    (if (<= t 4e-7) (/ (/ x_m z) 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 <= -1.6e-285) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 4e-7) {
		tmp = (x_m / z) / 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 <= (-1.6d-285)) then
        tmp = (x_m / y) / (t - z)
    else if (t <= 4d-7) then
        tmp = (x_m / z) / 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 <= -1.6e-285) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 4e-7) {
		tmp = (x_m / z) / 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 <= -1.6e-285:
		tmp = (x_m / y) / (t - z)
	elif t <= 4e-7:
		tmp = (x_m / z) / 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 <= -1.6e-285)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (t <= 4e-7)
		tmp = Float64(Float64(x_m / z) / 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 <= -1.6e-285)
		tmp = (x_m / y) / (t - z);
	elseif (t <= 4e-7)
		tmp = (x_m / z) / 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, -1.6e-285], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-7], N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.60000000000000008e-285

   1. Initial program 86.6%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(t - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\color{blue}{t} - z\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - z\right)\right) \]

      5. --lowering--.f64 97.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 97.5%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{\_.f64}\left(t, z\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 56.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{\_.f64}\left(\color{blue}{t}, z\right)\right) \]

   7. Simplified 56.4%

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

   if -1.60000000000000008e-285 < t < 3.9999999999999998e-7

   1. Initial program 90.9%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 62.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 62.2%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{z}\right) \]

      3. /-lowering-/.f64 65.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), z\right) \]

   7. Applied egg-rr 65.5%

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

   if 3.9999999999999998e-7 < t

   1. Initial program 82.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 77.2%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{t}\right), \color{blue}{\left(y - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), \left(\color{blue}{y} - z\right)\right) \]

      5. --lowering--.f64 87.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 87.2%

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

Alternative 6: 72.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-290}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 -1.4e-290)
    (/ x_m (* y (- t z)))
    (if (<= t 9.5e-7) (/ (/ x_m z) 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 <= -1.4e-290) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 9.5e-7) {
		tmp = (x_m / z) / 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 <= (-1.4d-290)) then
        tmp = x_m / (y * (t - z))
    else if (t <= 9.5d-7) then
        tmp = (x_m / z) / 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 <= -1.4e-290) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 9.5e-7) {
		tmp = (x_m / z) / 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 <= -1.4e-290:
		tmp = x_m / (y * (t - z))
	elif t <= 9.5e-7:
		tmp = (x_m / z) / 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 <= -1.4e-290)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (t <= 9.5e-7)
		tmp = Float64(Float64(x_m / z) / 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 <= -1.4e-290)
		tmp = x_m / (y * (t - z));
	elseif (t <= 9.5e-7)
		tmp = (x_m / z) / 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, -1.4e-290], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-7], N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.39999999999999998e-290

   1. Initial program 86.7%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(t - z\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\left(t - z\right) \cdot \color{blue}{y}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(t - z\right), \color{blue}{y}\right)\right) \]

      4. --lowering--.f64 54.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(t, z\right), y\right)\right) \]

   5. Simplified 54.8%

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

   if -1.39999999999999998e-290 < t < 9.5000000000000001e-7

   1. Initial program 90.7%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 63.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 63.2%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{z}\right) \]

      3. /-lowering-/.f64 66.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), z\right) \]

   7. Applied egg-rr 66.6%

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

   if 9.5000000000000001e-7 < t

   1. Initial program 82.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 77.2%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{t}\right), \color{blue}{\left(y - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), \left(\color{blue}{y} - z\right)\right) \]

      5. --lowering--.f64 87.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Applied egg-rr 87.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-289}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

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 -2.5e-289)
    (/ x_m (* y (- t z)))
    (if (<= t 1.1e-6) (/ (/ x_m z) z) (/ x_m (* (- y z) t))))))

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 <= -2.5e-289) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 1.1e-6) {
		tmp = (x_m / z) / z;
	} else {
		tmp = x_m / ((y - z) * t);
	}
	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 <= (-2.5d-289)) then
        tmp = x_m / (y * (t - z))
    else if (t <= 1.1d-6) then
        tmp = (x_m / z) / z
    else
        tmp = x_m / ((y - z) * t)
    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 <= -2.5e-289) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 1.1e-6) {
		tmp = (x_m / z) / z;
	} else {
		tmp = x_m / ((y - z) * t);
	}
	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 <= -2.5e-289:
		tmp = x_m / (y * (t - z))
	elif t <= 1.1e-6:
		tmp = (x_m / z) / z
	else:
		tmp = x_m / ((y - z) * t)
	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 <= -2.5e-289)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (t <= 1.1e-6)
		tmp = Float64(Float64(x_m / z) / z);
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	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 <= -2.5e-289)
		tmp = x_m / (y * (t - z));
	elseif (t <= 1.1e-6)
		tmp = (x_m / z) / z;
	else
		tmp = x_m / ((y - z) * t);
	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, -2.5e-289], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e-6], N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -2.50000000000000014e-289

   1. Initial program 86.7%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(t - z\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\left(t - z\right) \cdot \color{blue}{y}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(t - z\right), \color{blue}{y}\right)\right) \]

      4. --lowering--.f64 54.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(t, z\right), y\right)\right) \]

   5. Simplified 54.8%

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

   if -2.50000000000000014e-289 < t < 1.1000000000000001e-6

   1. Initial program 90.7%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 63.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 63.2%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{z}\right) \]

      3. /-lowering-/.f64 66.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), z\right) \]

   7. Applied egg-rr 66.6%

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

   if 1.1000000000000001e-6 < t

   1. Initial program 82.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 77.2%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-289}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{z}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -4.4e+114) t_1 (if (<= z 6.5e+57) (/ x_m (* y (- t 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 / z) / z;
	double tmp;
	if (z <= -4.4e+114) {
		tmp = t_1;
	} else if (z <= 6.5e+57) {
		tmp = x_m / (y * (t - 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 / z) / z
    if (z <= (-4.4d+114)) then
        tmp = t_1
    else if (z <= 6.5d+57) then
        tmp = x_m / (y * (t - 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 / z) / z;
	double tmp;
	if (z <= -4.4e+114) {
		tmp = t_1;
	} else if (z <= 6.5e+57) {
		tmp = x_m / (y * (t - 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 / z) / z
	tmp = 0
	if z <= -4.4e+114:
		tmp = t_1
	elif z <= 6.5e+57:
		tmp = x_m / (y * (t - 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(Float64(x_m / z) / z)
	tmp = 0.0
	if (z <= -4.4e+114)
		tmp = t_1;
	elseif (z <= 6.5e+57)
		tmp = Float64(x_m / Float64(y * Float64(t - 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 / z) / z;
	tmp = 0.0;
	if (z <= -4.4e+114)
		tmp = t_1;
	elseif (z <= 6.5e+57)
		tmp = x_m / (y * (t - 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[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -4.4e+114], t$95$1, If[LessEqual[z, 6.5e+57], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4000000000000001e114 or 6.4999999999999997e57 < z

   1. Initial program 77.9%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 67.3%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 67.3%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{z}\right) \]

      3. /-lowering-/.f64 74.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), z\right) \]

   7. Applied egg-rr 74.6%

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

   if -4.4000000000000001e114 < z < 6.4999999999999997e57

   1. Initial program 92.3%

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

   3. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(t - z\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\left(t - z\right) \cdot \color{blue}{y}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(t - z\right), \color{blue}{y}\right)\right) \]

      4. --lowering--.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(t, z\right), y\right)\right) \]

   5. Simplified 65.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{z}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3800000000000:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -2.45e+39)
      t_1
      (if (<= z 3800000000000.0) (/ (/ x_m y) t) 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 <= -2.45e+39) {
		tmp = t_1;
	} else if (z <= 3800000000000.0) {
		tmp = (x_m / y) / t;
	} 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 <= (-2.45d+39)) then
        tmp = t_1
    else if (z <= 3800000000000.0d0) then
        tmp = (x_m / y) / t
    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 <= -2.45e+39) {
		tmp = t_1;
	} else if (z <= 3800000000000.0) {
		tmp = (x_m / y) / t;
	} 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 <= -2.45e+39:
		tmp = t_1
	elif z <= 3800000000000.0:
		tmp = (x_m / y) / t
	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(Float64(x_m / z) / z)
	tmp = 0.0
	if (z <= -2.45e+39)
		tmp = t_1;
	elseif (z <= 3800000000000.0)
		tmp = Float64(Float64(x_m / y) / t);
	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 <= -2.45e+39)
		tmp = t_1;
	elseif (z <= 3800000000000.0)
		tmp = (x_m / y) / t;
	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[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -2.45e+39], t$95$1, If[LessEqual[z, 3800000000000.0], N[(N[(x$95$m / y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.44999999999999994e39 or 3.8e12 < z

   1. Initial program 78.3%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 59.1%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{z}\right) \]

      3. /-lowering-/.f64 64.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), z\right) \]

   7. Applied egg-rr 64.7%

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

   if -2.44999999999999994e39 < z < 3.8e12

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t \cdot y\right)}\right) \]

      2. *-lowering-*.f64 58.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{y}\right)\right) \]

   5. Simplified 58.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{t}\right) \]

      4. /-lowering-/.f64 61.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), t\right) \]

   7. Applied egg-rr 61.5%

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

Alternative 10: 64.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4000000000000:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -1.25e+39)
      t_1
      (if (<= z 4000000000000.0) (/ (/ x_m y) t) 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 <= -1.25e+39) {
		tmp = t_1;
	} else if (z <= 4000000000000.0) {
		tmp = (x_m / y) / t;
	} 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 <= (-1.25d+39)) then
        tmp = t_1
    else if (z <= 4000000000000.0d0) then
        tmp = (x_m / y) / t
    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 <= -1.25e+39) {
		tmp = t_1;
	} else if (z <= 4000000000000.0) {
		tmp = (x_m / y) / t;
	} 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 <= -1.25e+39:
		tmp = t_1
	elif z <= 4000000000000.0:
		tmp = (x_m / y) / t
	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 <= -1.25e+39)
		tmp = t_1;
	elseif (z <= 4000000000000.0)
		tmp = Float64(Float64(x_m / y) / t);
	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 <= -1.25e+39)
		tmp = t_1;
	elseif (z <= 4000000000000.0)
		tmp = (x_m / y) / t;
	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, -1.25e+39], t$95$1, If[LessEqual[z, 4000000000000.0], N[(N[(x$95$m / y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000004e39 or 4e12 < z

   1. Initial program 78.3%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 59.1%

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

   if -1.25000000000000004e39 < z < 4e12

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t \cdot y\right)}\right) \]

      2. *-lowering-*.f64 58.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{y}\right)\right) \]

   5. Simplified 58.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{t}\right) \]

      4. /-lowering-/.f64 61.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), t\right) \]

   7. Applied egg-rr 61.5%

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

Alternative 11: 64.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1460000000000:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -5.3e+32) t_1 (if (<= z 1460000000000.0) (/ (/ 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 <= -5.3e+32) {
		tmp = t_1;
	} else if (z <= 1460000000000.0) {
		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 <= (-5.3d+32)) then
        tmp = t_1
    else if (z <= 1460000000000.0d0) 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 <= -5.3e+32) {
		tmp = t_1;
	} else if (z <= 1460000000000.0) {
		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 <= -5.3e+32:
		tmp = t_1
	elif z <= 1460000000000.0:
		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 <= -5.3e+32)
		tmp = t_1;
	elseif (z <= 1460000000000.0)
		tmp = Float64(Float64(x_m / 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 <= -5.3e+32)
		tmp = t_1;
	elseif (z <= 1460000000000.0)
		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, -5.3e+32], t$95$1, If[LessEqual[z, 1460000000000.0], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2999999999999999e32 or 1.46e12 < z

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 59.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 59.4%

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

   if -5.2999999999999999e32 < z < 1.46e12

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t \cdot y\right)}\right) \]

      2. *-lowering-*.f64 58.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{y}\right)\right) \]

   5. Simplified 58.4%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{t}\right), \color{blue}{y}\right) \]

      3. /-lowering-/.f64 59.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, t\right), y\right) \]

   7. Applied egg-rr 59.9%

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

Alternative 12: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1500000000000:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -6.2e+30) t_1 (if (<= z 1500000000000.0) (/ x_m (* y t)) 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 <= -6.2e+30) {
		tmp = t_1;
	} else if (z <= 1500000000000.0) {
		tmp = x_m / (y * t);
	} 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 <= (-6.2d+30)) then
        tmp = t_1
    else if (z <= 1500000000000.0d0) then
        tmp = x_m / (y * t)
    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 <= -6.2e+30) {
		tmp = t_1;
	} else if (z <= 1500000000000.0) {
		tmp = x_m / (y * t);
	} 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 <= -6.2e+30:
		tmp = t_1
	elif z <= 1500000000000.0:
		tmp = x_m / (y * t)
	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 <= -6.2e+30)
		tmp = t_1;
	elseif (z <= 1500000000000.0)
		tmp = Float64(x_m / Float64(y * t));
	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 <= -6.2e+30)
		tmp = t_1;
	elseif (z <= 1500000000000.0)
		tmp = x_m / (y * t);
	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, -6.2e+30], t$95$1, If[LessEqual[z, 1500000000000.0], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.1999999999999995e30 or 1.5e12 < z

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 59.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 59.4%

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

   if -6.1999999999999995e30 < z < 1.5e12

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t \cdot y\right)}\right) \]

      2. *-lowering-*.f64 58.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{y}\right)\right) \]

   5. Simplified 58.4%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 1500000000000:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.0% accurate, 1.0× speedup

Math

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

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 (- y z)) (- t 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 / (y - z)) / (t - 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 / (y - z)) / (t - 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 / (y - z)) / (t - 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 / (y - z)) / (t - 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(Float64(x_m / Float64(y - z)) / Float64(t - 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 / (y - z)) / (t - 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[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.8%

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

   1. associate-/r* N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(t - z\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\color{blue}{t} - z\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - z\right)\right) \]

   5. --lowering--.f64 97.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{z}\right)\right) \]

4. Applied egg-rr 97.8%

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

Alternative 14: 39.5% accurate, 1.8× speedup

Math

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

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 (* y t))))

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 / (y * t));
}

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 / (y * t))
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 / (y * t));
}

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 / (y * t))

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(y * t)))
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 / (y * t));
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[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in z around 0

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

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

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t \cdot y\right)}\right) \]

   2. *-lowering-*.f64 34.3%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{y}\right)\right) \]

5. Simplified 34.3%

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

6. Final simplification 34.3%

   \[\leadsto \frac{x}{y \cdot t} \]
7. Add Preprocessing

Developer Target 1: 87.7% 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 2024161 
(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))))