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

Percentage Accurate: 88.8% → 97.4%

Time: 14.2s

Alternatives: 18

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% 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: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-243}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \end{array} \end{array} \end{array} \]

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-243) 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-243) {
		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-243)) 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-243) {
		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-243:
		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-243)
		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-243)
		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-243], 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))) < -9.99999999999999995e-244

   1. Initial program 96.0%

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

   if -9.99999999999999995e-244 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 86.1%

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

      1. associate-/l/ 98.0%

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

   3. Simplified 98.0%

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

Alternative 2: 69.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{t}}{y}\\ t_2 := \frac{x\_m}{z \cdot \left(z - y\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-25}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{1}{t}}{\frac{y}{x\_m}}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{1}{z}\\ \end{array} \end{array} \end{array} \]

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 t) y)) (t_2 (/ x_m (* z (- z y)))))
   (*
    x_s
    (if (<= z -3.8e-25)
      t_2
      (if (<= z -2.35e-122)
        t_1
        (if (<= z -1.5e-144)
          t_2
          (if (<= z 2.1e-202)
            (/ (/ 1.0 t) (/ y x_m))
            (if (<= z 3.2e-186)
              (/ x_m (* z (- z t)))
              (if (<= z 5.6e-39)
                t_1
                (if (<= z 3.6e+138) t_2 (* (/ x_m z) (/ 1.0 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 / t) / y;
	double t_2 = x_m / (z * (z - y));
	double tmp;
	if (z <= -3.8e-25) {
		tmp = t_2;
	} else if (z <= -2.35e-122) {
		tmp = t_1;
	} else if (z <= -1.5e-144) {
		tmp = t_2;
	} else if (z <= 2.1e-202) {
		tmp = (1.0 / t) / (y / x_m);
	} else if (z <= 3.2e-186) {
		tmp = x_m / (z * (z - t));
	} else if (z <= 5.6e-39) {
		tmp = t_1;
	} else if (z <= 3.6e+138) {
		tmp = t_2;
	} else {
		tmp = (x_m / z) * (1.0 / 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) :: t_2
    real(8) :: tmp
    t_1 = (x_m / t) / y
    t_2 = x_m / (z * (z - y))
    if (z <= (-3.8d-25)) then
        tmp = t_2
    else if (z <= (-2.35d-122)) then
        tmp = t_1
    else if (z <= (-1.5d-144)) then
        tmp = t_2
    else if (z <= 2.1d-202) then
        tmp = (1.0d0 / t) / (y / x_m)
    else if (z <= 3.2d-186) then
        tmp = x_m / (z * (z - t))
    else if (z <= 5.6d-39) then
        tmp = t_1
    else if (z <= 3.6d+138) then
        tmp = t_2
    else
        tmp = (x_m / z) * (1.0d0 / 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 / t) / y;
	double t_2 = x_m / (z * (z - y));
	double tmp;
	if (z <= -3.8e-25) {
		tmp = t_2;
	} else if (z <= -2.35e-122) {
		tmp = t_1;
	} else if (z <= -1.5e-144) {
		tmp = t_2;
	} else if (z <= 2.1e-202) {
		tmp = (1.0 / t) / (y / x_m);
	} else if (z <= 3.2e-186) {
		tmp = x_m / (z * (z - t));
	} else if (z <= 5.6e-39) {
		tmp = t_1;
	} else if (z <= 3.6e+138) {
		tmp = t_2;
	} else {
		tmp = (x_m / z) * (1.0 / 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 / t) / y
	t_2 = x_m / (z * (z - y))
	tmp = 0
	if z <= -3.8e-25:
		tmp = t_2
	elif z <= -2.35e-122:
		tmp = t_1
	elif z <= -1.5e-144:
		tmp = t_2
	elif z <= 2.1e-202:
		tmp = (1.0 / t) / (y / x_m)
	elif z <= 3.2e-186:
		tmp = x_m / (z * (z - t))
	elif z <= 5.6e-39:
		tmp = t_1
	elif z <= 3.6e+138:
		tmp = t_2
	else:
		tmp = (x_m / z) * (1.0 / 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(Float64(x_m / t) / y)
	t_2 = Float64(x_m / Float64(z * Float64(z - y)))
	tmp = 0.0
	if (z <= -3.8e-25)
		tmp = t_2;
	elseif (z <= -2.35e-122)
		tmp = t_1;
	elseif (z <= -1.5e-144)
		tmp = t_2;
	elseif (z <= 2.1e-202)
		tmp = Float64(Float64(1.0 / t) / Float64(y / x_m));
	elseif (z <= 3.2e-186)
		tmp = Float64(x_m / Float64(z * Float64(z - t)));
	elseif (z <= 5.6e-39)
		tmp = t_1;
	elseif (z <= 3.6e+138)
		tmp = t_2;
	else
		tmp = Float64(Float64(x_m / z) * Float64(1.0 / 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 / t) / y;
	t_2 = x_m / (z * (z - y));
	tmp = 0.0;
	if (z <= -3.8e-25)
		tmp = t_2;
	elseif (z <= -2.35e-122)
		tmp = t_1;
	elseif (z <= -1.5e-144)
		tmp = t_2;
	elseif (z <= 2.1e-202)
		tmp = (1.0 / t) / (y / x_m);
	elseif (z <= 3.2e-186)
		tmp = x_m / (z * (z - t));
	elseif (z <= 5.6e-39)
		tmp = t_1;
	elseif (z <= 3.6e+138)
		tmp = t_2;
	else
		tmp = (x_m / z) * (1.0 / 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[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m / N[(z * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3.8e-25], t$95$2, If[LessEqual[z, -2.35e-122], t$95$1, If[LessEqual[z, -1.5e-144], t$95$2, If[LessEqual[z, 2.1e-202], N[(N[(1.0 / t), $MachinePrecision] / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e-186], N[(x$95$m / N[(z * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.6e-39], t$95$1, If[LessEqual[z, 3.6e+138], t$95$2, N[(N[(x$95$m / z), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if z < -3.7999999999999998e-25 or -2.35e-122 < z < -1.4999999999999999e-144 or 5.6000000000000003e-39 < z < 3.6000000000000001e138

   1. Initial program 93.5%

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

   3. Taylor expanded in t around 0 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-rgt-neg-in 74.4%

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

      3. neg-sub0 74.4%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 74.4%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 74.4%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 74.4%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 74.4%

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

      8. remove-double-neg 74.4%

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

   5. Simplified 74.4%

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

   if -3.7999999999999998e-25 < z < -2.35e-122 or 3.2e-186 < z < 5.6000000000000003e-39

   1. Initial program 88.7%

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

      1. associate-/l/ 93.5%

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

      2. clear-num 93.6%

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

      3. inv-pow 93.6%

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

      4. div-inv 93.5%

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

      5. clear-num 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. unpow-1 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in z around 0 57.9%

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

      1. associate-/r* 64.7%

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

   9. Simplified 64.7%

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

   if -1.4999999999999999e-144 < z < 2.09999999999999985e-202

   1. Initial program 85.4%

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

      1. associate-/r* 91.5%

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

      2. div-inv 91.5%

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

   4. Applied egg-rr 91.5%

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

   5. Taylor expanded in t around inf 87.1%

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

   6. Taylor expanded in y around inf 78.9%

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

      1. *-commutative 78.9%

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

      2. clear-num 78.9%

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

      3. un-div-inv 79.6%

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

   8. Applied egg-rr 79.6%

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

   if 2.09999999999999985e-202 < z < 3.2e-186

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 50.3%

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

      1. mul-1-neg 50.3%

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

      2. distribute-rgt-neg-in 50.3%

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

      3. sub-neg 50.3%

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

      4. +-commutative 50.3%

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

      5. distribute-neg-in 50.3%

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

      6. remove-double-neg 50.3%

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

      7. unsub-neg 50.3%

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

   5. Simplified 50.3%

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

   if 3.6000000000000001e138 < z

   1. Initial program 71.6%

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

   3. Taylor expanded in t around 0 69.7%

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

      1. mul-1-neg 69.7%

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

      2. distribute-rgt-neg-in 69.7%

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

      3. neg-sub0 69.7%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 69.7%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 69.7%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 69.7%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 69.7%

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

      8. remove-double-neg 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in z around inf 69.7%

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

      1. associate-/r* 92.1%

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

      2. div-inv 92.1%

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

   8. Applied egg-rr 92.1%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{1}{z}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 69.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{t}}{y}\\ t_2 := \frac{x\_m}{z \cdot \left(z - y\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{-25}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-143}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{1}{z}\\ \end{array} \end{array} \end{array} \]

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 t) y)) (t_2 (/ x_m (* z (- z y)))))
   (*
    x_s
    (if (<= z -9.8e-25)
      t_2
      (if (<= z -2.45e-122)
        t_1
        (if (<= z -7.6e-143)
          t_2
          (if (<= z 2.1e-202)
            (/ (/ x_m y) t)
            (if (<= z 3.5e-186)
              (/ x_m (* z (- z t)))
              (if (<= z 3.4e-39)
                t_1
                (if (<= z 3.8e+138) t_2 (* (/ x_m z) (/ 1.0 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 / t) / y;
	double t_2 = x_m / (z * (z - y));
	double tmp;
	if (z <= -9.8e-25) {
		tmp = t_2;
	} else if (z <= -2.45e-122) {
		tmp = t_1;
	} else if (z <= -7.6e-143) {
		tmp = t_2;
	} else if (z <= 2.1e-202) {
		tmp = (x_m / y) / t;
	} else if (z <= 3.5e-186) {
		tmp = x_m / (z * (z - t));
	} else if (z <= 3.4e-39) {
		tmp = t_1;
	} else if (z <= 3.8e+138) {
		tmp = t_2;
	} else {
		tmp = (x_m / z) * (1.0 / 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) :: t_2
    real(8) :: tmp
    t_1 = (x_m / t) / y
    t_2 = x_m / (z * (z - y))
    if (z <= (-9.8d-25)) then
        tmp = t_2
    else if (z <= (-2.45d-122)) then
        tmp = t_1
    else if (z <= (-7.6d-143)) then
        tmp = t_2
    else if (z <= 2.1d-202) then
        tmp = (x_m / y) / t
    else if (z <= 3.5d-186) then
        tmp = x_m / (z * (z - t))
    else if (z <= 3.4d-39) then
        tmp = t_1
    else if (z <= 3.8d+138) then
        tmp = t_2
    else
        tmp = (x_m / z) * (1.0d0 / 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 / t) / y;
	double t_2 = x_m / (z * (z - y));
	double tmp;
	if (z <= -9.8e-25) {
		tmp = t_2;
	} else if (z <= -2.45e-122) {
		tmp = t_1;
	} else if (z <= -7.6e-143) {
		tmp = t_2;
	} else if (z <= 2.1e-202) {
		tmp = (x_m / y) / t;
	} else if (z <= 3.5e-186) {
		tmp = x_m / (z * (z - t));
	} else if (z <= 3.4e-39) {
		tmp = t_1;
	} else if (z <= 3.8e+138) {
		tmp = t_2;
	} else {
		tmp = (x_m / z) * (1.0 / 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 / t) / y
	t_2 = x_m / (z * (z - y))
	tmp = 0
	if z <= -9.8e-25:
		tmp = t_2
	elif z <= -2.45e-122:
		tmp = t_1
	elif z <= -7.6e-143:
		tmp = t_2
	elif z <= 2.1e-202:
		tmp = (x_m / y) / t
	elif z <= 3.5e-186:
		tmp = x_m / (z * (z - t))
	elif z <= 3.4e-39:
		tmp = t_1
	elif z <= 3.8e+138:
		tmp = t_2
	else:
		tmp = (x_m / z) * (1.0 / 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(Float64(x_m / t) / y)
	t_2 = Float64(x_m / Float64(z * Float64(z - y)))
	tmp = 0.0
	if (z <= -9.8e-25)
		tmp = t_2;
	elseif (z <= -2.45e-122)
		tmp = t_1;
	elseif (z <= -7.6e-143)
		tmp = t_2;
	elseif (z <= 2.1e-202)
		tmp = Float64(Float64(x_m / y) / t);
	elseif (z <= 3.5e-186)
		tmp = Float64(x_m / Float64(z * Float64(z - t)));
	elseif (z <= 3.4e-39)
		tmp = t_1;
	elseif (z <= 3.8e+138)
		tmp = t_2;
	else
		tmp = Float64(Float64(x_m / z) * Float64(1.0 / 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 / t) / y;
	t_2 = x_m / (z * (z - y));
	tmp = 0.0;
	if (z <= -9.8e-25)
		tmp = t_2;
	elseif (z <= -2.45e-122)
		tmp = t_1;
	elseif (z <= -7.6e-143)
		tmp = t_2;
	elseif (z <= 2.1e-202)
		tmp = (x_m / y) / t;
	elseif (z <= 3.5e-186)
		tmp = x_m / (z * (z - t));
	elseif (z <= 3.4e-39)
		tmp = t_1;
	elseif (z <= 3.8e+138)
		tmp = t_2;
	else
		tmp = (x_m / z) * (1.0 / 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[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m / N[(z * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -9.8e-25], t$95$2, If[LessEqual[z, -2.45e-122], t$95$1, If[LessEqual[z, -7.6e-143], t$95$2, If[LessEqual[z, 2.1e-202], N[(N[(x$95$m / y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.5e-186], N[(x$95$m / N[(z * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e-39], t$95$1, If[LessEqual[z, 3.8e+138], t$95$2, N[(N[(x$95$m / z), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if z < -9.7999999999999998e-25 or -2.4499999999999998e-122 < z < -7.59999999999999962e-143 or 3.3999999999999999e-39 < z < 3.80000000000000012e138

   1. Initial program 93.5%

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

   3. Taylor expanded in t around 0 74.4%

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

      1. mul-1-neg 74.4%

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

      2. distribute-rgt-neg-in 74.4%

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

      3. neg-sub0 74.4%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 74.4%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 74.4%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 74.4%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 74.4%

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

      8. remove-double-neg 74.4%

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

   5. Simplified 74.4%

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

   if -9.7999999999999998e-25 < z < -2.4499999999999998e-122 or 3.49999999999999989e-186 < z < 3.3999999999999999e-39

   1. Initial program 88.7%

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

      1. associate-/l/ 93.5%

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

      2. clear-num 93.6%

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

      3. inv-pow 93.6%

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

      4. div-inv 93.5%

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

      5. clear-num 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. unpow-1 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in z around 0 57.9%

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

      1. associate-/r* 64.7%

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

   9. Simplified 64.7%

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

   if -7.59999999999999962e-143 < z < 2.09999999999999985e-202

   1. Initial program 85.4%

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

   3. Taylor expanded in y around inf 75.2%

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

      1. associate-/r* 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in t around inf 79.0%

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

   if 2.09999999999999985e-202 < z < 3.49999999999999989e-186

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 50.3%

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

      1. mul-1-neg 50.3%

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

      2. distribute-rgt-neg-in 50.3%

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

      3. sub-neg 50.3%

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

      4. +-commutative 50.3%

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

      5. distribute-neg-in 50.3%

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

      6. remove-double-neg 50.3%

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

      7. unsub-neg 50.3%

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

   5. Simplified 50.3%

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

   if 3.80000000000000012e138 < z

   1. Initial program 71.6%

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

   3. Taylor expanded in t around 0 69.7%

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

      1. mul-1-neg 69.7%

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

      2. distribute-rgt-neg-in 69.7%

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

      3. neg-sub0 69.7%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 69.7%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 69.7%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 69.7%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 69.7%

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

      8. remove-double-neg 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in z around inf 69.7%

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

      1. associate-/r* 92.1%

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

      2. div-inv 92.1%

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

   8. Applied egg-rr 92.1%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{1}{z}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot \left(z - t\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \mathbf{elif}\;z \leq 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{1}{z}\\ \end{array} \end{array} \end{array} \]

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 (* z (- z t)))))
   (*
    x_s
    (if (<= z -4.5e-19)
      t_1
      (if (<= z 2.1e-202)
        (/ (/ x_m y) t)
        (if (<= z 3.2e-186)
          t_1
          (if (<= z 1.3e-38)
            (/ (/ x_m t) y)
            (if (<= z 1e+139) t_1 (* (/ x_m z) (/ 1.0 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 / (z * (z - t));
	double tmp;
	if (z <= -4.5e-19) {
		tmp = t_1;
	} else if (z <= 2.1e-202) {
		tmp = (x_m / y) / t;
	} else if (z <= 3.2e-186) {
		tmp = t_1;
	} else if (z <= 1.3e-38) {
		tmp = (x_m / t) / y;
	} else if (z <= 1e+139) {
		tmp = t_1;
	} else {
		tmp = (x_m / z) * (1.0 / 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 / (z * (z - t))
    if (z <= (-4.5d-19)) then
        tmp = t_1
    else if (z <= 2.1d-202) then
        tmp = (x_m / y) / t
    else if (z <= 3.2d-186) then
        tmp = t_1
    else if (z <= 1.3d-38) then
        tmp = (x_m / t) / y
    else if (z <= 1d+139) then
        tmp = t_1
    else
        tmp = (x_m / z) * (1.0d0 / 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 / (z * (z - t));
	double tmp;
	if (z <= -4.5e-19) {
		tmp = t_1;
	} else if (z <= 2.1e-202) {
		tmp = (x_m / y) / t;
	} else if (z <= 3.2e-186) {
		tmp = t_1;
	} else if (z <= 1.3e-38) {
		tmp = (x_m / t) / y;
	} else if (z <= 1e+139) {
		tmp = t_1;
	} else {
		tmp = (x_m / z) * (1.0 / 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 / (z * (z - t))
	tmp = 0
	if z <= -4.5e-19:
		tmp = t_1
	elif z <= 2.1e-202:
		tmp = (x_m / y) / t
	elif z <= 3.2e-186:
		tmp = t_1
	elif z <= 1.3e-38:
		tmp = (x_m / t) / y
	elif z <= 1e+139:
		tmp = t_1
	else:
		tmp = (x_m / z) * (1.0 / 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(z * Float64(z - t)))
	tmp = 0.0
	if (z <= -4.5e-19)
		tmp = t_1;
	elseif (z <= 2.1e-202)
		tmp = Float64(Float64(x_m / y) / t);
	elseif (z <= 3.2e-186)
		tmp = t_1;
	elseif (z <= 1.3e-38)
		tmp = Float64(Float64(x_m / t) / y);
	elseif (z <= 1e+139)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m / z) * Float64(1.0 / 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 / (z * (z - t));
	tmp = 0.0;
	if (z <= -4.5e-19)
		tmp = t_1;
	elseif (z <= 2.1e-202)
		tmp = (x_m / y) / t;
	elseif (z <= 3.2e-186)
		tmp = t_1;
	elseif (z <= 1.3e-38)
		tmp = (x_m / t) / y;
	elseif (z <= 1e+139)
		tmp = t_1;
	else
		tmp = (x_m / z) * (1.0 / 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[(z * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -4.5e-19], t$95$1, If[LessEqual[z, 2.1e-202], N[(N[(x$95$m / y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.2e-186], t$95$1, If[LessEqual[z, 1.3e-38], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 1e+139], t$95$1, N[(N[(x$95$m / z), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -4.50000000000000013e-19 or 2.09999999999999985e-202 < z < 3.2e-186 or 1.30000000000000005e-38 < z < 1.00000000000000003e139

   1. Initial program 93.5%

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

   3. Taylor expanded in y around 0 74.9%

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

      1. mul-1-neg 74.9%

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

      2. distribute-rgt-neg-in 74.9%

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

      3. sub-neg 74.9%

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

      4. +-commutative 74.9%

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

      5. distribute-neg-in 74.9%

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

      6. remove-double-neg 74.9%

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

      7. unsub-neg 74.9%

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

   5. Simplified 74.9%

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

   if -4.50000000000000013e-19 < z < 2.09999999999999985e-202

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf 69.5%

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

      1. associate-/r* 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in t around inf 69.1%

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

   if 3.2e-186 < z < 1.30000000000000005e-38

   1. Initial program 89.7%

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

      1. associate-/l/ 92.6%

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

      2. clear-num 92.6%

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

      3. inv-pow 92.6%

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

      4. div-inv 92.6%

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

      5. clear-num 92.8%

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

   4. Applied egg-rr 92.8%

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

      1. unpow-1 92.8%

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

   6. Simplified 92.8%

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

   7. Taylor expanded in z around 0 58.6%

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

      1. associate-/r* 62.0%

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

   9. Simplified 62.0%

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

   if 1.00000000000000003e139 < z

   1. Initial program 70.7%

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

   3. Taylor expanded in t around 0 70.7%

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

      1. mul-1-neg 70.7%

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

      2. distribute-rgt-neg-in 70.7%

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

      3. neg-sub0 70.7%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 70.7%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 70.7%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 70.7%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 70.7%

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

      8. remove-double-neg 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in z around inf 70.7%

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

      1. associate-/r* 93.8%

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

      2. div-inv 93.8%

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

   8. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{1}{z}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -82000000000000:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{x\_m}{y - z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 (<= y -82000000000000.0)
    (/ (/ x_m y) (- t z))
    (if (<= y -3.1e-59)
      (/ (/ x_m z) (- z y))
      (if (<= y -3.5e-80)
        (* (/ 1.0 t) (/ x_m (- y z)))
        (if (<= y 1.2e-100) (/ (/ x_m z) (- 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 (y <= -82000000000000.0) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= -3.1e-59) {
		tmp = (x_m / z) / (z - y);
	} else if (y <= -3.5e-80) {
		tmp = (1.0 / t) * (x_m / (y - z));
	} else if (y <= 1.2e-100) {
		tmp = (x_m / z) / (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 (y <= (-82000000000000.0d0)) then
        tmp = (x_m / y) / (t - z)
    else if (y <= (-3.1d-59)) then
        tmp = (x_m / z) / (z - y)
    else if (y <= (-3.5d-80)) then
        tmp = (1.0d0 / t) * (x_m / (y - z))
    else if (y <= 1.2d-100) then
        tmp = (x_m / z) / (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 (y <= -82000000000000.0) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= -3.1e-59) {
		tmp = (x_m / z) / (z - y);
	} else if (y <= -3.5e-80) {
		tmp = (1.0 / t) * (x_m / (y - z));
	} else if (y <= 1.2e-100) {
		tmp = (x_m / z) / (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 y <= -82000000000000.0:
		tmp = (x_m / y) / (t - z)
	elif y <= -3.1e-59:
		tmp = (x_m / z) / (z - y)
	elif y <= -3.5e-80:
		tmp = (1.0 / t) * (x_m / (y - z))
	elif y <= 1.2e-100:
		tmp = (x_m / z) / (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 (y <= -82000000000000.0)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (y <= -3.1e-59)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (y <= -3.5e-80)
		tmp = Float64(Float64(1.0 / t) * Float64(x_m / Float64(y - z)));
	elseif (y <= 1.2e-100)
		tmp = Float64(Float64(x_m / z) / Float64(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 (y <= -82000000000000.0)
		tmp = (x_m / y) / (t - z);
	elseif (y <= -3.1e-59)
		tmp = (x_m / z) / (z - y);
	elseif (y <= -3.5e-80)
		tmp = (1.0 / t) * (x_m / (y - z));
	elseif (y <= 1.2e-100)
		tmp = (x_m / z) / (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[y, -82000000000000.0], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.1e-59], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.5e-80], N[(N[(1.0 / t), $MachinePrecision] * N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-100], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if y < -8.2e13

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf 87.9%

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

      1. associate-/r* 93.9%

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

   5. Simplified 93.9%

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

   if -8.2e13 < y < -3.09999999999999999e-59

   1. Initial program 95.0%

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

   3. Taylor expanded in t around 0 57.0%

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

      1. mul-1-neg 57.0%

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

      2. associate-/r* 61.8%

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

      3. distribute-neg-frac2 61.8%

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

      4. neg-sub0 61.8%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{0 - \left(y - z\right)}} \]

      5. sub-neg 61.8%

         \[\leadsto \frac{\frac{x}{z}}{0 - \color{blue}{\left(y + \left(-z\right)\right)}} \]

      6. +-commutative 61.8%

         \[\leadsto \frac{\frac{x}{z}}{0 - \color{blue}{\left(\left(-z\right) + y\right)}} \]

      7. associate--r+ 61.8%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\left(0 - \left(-z\right)\right) - y}} \]

      8. neg-sub0 61.8%

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

      9. remove-double-neg 61.8%

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

   5. Simplified 61.8%

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

   if -3.09999999999999999e-59 < y < -3.50000000000000015e-80

   1. Initial program 81.6%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in t around inf 99.7%

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

   if -3.50000000000000015e-80 < y < 1.2000000000000001e-100

   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 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. associate-/r* 80.0%

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

      3. distribute-neg-frac2 80.0%

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

      4. sub-neg 80.0%

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

      5. +-commutative 80.0%

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

      6. distribute-neg-in 80.0%

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

      7. remove-double-neg 80.0%

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

      8. unsub-neg 80.0%

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

   5. Simplified 80.0%

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

   if 1.2000000000000001e-100 < y

   1. Initial program 78.9%

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

      1. associate-/l/ 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t around inf 61.3%

      \[\leadsto \frac{\frac{x}{\color{blue}{t}}}{y - z} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -82000000000000:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{x}{y - z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 (<= y -2.3e+14)
    (/ (/ x_m y) (- t z))
    (if (<= y -2.2e-76)
      (/ (/ x_m z) (- z y))
      (if (<= y -5.8e-79)
        (/ x_m (* y t))
        (if (<= y 5e-100) (/ (/ x_m z) (- 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 (y <= -2.3e+14) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= -2.2e-76) {
		tmp = (x_m / z) / (z - y);
	} else if (y <= -5.8e-79) {
		tmp = x_m / (y * t);
	} else if (y <= 5e-100) {
		tmp = (x_m / z) / (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 (y <= (-2.3d+14)) then
        tmp = (x_m / y) / (t - z)
    else if (y <= (-2.2d-76)) then
        tmp = (x_m / z) / (z - y)
    else if (y <= (-5.8d-79)) then
        tmp = x_m / (y * t)
    else if (y <= 5d-100) then
        tmp = (x_m / z) / (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 (y <= -2.3e+14) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= -2.2e-76) {
		tmp = (x_m / z) / (z - y);
	} else if (y <= -5.8e-79) {
		tmp = x_m / (y * t);
	} else if (y <= 5e-100) {
		tmp = (x_m / z) / (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 y <= -2.3e+14:
		tmp = (x_m / y) / (t - z)
	elif y <= -2.2e-76:
		tmp = (x_m / z) / (z - y)
	elif y <= -5.8e-79:
		tmp = x_m / (y * t)
	elif y <= 5e-100:
		tmp = (x_m / z) / (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 (y <= -2.3e+14)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (y <= -2.2e-76)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (y <= -5.8e-79)
		tmp = Float64(x_m / Float64(y * t));
	elseif (y <= 5e-100)
		tmp = Float64(Float64(x_m / z) / Float64(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 (y <= -2.3e+14)
		tmp = (x_m / y) / (t - z);
	elseif (y <= -2.2e-76)
		tmp = (x_m / z) / (z - y);
	elseif (y <= -5.8e-79)
		tmp = x_m / (y * t);
	elseif (y <= 5e-100)
		tmp = (x_m / z) / (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[y, -2.3e+14], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.2e-76], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.8e-79], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-100], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if y < -2.3e14

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf 87.9%

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

      1. associate-/r* 93.9%

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

   5. Simplified 93.9%

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

   if -2.3e14 < y < -2.19999999999999999e-76

   1. Initial program 95.6%

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

   3. Taylor expanded in t around 0 58.4%

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

      1. mul-1-neg 58.4%

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

      2. associate-/r* 62.7%

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

      3. distribute-neg-frac2 62.7%

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

      4. neg-sub0 62.7%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{0 - \left(y - z\right)}} \]

      5. sub-neg 62.7%

         \[\leadsto \frac{\frac{x}{z}}{0 - \color{blue}{\left(y + \left(-z\right)\right)}} \]

      6. +-commutative 62.7%

         \[\leadsto \frac{\frac{x}{z}}{0 - \color{blue}{\left(\left(-z\right) + y\right)}} \]

      7. associate--r+ 62.7%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\left(0 - \left(-z\right)\right) - y}} \]

      8. neg-sub0 62.7%

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

      9. remove-double-neg 62.7%

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

   5. Simplified 62.7%

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

   if -2.19999999999999999e-76 < y < -5.8000000000000001e-79

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

   if -5.8000000000000001e-79 < y < 5.0000000000000001e-100

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0 79.1%

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

      1. mul-1-neg 79.1%

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

      2. associate-/r* 80.2%

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

      3. distribute-neg-frac2 80.2%

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

      4. sub-neg 80.2%

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

      5. +-commutative 80.2%

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

      6. distribute-neg-in 80.2%

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

      7. remove-double-neg 80.2%

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

      8. unsub-neg 80.2%

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

   5. Simplified 80.2%

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

   if 5.0000000000000001e-100 < y

   1. Initial program 78.9%

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

      1. associate-/l/ 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t around inf 61.3%

      \[\leadsto \frac{\frac{x}{\color{blue}{t}}}{y - z} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -480 \lor \neg \left(t \leq 5.2 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= t -480.0) (not (<= t 5.2e-61)))
    (/ (/ x_m t) (- y z))
    (/ x_m (* z (- z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -480.0) || !(t <= 5.2e-61)) {
		tmp = (x_m / t) / (y - z);
	} else {
		tmp = x_m / (z * (z - y));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-480.0d0)) .or. (.not. (t <= 5.2d-61))) then
        tmp = (x_m / t) / (y - z)
    else
        tmp = x_m / (z * (z - y))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -480.0) || !(t <= 5.2e-61)) {
		tmp = (x_m / t) / (y - z);
	} else {
		tmp = x_m / (z * (z - y));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (t <= -480.0) or not (t <= 5.2e-61):
		tmp = (x_m / t) / (y - z)
	else:
		tmp = x_m / (z * (z - y))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((t <= -480.0) || !(t <= 5.2e-61))
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	else
		tmp = Float64(x_m / Float64(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 ((t <= -480.0) || ~((t <= 5.2e-61)))
		tmp = (x_m / t) / (y - z);
	else
		tmp = x_m / (z * (z - y));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[t, -480.0], N[Not[LessEqual[t, 5.2e-61]], $MachinePrecision]], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(z * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -480 or 5.20000000000000021e-61 < t

   1. Initial program 86.2%

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

      1. associate-/l/ 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in t around inf 85.8%

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

   if -480 < t < 5.20000000000000021e-61

   1. Initial program 91.0%

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

   3. Taylor expanded in t around 0 77.1%

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

      1. mul-1-neg 77.1%

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

      2. distribute-rgt-neg-in 77.1%

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

      3. neg-sub0 77.1%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 77.1%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 77.1%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 77.1%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 77.1%

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

      8. remove-double-neg 77.1%

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

   5. Simplified 77.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -480 \lor \neg \left(t \leq 5.2 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -130000000 \lor \neg \left(z \leq 9.2 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t}\\ \end{array} \end{array} \]

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 (or (<= z -130000000.0) (not (<= z 9.2e+25)))
    (* (/ x_m z) (/ 1.0 z))
    (/ (/ 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) {
	double tmp;
	if ((z <= -130000000.0) || !(z <= 9.2e+25)) {
		tmp = (x_m / z) * (1.0 / z);
	} else {
		tmp = (x_m / y) / 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 ((z <= (-130000000.0d0)) .or. (.not. (z <= 9.2d+25))) then
        tmp = (x_m / z) * (1.0d0 / z)
    else
        tmp = (x_m / y) / 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 ((z <= -130000000.0) || !(z <= 9.2e+25)) {
		tmp = (x_m / z) * (1.0 / z);
	} else {
		tmp = (x_m / y) / 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 (z <= -130000000.0) or not (z <= 9.2e+25):
		tmp = (x_m / z) * (1.0 / z)
	else:
		tmp = (x_m / y) / 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 ((z <= -130000000.0) || !(z <= 9.2e+25))
		tmp = Float64(Float64(x_m / z) * Float64(1.0 / z));
	else
		tmp = Float64(Float64(x_m / y) / 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 ((z <= -130000000.0) || ~((z <= 9.2e+25)))
		tmp = (x_m / z) * (1.0 / z);
	else
		tmp = (x_m / y) / 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[Or[LessEqual[z, -130000000.0], N[Not[LessEqual[z, 9.2e+25]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / y), $MachinePrecision] / t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e8 or 9.1999999999999992e25 < z

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. distribute-rgt-neg-in 80.3%

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

      3. neg-sub0 80.3%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 80.3%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 80.3%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 80.3%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 80.3%

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

      8. remove-double-neg 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in z around inf 70.5%

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

      1. associate-/r* 79.2%

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

      2. div-inv 79.2%

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

   8. Applied egg-rr 79.2%

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

   if -1.3e8 < z < 9.1999999999999992e25

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf 63.0%

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

      1. associate-/r* 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in t around inf 58.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000000 \lor \neg \left(z \leq 9.2 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 (<= y -5.8e-22)
    (/ (/ x_m y) (- t z))
    (if (<= y 2.1e-100) (/ (/ x_m z) (- 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 (y <= -5.8e-22) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= 2.1e-100) {
		tmp = (x_m / z) / (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 (y <= (-5.8d-22)) then
        tmp = (x_m / y) / (t - z)
    else if (y <= 2.1d-100) then
        tmp = (x_m / z) / (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 (y <= -5.8e-22) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= 2.1e-100) {
		tmp = (x_m / z) / (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 y <= -5.8e-22:
		tmp = (x_m / y) / (t - z)
	elif y <= 2.1e-100:
		tmp = (x_m / z) / (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 (y <= -5.8e-22)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (y <= 2.1e-100)
		tmp = Float64(Float64(x_m / z) / Float64(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 (y <= -5.8e-22)
		tmp = (x_m / y) / (t - z);
	elseif (y <= 2.1e-100)
		tmp = (x_m / z) / (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[y, -5.8e-22], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-100], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.8000000000000003e-22

   1. Initial program 90.4%

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

   3. Taylor expanded in y around inf 84.8%

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

      1. associate-/r* 89.9%

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

   5. Simplified 89.9%

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

   if -5.8000000000000003e-22 < y < 2.10000000000000009e-100

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0 76.8%

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

      1. mul-1-neg 76.8%

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

      2. associate-/r* 77.7%

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

      3. distribute-neg-frac2 77.7%

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

      4. sub-neg 77.7%

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

      5. +-commutative 77.7%

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

      6. distribute-neg-in 77.7%

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

      7. remove-double-neg 77.7%

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

      8. unsub-neg 77.7%

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

   5. Simplified 77.7%

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

   if 2.10000000000000009e-100 < y

   1. Initial program 78.9%

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

      1. associate-/l/ 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t around inf 61.3%

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

Alternative 10: 78.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-256}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -1.25e-23)
    (/ (/ x_m y) (- t z))
    (if (<= y 4.6e-256) (/ x_m (* z (- 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 (y <= -1.25e-23) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= 4.6e-256) {
		tmp = x_m / (z * (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 (y <= (-1.25d-23)) then
        tmp = (x_m / y) / (t - z)
    else if (y <= 4.6d-256) then
        tmp = x_m / (z * (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 (y <= -1.25e-23) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= 4.6e-256) {
		tmp = x_m / (z * (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 y <= -1.25e-23:
		tmp = (x_m / y) / (t - z)
	elif y <= 4.6e-256:
		tmp = x_m / (z * (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 (y <= -1.25e-23)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (y <= 4.6e-256)
		tmp = Float64(x_m / Float64(z * Float64(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 (y <= -1.25e-23)
		tmp = (x_m / y) / (t - z);
	elseif (y <= 4.6e-256)
		tmp = x_m / (z * (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[y, -1.25e-23], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e-256], N[(x$95$m / N[(z * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.2500000000000001e-23

   1. Initial program 90.4%

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

   3. Taylor expanded in y around inf 84.8%

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

      1. associate-/r* 89.9%

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

   5. Simplified 89.9%

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

   if -1.2500000000000001e-23 < y < 4.6e-256

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0 82.1%

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

      1. mul-1-neg 82.1%

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

      2. distribute-rgt-neg-in 82.1%

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

      3. sub-neg 82.1%

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

      4. +-commutative 82.1%

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

      5. distribute-neg-in 82.1%

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

      6. remove-double-neg 82.1%

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

      7. unsub-neg 82.1%

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

   5. Simplified 82.1%

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

   if 4.6e-256 < y

   1. Initial program 83.2%

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

      1. associate-/l/ 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in t around inf 59.8%

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

Alternative 11: 66.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -98000000 \lor \neg \left(z \leq 9.2 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t}\\ \end{array} \end{array} \]

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 (or (<= z -98000000.0) (not (<= z 9.2e+21)))
    (/ (/ x_m z) z)
    (/ (/ 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) {
	double tmp;
	if ((z <= -98000000.0) || !(z <= 9.2e+21)) {
		tmp = (x_m / z) / z;
	} else {
		tmp = (x_m / y) / 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 ((z <= (-98000000.0d0)) .or. (.not. (z <= 9.2d+21))) then
        tmp = (x_m / z) / z
    else
        tmp = (x_m / y) / 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 ((z <= -98000000.0) || !(z <= 9.2e+21)) {
		tmp = (x_m / z) / z;
	} else {
		tmp = (x_m / y) / 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 (z <= -98000000.0) or not (z <= 9.2e+21):
		tmp = (x_m / z) / z
	else:
		tmp = (x_m / y) / 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 ((z <= -98000000.0) || !(z <= 9.2e+21))
		tmp = Float64(Float64(x_m / z) / z);
	else
		tmp = Float64(Float64(x_m / y) / 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 ((z <= -98000000.0) || ~((z <= 9.2e+21)))
		tmp = (x_m / z) / z;
	else
		tmp = (x_m / y) / 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[Or[LessEqual[z, -98000000.0], N[Not[LessEqual[z, 9.2e+21]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / y), $MachinePrecision] / t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -9.8e7 or 9.2e21 < z

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. associate-/r* 90.7%

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

      3. distribute-neg-frac2 90.7%

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

      4. neg-sub0 90.7%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{0 - \left(y - z\right)}} \]

      5. sub-neg 90.7%

         \[\leadsto \frac{\frac{x}{z}}{0 - \color{blue}{\left(y + \left(-z\right)\right)}} \]

      6. +-commutative 90.7%

         \[\leadsto \frac{\frac{x}{z}}{0 - \color{blue}{\left(\left(-z\right) + y\right)}} \]

      7. associate--r+ 90.7%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\left(0 - \left(-z\right)\right) - y}} \]

      8. neg-sub0 90.7%

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

      9. remove-double-neg 90.7%

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

   5. Simplified 90.7%

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

   6. Taylor expanded in z around inf 79.2%

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

   if -9.8e7 < z < 9.2e21

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf 63.0%

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

      1. associate-/r* 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in t around inf 58.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -98000000 \lor \neg \left(z \leq 9.2 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -145000000 \lor \neg \left(z \leq 3.8 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t}\\ \end{array} \end{array} \]

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 (or (<= z -145000000.0) (not (<= z 3.8e+27)))
    (/ x_m (* z z))
    (/ (/ 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) {
	double tmp;
	if ((z <= -145000000.0) || !(z <= 3.8e+27)) {
		tmp = x_m / (z * z);
	} else {
		tmp = (x_m / y) / 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 ((z <= (-145000000.0d0)) .or. (.not. (z <= 3.8d+27))) then
        tmp = x_m / (z * z)
    else
        tmp = (x_m / y) / 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 ((z <= -145000000.0) || !(z <= 3.8e+27)) {
		tmp = x_m / (z * z);
	} else {
		tmp = (x_m / y) / 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 (z <= -145000000.0) or not (z <= 3.8e+27):
		tmp = x_m / (z * z)
	else:
		tmp = (x_m / y) / 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 ((z <= -145000000.0) || !(z <= 3.8e+27))
		tmp = Float64(x_m / Float64(z * z));
	else
		tmp = Float64(Float64(x_m / y) / 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 ((z <= -145000000.0) || ~((z <= 3.8e+27)))
		tmp = x_m / (z * z);
	else
		tmp = (x_m / y) / 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[Or[LessEqual[z, -145000000.0], N[Not[LessEqual[z, 3.8e+27]], $MachinePrecision]], N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / y), $MachinePrecision] / t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e8 or 3.80000000000000022e27 < z

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. distribute-rgt-neg-in 80.3%

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

      3. neg-sub0 80.3%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 80.3%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 80.3%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 80.3%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 80.3%

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

      8. remove-double-neg 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in z around inf 70.5%

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

   if -1.45e8 < z < 3.80000000000000022e27

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf 63.0%

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

      1. associate-/r* 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in t around inf 58.5%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -145000000 \lor \neg \left(z \leq 3.8 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -106000000 \lor \neg \left(z \leq 1.05 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \end{array} \end{array} \]

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 (or (<= z -106000000.0) (not (<= z 1.05e+23)))
    (/ x_m (* z z))
    (/ (/ x_m t) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -106000000.0) || !(z <= 1.05e+23)) {
		tmp = x_m / (z * z);
	} else {
		tmp = (x_m / t) / 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 <= (-106000000.0d0)) .or. (.not. (z <= 1.05d+23))) then
        tmp = x_m / (z * z)
    else
        tmp = (x_m / t) / 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 <= -106000000.0) || !(z <= 1.05e+23)) {
		tmp = x_m / (z * z);
	} else {
		tmp = (x_m / t) / 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 <= -106000000.0) or not (z <= 1.05e+23):
		tmp = x_m / (z * z)
	else:
		tmp = (x_m / t) / 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 <= -106000000.0) || !(z <= 1.05e+23))
		tmp = Float64(x_m / Float64(z * z));
	else
		tmp = Float64(Float64(x_m / t) / 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 <= -106000000.0) || ~((z <= 1.05e+23)))
		tmp = x_m / (z * z);
	else
		tmp = (x_m / t) / 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[Or[LessEqual[z, -106000000.0], N[Not[LessEqual[z, 1.05e+23]], $MachinePrecision]], N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.06e8 or 1.0500000000000001e23 < z

   1. Initial program 86.0%

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

   3. Taylor expanded in t around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. distribute-rgt-neg-in 80.3%

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

      3. neg-sub0 80.3%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 80.3%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 80.3%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 80.3%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 80.3%

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

      8. remove-double-neg 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in z around inf 70.5%

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

   if -1.06e8 < z < 1.0500000000000001e23

   1. Initial program 89.9%

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

      1. associate-/l/ 95.9%

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

      2. clear-num 95.9%

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

      3. inv-pow 95.9%

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

      4. div-inv 95.8%

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

      5. clear-num 95.9%

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

   4. Applied egg-rr 95.9%

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

      1. unpow-1 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in z around 0 53.4%

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

      1. associate-/r* 60.7%

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

   9. Simplified 60.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -106000000 \lor \neg \left(z \leq 1.05 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -100000000 \lor \neg \left(z \leq 2.05 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \end{array} \end{array} \]

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 (or (<= z -100000000.0) (not (<= z 2.05e-38)))
    (/ x_m (* z z))
    (/ 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) {
	double tmp;
	if ((z <= -100000000.0) || !(z <= 2.05e-38)) {
		tmp = x_m / (z * z);
	} else {
		tmp = x_m / (y * 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 ((z <= (-100000000.0d0)) .or. (.not. (z <= 2.05d-38))) then
        tmp = x_m / (z * z)
    else
        tmp = x_m / (y * 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 ((z <= -100000000.0) || !(z <= 2.05e-38)) {
		tmp = x_m / (z * z);
	} else {
		tmp = x_m / (y * 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 (z <= -100000000.0) or not (z <= 2.05e-38):
		tmp = x_m / (z * z)
	else:
		tmp = x_m / (y * 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 ((z <= -100000000.0) || !(z <= 2.05e-38))
		tmp = Float64(x_m / Float64(z * z));
	else
		tmp = Float64(x_m / Float64(y * 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 ((z <= -100000000.0) || ~((z <= 2.05e-38)))
		tmp = x_m / (z * z);
	else
		tmp = x_m / (y * 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[Or[LessEqual[z, -100000000.0], N[Not[LessEqual[z, 2.05e-38]], $MachinePrecision]], N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1e8 or 2.0499999999999999e-38 < z

   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 0 76.9%

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

      1. mul-1-neg 76.9%

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

      2. distribute-rgt-neg-in 76.9%

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

      3. neg-sub0 76.9%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(0 - \left(y - z\right)\right)}} \]

      4. sub-neg 76.9%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right)} \]

      5. +-commutative 76.9%

         \[\leadsto \frac{x}{z \cdot \left(0 - \color{blue}{\left(\left(-z\right) + y\right)}\right)} \]

      6. associate--r+ 76.9%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\left(\left(0 - \left(-z\right)\right) - y\right)}} \]

      7. neg-sub0 76.9%

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

      8. remove-double-neg 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in z around inf 65.1%

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

   if -1e8 < z < 2.0499999999999999e-38

   1. Initial program 89.2%

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

   3. Taylor expanded in z around 0 57.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -100000000 \lor \neg \left(z \leq 2.05 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -150000000 \lor \neg \left(z \leq 2.4 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \end{array} \end{array} \]

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 (or (<= z -150000000.0) (not (<= z 2.4e+90)))
    (/ x_m (* y z))
    (/ 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) {
	double tmp;
	if ((z <= -150000000.0) || !(z <= 2.4e+90)) {
		tmp = x_m / (y * z);
	} else {
		tmp = x_m / (y * 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 ((z <= (-150000000.0d0)) .or. (.not. (z <= 2.4d+90))) then
        tmp = x_m / (y * z)
    else
        tmp = x_m / (y * 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 ((z <= -150000000.0) || !(z <= 2.4e+90)) {
		tmp = x_m / (y * z);
	} else {
		tmp = x_m / (y * 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 (z <= -150000000.0) or not (z <= 2.4e+90):
		tmp = x_m / (y * z)
	else:
		tmp = x_m / (y * 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 ((z <= -150000000.0) || !(z <= 2.4e+90))
		tmp = Float64(x_m / Float64(y * z));
	else
		tmp = Float64(x_m / Float64(y * 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 ((z <= -150000000.0) || ~((z <= 2.4e+90)))
		tmp = x_m / (y * z);
	else
		tmp = x_m / (y * 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[Or[LessEqual[z, -150000000.0], N[Not[LessEqual[z, 2.4e+90]], $MachinePrecision]], N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.5e8 or 2.4000000000000001e90 < z

   1. Initial program 85.6%

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

   3. Taylor expanded in t around 0 81.3%

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

      1. mul-1-neg 81.3%

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

      2. associate-/r* 92.7%

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

      3. distribute-neg-frac2 92.7%

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

      4. neg-sub0 92.7%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{0 - \left(y - z\right)}} \]

      5. sub-neg 92.7%

         \[\leadsto \frac{\frac{x}{z}}{0 - \color{blue}{\left(y + \left(-z\right)\right)}} \]

      6. +-commutative 92.7%

         \[\leadsto \frac{\frac{x}{z}}{0 - \color{blue}{\left(\left(-z\right) + y\right)}} \]

      7. associate--r+ 92.7%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\left(0 - \left(-z\right)\right) - y}} \]

      8. neg-sub0 92.7%

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

      9. remove-double-neg 92.7%

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

   5. Simplified 92.7%

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

   6. Taylor expanded in z around 0 49.3%

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

      1. neg-mul-1 49.3%

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

   8. Simplified 49.3%

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

      1. div-inv 49.3%

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

      2. associate-/l* 37.5%

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

      3. add-sqr-sqrt 23.5%

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

      4. sqrt-unprod 32.8%

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

      5. sqr-neg 32.8%

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

      6. sqrt-unprod 11.5%

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

      7. add-sqr-sqrt 32.4%

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

   10. Applied egg-rr 32.4%

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

       1. associate-/l/ 32.4%

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

       2. associate-*r/ 32.4%

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

       3. associate-*l/ 32.4%

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

       4. *-rgt-identity 32.4%

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

       5. *-commutative 32.4%

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

   12. Simplified 32.4%

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

   if -1.5e8 < z < 2.4000000000000001e90

   1. Initial program 89.9%

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

   3. Taylor expanded in z around 0 50.5%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -150000000 \lor \neg \left(z \leq 2.4 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 90.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t}}{\frac{y - z}{x\_m}}\\ \end{array} \end{array} \]

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 8.4e+168)
    (/ x_m (* (- y z) (- t z)))
    (/ (/ 1.0 t) (/ (- y z) x_m)))))

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 <= 8.4e+168) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (1.0 / t) / ((y - z) / x_m);
	}
	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 <= 8.4d+168) then
        tmp = x_m / ((y - z) * (t - z))
    else
        tmp = (1.0d0 / t) / ((y - z) / x_m)
    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 <= 8.4e+168) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (1.0 / t) / ((y - z) / x_m);
	}
	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 <= 8.4e+168:
		tmp = x_m / ((y - z) * (t - z))
	else:
		tmp = (1.0 / t) / ((y - z) / x_m)
	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 <= 8.4e+168)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = Float64(Float64(1.0 / t) / Float64(Float64(y - z) / x_m));
	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 <= 8.4e+168)
		tmp = x_m / ((y - z) * (t - z));
	else
		tmp = (1.0 / t) / ((y - z) / x_m);
	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, 8.4e+168], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t), $MachinePrecision] / N[(N[(y - z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.40000000000000011e168

   1. Initial program 90.3%

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

   if 8.40000000000000011e168 < t

   1. Initial program 69.4%

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

      1. associate-/r* 96.0%

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

      2. div-inv 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in t around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. clear-num 95.9%

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

      3. un-div-inv 95.9%

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

   7. Applied egg-rr 95.9%

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

Alternative 17: 90.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{+166}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{x\_m}{y - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t 2.35e+166)
    (/ x_m (* (- y z) (- t z)))
    (* (/ 1.0 t) (/ x_m (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 2.35e+166) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (1.0 / t) * (x_m / (y - z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.35d+166) then
        tmp = x_m / ((y - z) * (t - z))
    else
        tmp = (1.0d0 / t) * (x_m / (y - z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 2.35e+166) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (1.0 / t) * (x_m / (y - z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 2.35e+166:
		tmp = x_m / ((y - z) * (t - z))
	else:
		tmp = (1.0 / t) * (x_m / (y - z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 2.35e+166)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = Float64(Float64(1.0 / t) * Float64(x_m / Float64(y - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 2.35e+166)
		tmp = x_m / ((y - z) * (t - z));
	else
		tmp = (1.0 / t) * (x_m / (y - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 2.35e+166], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t), $MachinePrecision] * N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.35e166

   1. Initial program 90.3%

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

   if 2.35e166 < t

   1. Initial program 69.4%

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

      1. associate-/r* 96.0%

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

      2. div-inv 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in t around inf 95.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{+166}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{x}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 39.4% 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 88.2%

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

3. Taylor expanded in z around 0 39.0%

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

4. Final simplification 39.0%

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

Developer target: 87.8% 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 2024110 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :alt
  (if (< (/ x (* (- y z) (- t z))) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 (* (- y z) (- t z)))))

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