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

Percentage Accurate: 89.2% → 97.8%

Time: 11.3s

Alternatives: 13

Speedup: 0.5×

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: 89.2% 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: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \end{array} \end{array} \end{array} \]

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 -1e-299) t_1 (/ (/ x_m (- t z)) (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / ((y - z) * (t - z));
	double tmp;
	if (t_1 <= -1e-299) {
		tmp = t_1;
	} else {
		tmp = (x_m / (t - z)) / (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) :: t_1
    real(8) :: tmp
    t_1 = x_m / ((y - z) * (t - z))
    if (t_1 <= (-1d-299)) then
        tmp = t_1
    else
        tmp = (x_m / (t - z)) / (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 t_1 = x_m / ((y - z) * (t - z));
	double tmp;
	if (t_1 <= -1e-299) {
		tmp = t_1;
	} else {
		tmp = (x_m / (t - z)) / (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):
	t_1 = x_m / ((y - z) * (t - z))
	tmp = 0
	if t_1 <= -1e-299:
		tmp = t_1
	else:
		tmp = (x_m / (t - z)) / (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)
	t_1 = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)))
	tmp = 0.0
	if (t_1 <= -1e-299)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) / 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)
	t_1 = x_m / ((y - z) * (t - z));
	tmp = 0.0;
	if (t_1 <= -1e-299)
		tmp = t_1;
	else
		tmp = (x_m / (t - z)) / (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_] := 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, -1e-299], t$95$1, N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < -9.99999999999999992e-300

   1. Initial program 99.7%

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

   if -9.99999999999999992e-300 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 84.4%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 99.3%

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

   4. Applied egg-rr 99.3%

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

Alternative 2: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+236}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 -1.02e-174)
    (/ (/ x_m (- t z)) y)
    (if (<= t 4.8e-24)
      (/ (/ x_m z) (- z y))
      (if (<= t 1.35e+236) (/ x_m (* (- y z) t)) (/ (/ 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 <= -1.02e-174) {
		tmp = (x_m / (t - z)) / y;
	} else if (t <= 4.8e-24) {
		tmp = (x_m / z) / (z - y);
	} else if (t <= 1.35e+236) {
		tmp = x_m / ((y - z) * t);
	} 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 <= (-1.02d-174)) then
        tmp = (x_m / (t - z)) / y
    else if (t <= 4.8d-24) then
        tmp = (x_m / z) / (z - y)
    else if (t <= 1.35d+236) then
        tmp = x_m / ((y - z) * t)
    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 <= -1.02e-174) {
		tmp = (x_m / (t - z)) / y;
	} else if (t <= 4.8e-24) {
		tmp = (x_m / z) / (z - y);
	} else if (t <= 1.35e+236) {
		tmp = x_m / ((y - z) * t);
	} 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 <= -1.02e-174:
		tmp = (x_m / (t - z)) / y
	elif t <= 4.8e-24:
		tmp = (x_m / z) / (z - y)
	elif t <= 1.35e+236:
		tmp = x_m / ((y - z) * t)
	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 <= -1.02e-174)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	elseif (t <= 4.8e-24)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (t <= 1.35e+236)
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	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 <= -1.02e-174)
		tmp = (x_m / (t - z)) / y;
	elseif (t <= 4.8e-24)
		tmp = (x_m / z) / (z - y);
	elseif (t <= 1.35e+236)
		tmp = x_m / ((y - z) * t);
	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, -1.02e-174], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t, 4.8e-24], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+236], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < -1.02000000000000011e-174

   1. Initial program 85.7%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 61.0%

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

   5. Simplified 61.0%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. --lowering--.f64 68.4%

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

   7. Applied egg-rr 68.4%

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

   if -1.02000000000000011e-174 < t < 4.7999999999999996e-24

   1. Initial program 89.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\frac{{t}^{3} - {z}^{3}}{t \cdot t + \left(z \cdot z + t \cdot z\right)}}}{y - z} \cdot x \]

      5. clear-num N/A

         \[\leadsto \frac{\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}}}{y - z} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}} \cdot x}{\color{blue}{y - z}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}} \cdot x\right), \color{blue}{\left(y - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}}\right), x\right), \left(\color{blue}{y} - z\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{{t}^{3} - {z}^{3}}{t \cdot t + \left(z \cdot z + t \cdot z\right)}}\right), x\right), \left(y - z\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{t - z}\right), x\right), \left(y - z\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(t - z\right)\right), x\right), \left(y - z\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(t, z\right)\right), x\right), \left(y - z\right)\right) \]

      13. --lowering--.f64 98.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(t, z\right)\right), x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 83.8%

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

   7. Simplified 83.8%

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

   if 4.7999999999999996e-24 < t < 1.3500000000000001e236

   1. Initial program 94.6%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 85.1%

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

   if 1.3500000000000001e236 < t

   1. Initial program 78.3%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 88.7%

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

   5. Simplified 88.7%

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

      1. associate-/l/ N/A

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

      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 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+236}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-264}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+234}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 -1.15e-264)
    (/ (/ x_m (- t z)) y)
    (if (<= t 3.6e-37)
      (/ (/ x_m z) z)
      (if (<= t 7e+234) (/ x_m (* (- y z) t)) (/ (/ 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 <= -1.15e-264) {
		tmp = (x_m / (t - z)) / y;
	} else if (t <= 3.6e-37) {
		tmp = (x_m / z) / z;
	} else if (t <= 7e+234) {
		tmp = x_m / ((y - z) * t);
	} 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 <= (-1.15d-264)) then
        tmp = (x_m / (t - z)) / y
    else if (t <= 3.6d-37) then
        tmp = (x_m / z) / z
    else if (t <= 7d+234) then
        tmp = x_m / ((y - z) * t)
    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 <= -1.15e-264) {
		tmp = (x_m / (t - z)) / y;
	} else if (t <= 3.6e-37) {
		tmp = (x_m / z) / z;
	} else if (t <= 7e+234) {
		tmp = x_m / ((y - z) * t);
	} 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 <= -1.15e-264:
		tmp = (x_m / (t - z)) / y
	elif t <= 3.6e-37:
		tmp = (x_m / z) / z
	elif t <= 7e+234:
		tmp = x_m / ((y - z) * t)
	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 <= -1.15e-264)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	elseif (t <= 3.6e-37)
		tmp = Float64(Float64(x_m / z) / z);
	elseif (t <= 7e+234)
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	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 <= -1.15e-264)
		tmp = (x_m / (t - z)) / y;
	elseif (t <= 3.6e-37)
		tmp = (x_m / z) / z;
	elseif (t <= 7e+234)
		tmp = x_m / ((y - z) * t);
	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, -1.15e-264], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t, 3.6e-37], N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 7e+234], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < -1.15000000000000006e-264

   1. Initial program 85.5%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 61.3%

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

   5. Simplified 61.3%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. --lowering--.f64 69.2%

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

   7. Applied egg-rr 69.2%

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

   if -1.15000000000000006e-264 < t < 3.60000000000000007e-37

   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 35.5%

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

   5. Simplified 35.5%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z}} \]
   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 44.4%

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

   7. Applied egg-rr 44.4%

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

   if 3.60000000000000007e-37 < t < 7.00000000000000065e234

   1. Initial program 94.7%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 83.8%

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

   if 7.00000000000000065e234 < t

   1. Initial program 78.3%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 88.7%

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

   5. Simplified 88.7%

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

      1. associate-/l/ N/A

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

      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 99.9%

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

   7. Applied egg-rr 99.9%

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

Alternative 4: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-273}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+234}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 -4.3e-273)
    (/ (/ x_m y) (- t z))
    (if (<= t 4.5e-37)
      (/ (/ x_m z) z)
      (if (<= t 3.3e+234) (/ x_m (* (- y z) t)) (/ (/ 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 <= -4.3e-273) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 4.5e-37) {
		tmp = (x_m / z) / z;
	} else if (t <= 3.3e+234) {
		tmp = x_m / ((y - z) * t);
	} 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 <= (-4.3d-273)) then
        tmp = (x_m / y) / (t - z)
    else if (t <= 4.5d-37) then
        tmp = (x_m / z) / z
    else if (t <= 3.3d+234) then
        tmp = x_m / ((y - z) * t)
    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 <= -4.3e-273) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 4.5e-37) {
		tmp = (x_m / z) / z;
	} else if (t <= 3.3e+234) {
		tmp = x_m / ((y - z) * t);
	} 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 <= -4.3e-273:
		tmp = (x_m / y) / (t - z)
	elif t <= 4.5e-37:
		tmp = (x_m / z) / z
	elif t <= 3.3e+234:
		tmp = x_m / ((y - z) * t)
	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 <= -4.3e-273)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (t <= 4.5e-37)
		tmp = Float64(Float64(x_m / z) / z);
	elseif (t <= 3.3e+234)
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	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 <= -4.3e-273)
		tmp = (x_m / y) / (t - z);
	elseif (t <= 4.5e-37)
		tmp = (x_m / z) / z;
	elseif (t <= 3.3e+234)
		tmp = x_m / ((y - z) * t);
	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, -4.3e-273], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e-37], N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 3.3e+234], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < -4.3000000000000004e-273

   1. Initial program 85.3%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 62.0%

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

   5. Simplified 62.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. --lowering--.f64 64.7%

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

   7. Applied egg-rr 64.7%

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

   if -4.3000000000000004e-273 < t < 4.5000000000000004e-37

   1. Initial program 91.7%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 38.3%

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

   5. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z}} \]
   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 46.4%

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

   7. Applied egg-rr 46.4%

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

   if 4.5000000000000004e-37 < t < 3.3000000000000003e234

   1. Initial program 94.7%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 83.8%

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

   if 3.3000000000000003e234 < t

   1. Initial program 78.3%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 88.7%

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

   5. Simplified 88.7%

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

      1. associate-/l/ N/A

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

      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 99.9%

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

   7. Applied egg-rr 99.9%

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

Alternative 5: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+234}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 -9.5e-271)
    (/ x_m (* y (- t z)))
    (if (<= t 3.6e-37)
      (/ (/ x_m z) z)
      (if (<= t 2.7e+234) (/ x_m (* (- y z) t)) (/ (/ 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 <= -9.5e-271) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 3.6e-37) {
		tmp = (x_m / z) / z;
	} else if (t <= 2.7e+234) {
		tmp = x_m / ((y - z) * t);
	} 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 <= (-9.5d-271)) then
        tmp = x_m / (y * (t - z))
    else if (t <= 3.6d-37) then
        tmp = (x_m / z) / z
    else if (t <= 2.7d+234) then
        tmp = x_m / ((y - z) * t)
    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 <= -9.5e-271) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 3.6e-37) {
		tmp = (x_m / z) / z;
	} else if (t <= 2.7e+234) {
		tmp = x_m / ((y - z) * t);
	} 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 <= -9.5e-271:
		tmp = x_m / (y * (t - z))
	elif t <= 3.6e-37:
		tmp = (x_m / z) / z
	elif t <= 2.7e+234:
		tmp = x_m / ((y - z) * t)
	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 <= -9.5e-271)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (t <= 3.6e-37)
		tmp = Float64(Float64(x_m / z) / z);
	elseif (t <= 2.7e+234)
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	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 <= -9.5e-271)
		tmp = x_m / (y * (t - z));
	elseif (t <= 3.6e-37)
		tmp = (x_m / z) / z;
	elseif (t <= 2.7e+234)
		tmp = x_m / ((y - z) * t);
	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, -9.5e-271], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e-37], N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 2.7e+234], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < -9.50000000000000103e-271

   1. Initial program 84.9%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 61.1%

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

   5. Simplified 61.1%

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

   if -9.50000000000000103e-271 < t < 3.60000000000000007e-37

   1. Initial program 92.1%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 36.5%

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

   5. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z}} \]
   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 44.2%

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

   7. Applied egg-rr 44.2%

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

   if 3.60000000000000007e-37 < t < 2.7000000000000002e234

   1. Initial program 94.7%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 83.8%

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

   if 2.7000000000000002e234 < t

   1. Initial program 78.3%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 88.7%

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

   5. Simplified 88.7%

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

      1. associate-/l/ N/A

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

      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 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+234}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -4.8e+151)
    (/ (/ x_m z) (- z t))
    (if (<= z 2.1e+146) (/ x_m (* (- y z) (- t z))) (/ (/ x_m z) (- z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -4.8e+151) {
		tmp = (x_m / z) / (z - t);
	} else if (z <= 2.1e+146) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / z) / (z - y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-4.8d+151)) then
        tmp = (x_m / z) / (z - t)
    else if (z <= 2.1d+146) then
        tmp = x_m / ((y - z) * (t - z))
    else
        tmp = (x_m / z) / (z - y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -4.8e+151) {
		tmp = (x_m / z) / (z - t);
	} else if (z <= 2.1e+146) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / z) / (z - y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -4.8e+151:
		tmp = (x_m / z) / (z - t)
	elif z <= 2.1e+146:
		tmp = x_m / ((y - z) * (t - z))
	else:
		tmp = (x_m / z) / (z - y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -4.8e+151)
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	elseif (z <= 2.1e+146)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -4.8e+151)
		tmp = (x_m / z) / (z - t);
	elseif (z <= 2.1e+146)
		tmp = x_m / ((y - z) * (t - z));
	else
		tmp = (x_m / z) / (z - y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -4.8e+151], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e+146], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -4.8000000000000002e151

   1. Initial program 64.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\frac{{t}^{3} - {z}^{3}}{t \cdot t + \left(z \cdot z + t \cdot z\right)}}}{y - z} \cdot x \]

      5. clear-num N/A

         \[\leadsto \frac{\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}}}{y - z} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}} \cdot x}{\color{blue}{y - z}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}} \cdot x\right), \color{blue}{\left(y - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}}\right), x\right), \left(\color{blue}{y} - z\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{{t}^{3} - {z}^{3}}{t \cdot t + \left(z \cdot z + t \cdot z\right)}}\right), x\right), \left(y - z\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{t - z}\right), x\right), \left(y - z\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(t - z\right)\right), x\right), \left(y - z\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(t, z\right)\right), x\right), \left(y - z\right)\right) \]

      13. --lowering--.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(t, z\right)\right), x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 96.1%

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

   7. Simplified 96.1%

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

   if -4.8000000000000002e151 < z < 2.1000000000000001e146

   1. Initial program 93.4%

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

   if 2.1000000000000001e146 < z

   1. Initial program 75.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\frac{{t}^{3} - {z}^{3}}{t \cdot t + \left(z \cdot z + t \cdot z\right)}}}{y - z} \cdot x \]

      5. clear-num N/A

         \[\leadsto \frac{\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}}}{y - z} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}} \cdot x}{\color{blue}{y - z}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}} \cdot x\right), \color{blue}{\left(y - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}}\right), x\right), \left(\color{blue}{y} - z\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{{t}^{3} - {z}^{3}}{t \cdot t + \left(z \cdot z + t \cdot z\right)}}\right), x\right), \left(y - z\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{t - z}\right), x\right), \left(y - z\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(t - z\right)\right), x\right), \left(y - z\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(t, z\right)\right), x\right), \left(y - z\right)\right) \]

      13. --lowering--.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(t, z\right)\right), x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 92.8%

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

   7. Simplified 92.8%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t}\\ \end{array} \end{array} \]

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 (<= y -9.8e-73)
    (/ (/ x_m (- t z)) y)
    (if (<= y 2.9e-124) (/ (/ x_m z) (- z t)) (/ (/ 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 (y <= -9.8e-73) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= 2.9e-124) {
		tmp = (x_m / z) / (z - t);
	} 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 (y <= (-9.8d-73)) then
        tmp = (x_m / (t - z)) / y
    else if (y <= 2.9d-124) then
        tmp = (x_m / z) / (z - t)
    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 (y <= -9.8e-73) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= 2.9e-124) {
		tmp = (x_m / z) / (z - t);
	} 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 y <= -9.8e-73:
		tmp = (x_m / (t - z)) / y
	elif y <= 2.9e-124:
		tmp = (x_m / z) / (z - t)
	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 (y <= -9.8e-73)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	elseif (y <= 2.9e-124)
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	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 (y <= -9.8e-73)
		tmp = (x_m / (t - z)) / y;
	elseif (y <= 2.9e-124)
		tmp = (x_m / z) / (z - t);
	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[y, -9.8e-73], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2.9e-124], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -9.80000000000000057e-73

   1. Initial program 87.3%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 75.5%

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

   5. Simplified 75.5%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. --lowering--.f64 78.7%

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

   7. Applied egg-rr 78.7%

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

   if -9.80000000000000057e-73 < y < 2.9000000000000002e-124

   1. Initial program 90.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\frac{{t}^{3} - {z}^{3}}{t \cdot t + \left(z \cdot z + t \cdot z\right)}}}{y - z} \cdot x \]

      5. clear-num N/A

         \[\leadsto \frac{\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}}}{y - z} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}} \cdot x}{\color{blue}{y - z}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}} \cdot x\right), \color{blue}{\left(y - z\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot t + \left(z \cdot z + t \cdot z\right)}{{t}^{3} - {z}^{3}}\right), x\right), \left(\color{blue}{y} - z\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{{t}^{3} - {z}^{3}}{t \cdot t + \left(z \cdot z + t \cdot z\right)}}\right), x\right), \left(y - z\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{t - z}\right), x\right), \left(y - z\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(t - z\right)\right), x\right), \left(y - z\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(t, z\right)\right), x\right), \left(y - z\right)\right) \]

      13. --lowering--.f64 97.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(t, z\right)\right), x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 83.3%

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

   7. Simplified 83.3%

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

   if 2.9000000000000002e-124 < y

   1. Initial program 87.3%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 65.2%

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

   5. Simplified 65.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-273}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

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 -3.8e-273)
    (/ x_m (* y (- t z)))
    (if (<= t 7e-37) (/ (/ 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 <= -3.8e-273) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 7e-37) {
		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 <= (-3.8d-273)) then
        tmp = x_m / (y * (t - z))
    else if (t <= 7d-37) 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 <= -3.8e-273) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 7e-37) {
		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 <= -3.8e-273:
		tmp = x_m / (y * (t - z))
	elif t <= 7e-37:
		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 <= -3.8e-273)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (t <= 7e-37)
		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 <= -3.8e-273)
		tmp = x_m / (y * (t - z));
	elseif (t <= 7e-37)
		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, -3.8e-273], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e-37], 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 < -3.8000000000000004e-273

   1. Initial program 85.3%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 62.0%

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

   5. Simplified 62.0%

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

   if -3.8000000000000004e-273 < t < 7.0000000000000003e-37

   1. Initial program 91.7%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 38.3%

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

   5. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z}} \]
   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 46.4%

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

   7. Applied egg-rr 46.4%

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

   if 7.0000000000000003e-37 < t

   1. Initial program 90.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 82.5%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-273}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{z}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -6e+44) t_1 (if (<= z 6.5e-26) (/ 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 <= -6e+44) {
		tmp = t_1;
	} else if (z <= 6.5e-26) {
		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 <= (-6d+44)) then
        tmp = t_1
    else if (z <= 6.5d-26) 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 <= -6e+44) {
		tmp = t_1;
	} else if (z <= 6.5e-26) {
		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 <= -6e+44:
		tmp = t_1
	elif z <= 6.5e-26:
		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 <= -6e+44)
		tmp = t_1;
	elseif (z <= 6.5e-26)
		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 <= -6e+44)
		tmp = t_1;
	elseif (z <= 6.5e-26)
		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, -6e+44], t$95$1, If[LessEqual[z, 6.5e-26], 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 < -5.99999999999999974e44 or 6.5e-26 < z

   1. Initial program 81.9%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 60.5%

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

   5. Simplified 60.5%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z}} \]
   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 70.4%

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

   7. Applied egg-rr 70.4%

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

   if -5.99999999999999974e44 < z < 6.5e-26

   1. Initial program 93.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 73.4%

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

   5. Simplified 73.4%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{z}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -9e+31) t_1 (if (<= z 7.4e-26) (/ (/ 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 <= -9e+31) {
		tmp = t_1;
	} else if (z <= 7.4e-26) {
		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 <= (-9d+31)) then
        tmp = t_1
    else if (z <= 7.4d-26) 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 <= -9e+31) {
		tmp = t_1;
	} else if (z <= 7.4e-26) {
		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 <= -9e+31:
		tmp = t_1
	elif z <= 7.4e-26:
		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(Float64(x_m / z) / z)
	tmp = 0.0
	if (z <= -9e+31)
		tmp = t_1;
	elseif (z <= 7.4e-26)
		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 <= -9e+31)
		tmp = t_1;
	elseif (z <= 7.4e-26)
		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[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -9e+31], t$95$1, If[LessEqual[z, 7.4e-26], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.9999999999999992e31 or 7.3999999999999997e-26 < z

   1. Initial program 82.0%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 60.1%

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

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z}} \]
   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 69.9%

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

   7. Applied egg-rr 69.9%

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

   if -8.9999999999999992e31 < z < 7.3999999999999997e-26

   1. Initial program 93.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 56.3%

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

   5. Simplified 56.3%

      \[\leadsto \color{blue}{\frac{x}{t \cdot y}} \]
   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 60.4%

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

   7. Applied egg-rr 60.4%

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

Alternative 11: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -6.2e+31) t_1 (if (<= z 7.4e-26) (/ (/ 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 <= -6.2e+31) {
		tmp = t_1;
	} else if (z <= 7.4e-26) {
		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 <= (-6.2d+31)) then
        tmp = t_1
    else if (z <= 7.4d-26) 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 <= -6.2e+31) {
		tmp = t_1;
	} else if (z <= 7.4e-26) {
		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 <= -6.2e+31:
		tmp = t_1
	elif z <= 7.4e-26:
		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 <= -6.2e+31)
		tmp = t_1;
	elseif (z <= 7.4e-26)
		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 <= -6.2e+31)
		tmp = t_1;
	elseif (z <= 7.4e-26)
		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, -6.2e+31], t$95$1, If[LessEqual[z, 7.4e-26], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000004e31 or 7.3999999999999997e-26 < z

   1. Initial program 82.0%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 60.1%

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

   5. Simplified 60.1%

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

   if -6.2000000000000004e31 < z < 7.3999999999999997e-26

   1. Initial program 93.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 56.3%

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

   5. Simplified 56.3%

      \[\leadsto \color{blue}{\frac{x}{t \cdot y}} \]
   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 60.4%

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

   7. Applied egg-rr 60.4%

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

Alternative 12: 61.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -3e+28) t_1 (if (<= z 4.7e-26) (/ 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 <= -3e+28) {
		tmp = t_1;
	} else if (z <= 4.7e-26) {
		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 <= (-3d+28)) then
        tmp = t_1
    else if (z <= 4.7d-26) 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 <= -3e+28) {
		tmp = t_1;
	} else if (z <= 4.7e-26) {
		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 <= -3e+28:
		tmp = t_1
	elif z <= 4.7e-26:
		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 <= -3e+28)
		tmp = t_1;
	elseif (z <= 4.7e-26)
		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 <= -3e+28)
		tmp = t_1;
	elseif (z <= 4.7e-26)
		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, -3e+28], t$95$1, If[LessEqual[z, 4.7e-26], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0000000000000001e28 or 4.69999999999999989e-26 < z

   1. Initial program 82.0%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 60.1%

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

   5. Simplified 60.1%

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

   if -3.0000000000000001e28 < z < 4.69999999999999989e-26

   1. Initial program 93.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 56.3%

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

   5. Simplified 56.3%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.7% 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 88.3%

   \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
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 36.6%

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

5. Simplified 36.6%

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

6. Final simplification 36.6%

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

Developer Target 1: 88.2% 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 2024140 
(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))))