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

Percentage Accurate: 89.3% → 97.9%

Time: 11.8s

Alternatives: 12

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \end{array} \end{array} \end{array} \]

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 -2e-307) 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 <= -2e-307) {
		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 <= (-2d-307)) 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 <= -2e-307) {
		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 <= -2e-307:
		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 <= -2e-307)
		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 <= -2e-307)
		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, -2e-307], 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))) < -1.99999999999999982e-307

   1. Initial program 98.1%

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

   if -1.99999999999999982e-307 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 85.6%

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

      1. associate-/l/ 97.8%

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

   3. Simplified 97.8%

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

Alternative 2: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{z}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-78}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{+158}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x_m z) z)))
   (*
    x_s
    (if (<= z -1.65e+64)
      t_1
      (if (<= z -2.1e-190)
        (/ (/ x_m t) (- y z))
        (if (<= z 6e-78)
          (/ x_m (* y (- t z)))
          (if (<= z 3.25e+158) (/ x_m (* z (- z t))) t_1)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / z) / z;
	double tmp;
	if (z <= -1.65e+64) {
		tmp = t_1;
	} else if (z <= -2.1e-190) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 6e-78) {
		tmp = x_m / (y * (t - z));
	} else if (z <= 3.25e+158) {
		tmp = x_m / (z * (z - t));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / z) / z
    if (z <= (-1.65d+64)) then
        tmp = t_1
    else if (z <= (-2.1d-190)) then
        tmp = (x_m / t) / (y - z)
    else if (z <= 6d-78) then
        tmp = x_m / (y * (t - z))
    else if (z <= 3.25d+158) then
        tmp = x_m / (z * (z - t))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / z) / z;
	double tmp;
	if (z <= -1.65e+64) {
		tmp = t_1;
	} else if (z <= -2.1e-190) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 6e-78) {
		tmp = x_m / (y * (t - z));
	} else if (z <= 3.25e+158) {
		tmp = x_m / (z * (z - t));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / z) / z
	tmp = 0
	if z <= -1.65e+64:
		tmp = t_1
	elif z <= -2.1e-190:
		tmp = (x_m / t) / (y - z)
	elif z <= 6e-78:
		tmp = x_m / (y * (t - z))
	elif z <= 3.25e+158:
		tmp = x_m / (z * (z - t))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / z) / z)
	tmp = 0.0
	if (z <= -1.65e+64)
		tmp = t_1;
	elseif (z <= -2.1e-190)
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	elseif (z <= 6e-78)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (z <= 3.25e+158)
		tmp = Float64(x_m / Float64(z * Float64(z - t)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / z) / z;
	tmp = 0.0;
	if (z <= -1.65e+64)
		tmp = t_1;
	elseif (z <= -2.1e-190)
		tmp = (x_m / t) / (y - z);
	elseif (z <= 6e-78)
		tmp = x_m / (y * (t - z));
	elseif (z <= 3.25e+158)
		tmp = x_m / (z * (z - t));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.65e+64], t$95$1, If[LessEqual[z, -2.1e-190], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e-78], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.25e+158], N[(x$95$m / N[(z * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -1.64999999999999994e64 or 3.2500000000000001e158 < z

   1. Initial program 78.7%

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

   3. Taylor expanded in t around 0 75.1%

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

      1. mul-1-neg 75.1%

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

      2. associate-/r* 93.3%

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

      3. distribute-neg-frac2 93.3%

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

      4. neg-sub0 93.3%

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

      5. sub-neg 93.3%

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

      6. +-commutative 93.3%

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

      7. associate--r+ 93.3%

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

      8. neg-sub0 93.3%

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

      9. remove-double-neg 93.3%

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

   5. Simplified 93.3%

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

   6. Taylor expanded in z around inf 89.1%

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

   if -1.64999999999999994e64 < z < -2.09999999999999991e-190

   1. Initial program 93.8%

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

      1. associate-/l/ 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t around inf 70.9%

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

   if -2.09999999999999991e-190 < z < 5.99999999999999975e-78

   1. Initial program 97.2%

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

   3. Taylor expanded in y around inf 88.2%

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

      1. *-commutative 88.2%

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

   5. Simplified 88.2%

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

   if 5.99999999999999975e-78 < z < 3.2500000000000001e158

   1. Initial program 90.1%

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

   3. Taylor expanded in y around 0 65.6%

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

      1. mul-1-neg 65.6%

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

      2. distribute-rgt-neg-in 65.6%

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

      3. sub-neg 65.6%

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

      4. +-commutative 65.6%

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

      5. distribute-neg-in 65.6%

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

      6. remove-double-neg 65.6%

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

      7. unsub-neg 65.6%

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

   5. Simplified 65.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{+158}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot \left(z - t\right)}\\ t_2 := \frac{\frac{x\_m}{z}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+164}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-75}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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)))) (t_2 (/ (/ x_m z) z)))
   (*
    x_s
    (if (<= z -2.25e+164)
      t_2
      (if (<= z -1e-19)
        t_1
        (if (<= z 1.85e-75)
          (/ x_m (* y (- t z)))
          (if (<= z 3.25e+158) t_1 t_2)))))))

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 t_2 = (x_m / z) / z;
	double tmp;
	if (z <= -2.25e+164) {
		tmp = t_2;
	} else if (z <= -1e-19) {
		tmp = t_1;
	} else if (z <= 1.85e-75) {
		tmp = x_m / (y * (t - z));
	} else if (z <= 3.25e+158) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 / (z * (z - t))
    t_2 = (x_m / z) / z
    if (z <= (-2.25d+164)) then
        tmp = t_2
    else if (z <= (-1d-19)) then
        tmp = t_1
    else if (z <= 1.85d-75) then
        tmp = x_m / (y * (t - z))
    else if (z <= 3.25d+158) then
        tmp = t_1
    else
        tmp = t_2
    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 t_2 = (x_m / z) / z;
	double tmp;
	if (z <= -2.25e+164) {
		tmp = t_2;
	} else if (z <= -1e-19) {
		tmp = t_1;
	} else if (z <= 1.85e-75) {
		tmp = x_m / (y * (t - z));
	} else if (z <= 3.25e+158) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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))
	t_2 = (x_m / z) / z
	tmp = 0
	if z <= -2.25e+164:
		tmp = t_2
	elif z <= -1e-19:
		tmp = t_1
	elif z <= 1.85e-75:
		tmp = x_m / (y * (t - z))
	elif z <= 3.25e+158:
		tmp = t_1
	else:
		tmp = t_2
	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)))
	t_2 = Float64(Float64(x_m / z) / z)
	tmp = 0.0
	if (z <= -2.25e+164)
		tmp = t_2;
	elseif (z <= -1e-19)
		tmp = t_1;
	elseif (z <= 1.85e-75)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (z <= 3.25e+158)
		tmp = t_1;
	else
		tmp = t_2;
	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));
	t_2 = (x_m / z) / z;
	tmp = 0.0;
	if (z <= -2.25e+164)
		tmp = t_2;
	elseif (z <= -1e-19)
		tmp = t_1;
	elseif (z <= 1.85e-75)
		tmp = x_m / (y * (t - z));
	elseif (z <= 3.25e+158)
		tmp = t_1;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -2.25e+164], t$95$2, If[LessEqual[z, -1e-19], t$95$1, If[LessEqual[z, 1.85e-75], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.25e+158], t$95$1, t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.24999999999999988e164 or 3.2500000000000001e158 < z

   1. Initial program 74.1%

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

   3. Taylor expanded in t around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. associate-/r* 98.1%

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

      3. distribute-neg-frac2 98.1%

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

      4. neg-sub0 98.1%

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

      5. sub-neg 98.1%

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

      6. +-commutative 98.1%

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

      7. associate--r+ 98.1%

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

      8. neg-sub0 98.1%

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

      9. remove-double-neg 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in z around inf 98.1%

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

   if -2.24999999999999988e164 < z < -9.9999999999999998e-20 or 1.85000000000000012e-75 < z < 3.2500000000000001e158

   1. Initial program 89.2%

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

   3. Taylor expanded in y around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. distribute-rgt-neg-in 68.1%

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

      3. sub-neg 68.1%

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

      4. +-commutative 68.1%

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

      5. distribute-neg-in 68.1%

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

      6. remove-double-neg 68.1%

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

      7. unsub-neg 68.1%

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

   5. Simplified 68.1%

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

   if -9.9999999999999998e-20 < z < 1.85000000000000012e-75

   1. Initial program 97.1%

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

   3. Taylor expanded in y around inf 84.0%

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

      1. *-commutative 84.0%

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

   5. Simplified 84.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{+158}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot \left(z - t\right)}\\ t_2 := \frac{\frac{x\_m}{z}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+164}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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)))) (t_2 (/ (/ x_m z) z)))
   (*
    x_s
    (if (<= z -2.25e+164)
      t_2
      (if (<= z -1.45e-63)
        t_1
        (if (<= z 2.8e-76) (/ x_m (* y t)) (if (<= z 3.3e+158) t_1 t_2)))))))

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 t_2 = (x_m / z) / z;
	double tmp;
	if (z <= -2.25e+164) {
		tmp = t_2;
	} else if (z <= -1.45e-63) {
		tmp = t_1;
	} else if (z <= 2.8e-76) {
		tmp = x_m / (y * t);
	} else if (z <= 3.3e+158) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 / (z * (z - t))
    t_2 = (x_m / z) / z
    if (z <= (-2.25d+164)) then
        tmp = t_2
    else if (z <= (-1.45d-63)) then
        tmp = t_1
    else if (z <= 2.8d-76) then
        tmp = x_m / (y * t)
    else if (z <= 3.3d+158) then
        tmp = t_1
    else
        tmp = t_2
    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 t_2 = (x_m / z) / z;
	double tmp;
	if (z <= -2.25e+164) {
		tmp = t_2;
	} else if (z <= -1.45e-63) {
		tmp = t_1;
	} else if (z <= 2.8e-76) {
		tmp = x_m / (y * t);
	} else if (z <= 3.3e+158) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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))
	t_2 = (x_m / z) / z
	tmp = 0
	if z <= -2.25e+164:
		tmp = t_2
	elif z <= -1.45e-63:
		tmp = t_1
	elif z <= 2.8e-76:
		tmp = x_m / (y * t)
	elif z <= 3.3e+158:
		tmp = t_1
	else:
		tmp = t_2
	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)))
	t_2 = Float64(Float64(x_m / z) / z)
	tmp = 0.0
	if (z <= -2.25e+164)
		tmp = t_2;
	elseif (z <= -1.45e-63)
		tmp = t_1;
	elseif (z <= 2.8e-76)
		tmp = Float64(x_m / Float64(y * t));
	elseif (z <= 3.3e+158)
		tmp = t_1;
	else
		tmp = t_2;
	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));
	t_2 = (x_m / z) / z;
	tmp = 0.0;
	if (z <= -2.25e+164)
		tmp = t_2;
	elseif (z <= -1.45e-63)
		tmp = t_1;
	elseif (z <= 2.8e-76)
		tmp = x_m / (y * t);
	elseif (z <= 3.3e+158)
		tmp = t_1;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -2.25e+164], t$95$2, If[LessEqual[z, -1.45e-63], t$95$1, If[LessEqual[z, 2.8e-76], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e+158], t$95$1, t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.24999999999999988e164 or 3.30000000000000017e158 < z

   1. Initial program 74.1%

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

   3. Taylor expanded in t around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. associate-/r* 98.1%

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

      3. distribute-neg-frac2 98.1%

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

      4. neg-sub0 98.1%

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

      5. sub-neg 98.1%

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

      6. +-commutative 98.1%

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

      7. associate--r+ 98.1%

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

      8. neg-sub0 98.1%

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

      9. remove-double-neg 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in z around inf 98.1%

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

   if -2.24999999999999988e164 < z < -1.44999999999999987e-63 or 2.8000000000000001e-76 < z < 3.30000000000000017e158

   1. Initial program 90.3%

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

   3. Taylor expanded in y around 0 68.3%

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

      1. mul-1-neg 68.3%

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

      2. distribute-rgt-neg-in 68.3%

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

      3. sub-neg 68.3%

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

      4. +-commutative 68.3%

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

      5. distribute-neg-in 68.3%

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

      6. remove-double-neg 68.3%

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

      7. unsub-neg 68.3%

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

   5. Simplified 68.3%

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

   if -1.44999999999999987e-63 < z < 2.8000000000000001e-76

   1. Initial program 96.8%

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

   3. Taylor expanded in z around 0 79.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+158}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-75}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -9.5e+62)
    (/ (/ x_m z) (- z y))
    (if (<= z -8.2e-191)
      (/ (/ x_m t) (- y z))
      (if (<= z 1.65e-75) (/ x_m (* y (- t z))) (/ (/ x_m z) (- z t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e+62) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= -8.2e-191) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 1.65e-75) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-9.5d+62)) then
        tmp = (x_m / z) / (z - y)
    else if (z <= (-8.2d-191)) then
        tmp = (x_m / t) / (y - z)
    else if (z <= 1.65d-75) then
        tmp = x_m / (y * (t - z))
    else
        tmp = (x_m / z) / (z - t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e+62) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= -8.2e-191) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 1.65e-75) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -9.5e+62:
		tmp = (x_m / z) / (z - y)
	elif z <= -8.2e-191:
		tmp = (x_m / t) / (y - z)
	elif z <= 1.65e-75:
		tmp = x_m / (y * (t - z))
	else:
		tmp = (x_m / z) / (z - t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -9.5e+62)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (z <= -8.2e-191)
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	elseif (z <= 1.65e-75)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -9.5e+62)
		tmp = (x_m / z) / (z - y);
	elseif (z <= -8.2e-191)
		tmp = (x_m / t) / (y - z);
	elseif (z <= 1.65e-75)
		tmp = x_m / (y * (t - z));
	else
		tmp = (x_m / z) / (z - t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -9.5e+62], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -8.2e-191], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.65e-75], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -9.5000000000000003e62

   1. Initial program 79.8%

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

   3. Taylor expanded in t around 0 73.9%

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

      1. mul-1-neg 73.9%

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

      2. associate-/r* 91.1%

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

      3. distribute-neg-frac2 91.1%

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

      4. neg-sub0 91.1%

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

      5. sub-neg 91.1%

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

      6. +-commutative 91.1%

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

      7. associate--r+ 91.1%

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

      8. neg-sub0 91.1%

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

      9. remove-double-neg 91.1%

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

   5. Simplified 91.1%

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

   if -9.5000000000000003e62 < z < -8.2000000000000004e-191

   1. Initial program 93.8%

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

      1. associate-/l/ 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t around inf 70.9%

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

   if -8.2000000000000004e-191 < z < 1.65e-75

   1. Initial program 97.2%

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

   3. Taylor expanded in y around inf 88.2%

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

      1. *-commutative 88.2%

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

   5. Simplified 88.2%

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

   if 1.65e-75 < z

   1. Initial program 84.9%

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

   3. Taylor expanded in y around 0 70.1%

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

      1. mul-1-neg 70.1%

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

      2. associate-/r* 81.3%

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

      3. distribute-neg-frac2 81.3%

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

      4. sub-neg 81.3%

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

      5. +-commutative 81.3%

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

      6. distribute-neg-in 81.3%

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

      7. remove-double-neg 81.3%

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

      8. unsub-neg 81.3%

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

   5. Simplified 81.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{z}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{-z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-73}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x_m z) z)))
   (*
    x_s
    (if (<= z -4e+60)
      t_1
      (if (<= z -6e-63)
        (/ (/ x_m t) (- z))
        (if (<= z 3e-73) (/ x_m (* y t)) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / z) / z;
	double tmp;
	if (z <= -4e+60) {
		tmp = t_1;
	} else if (z <= -6e-63) {
		tmp = (x_m / t) / -z;
	} else if (z <= 3e-73) {
		tmp = x_m / (y * t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / z) / z
    if (z <= (-4d+60)) then
        tmp = t_1
    else if (z <= (-6d-63)) then
        tmp = (x_m / t) / -z
    else if (z <= 3d-73) then
        tmp = x_m / (y * t)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / z) / z;
	double tmp;
	if (z <= -4e+60) {
		tmp = t_1;
	} else if (z <= -6e-63) {
		tmp = (x_m / t) / -z;
	} else if (z <= 3e-73) {
		tmp = x_m / (y * t);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / z) / z
	tmp = 0
	if z <= -4e+60:
		tmp = t_1
	elif z <= -6e-63:
		tmp = (x_m / t) / -z
	elif z <= 3e-73:
		tmp = x_m / (y * t)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / z) / z)
	tmp = 0.0
	if (z <= -4e+60)
		tmp = t_1;
	elseif (z <= -6e-63)
		tmp = Float64(Float64(x_m / t) / Float64(-z));
	elseif (z <= 3e-73)
		tmp = Float64(x_m / Float64(y * t));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / z) / z;
	tmp = 0.0;
	if (z <= -4e+60)
		tmp = t_1;
	elseif (z <= -6e-63)
		tmp = (x_m / t) / -z;
	elseif (z <= 3e-73)
		tmp = x_m / (y * t);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -4e+60], t$95$1, If[LessEqual[z, -6e-63], N[(N[(x$95$m / t), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[z, 3e-73], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -3.9999999999999998e60 or 3e-73 < z

   1. Initial program 82.5%

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

   3. Taylor expanded in t around 0 72.8%

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

      1. mul-1-neg 72.8%

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

      2. associate-/r* 83.9%

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

      3. distribute-neg-frac2 83.9%

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

      4. neg-sub0 83.9%

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

      5. sub-neg 83.9%

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

      6. +-commutative 83.9%

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

      7. associate--r+ 83.9%

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

      8. neg-sub0 83.9%

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

      9. remove-double-neg 83.9%

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

   5. Simplified 83.9%

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

   6. Taylor expanded in z around inf 72.7%

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

   if -3.9999999999999998e60 < z < -5.99999999999999959e-63

   1. Initial program 96.0%

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

   3. Taylor expanded in y around 0 59.8%

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

      1. mul-1-neg 59.8%

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

      2. distribute-rgt-neg-in 59.8%

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

      3. sub-neg 59.8%

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

      4. +-commutative 59.8%

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

      5. distribute-neg-in 59.8%

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

      6. remove-double-neg 59.8%

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

      7. unsub-neg 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in z around 0 55.2%

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

      1. associate-*r/ 55.2%

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

      2. neg-mul-1 55.2%

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

      3. *-commutative 55.2%

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

   8. Simplified 55.2%

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

   9. Taylor expanded in x around 0 55.2%

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

       1. mul-1-neg 55.2%

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

       2. associate-/r* 55.2%

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

   11. Simplified 55.2%

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

   if -5.99999999999999959e-63 < z < 3e-73

   1. Initial program 96.8%

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

   3. Taylor expanded in z around 0 79.3%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+141}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -9.8e+86)
    (/ (/ x_m z) (- z y))
    (if (<= z 2.55e+141) (/ x_m (* (- y z) (- t z))) (/ (/ x_m z) (- z t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9.8e+86) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.55e+141) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-9.8d+86)) then
        tmp = (x_m / z) / (z - y)
    else if (z <= 2.55d+141) then
        tmp = x_m / ((y - z) * (t - z))
    else
        tmp = (x_m / z) / (z - t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9.8e+86) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.55e+141) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -9.8e+86:
		tmp = (x_m / z) / (z - y)
	elif z <= 2.55e+141:
		tmp = x_m / ((y - z) * (t - z))
	else:
		tmp = (x_m / z) / (z - t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -9.8e+86)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (z <= 2.55e+141)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -9.8e+86)
		tmp = (x_m / z) / (z - y);
	elseif (z <= 2.55e+141)
		tmp = x_m / ((y - z) * (t - z));
	else
		tmp = (x_m / z) / (z - t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -9.8e+86], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.55e+141], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -9.7999999999999999e86

   1. Initial program 75.3%

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

   3. Taylor expanded in t around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. associate-/r* 91.6%

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

      3. distribute-neg-frac2 91.6%

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

      4. neg-sub0 91.6%

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

      5. sub-neg 91.6%

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

      6. +-commutative 91.6%

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

      7. associate--r+ 91.6%

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

      8. neg-sub0 91.6%

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

      9. remove-double-neg 91.6%

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

   5. Simplified 91.6%

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

   if -9.7999999999999999e86 < z < 2.5499999999999999e141

   1. Initial program 96.0%

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

   if 2.5499999999999999e141 < z

   1. Initial program 72.5%

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

   3. Taylor expanded in y around 0 70.9%

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

      1. mul-1-neg 70.9%

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

      2. associate-/r* 95.8%

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

      3. distribute-neg-frac2 95.8%

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

      4. sub-neg 95.8%

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

      5. +-commutative 95.8%

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

      6. distribute-neg-in 95.8%

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

      7. remove-double-neg 95.8%

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

      8. unsub-neg 95.8%

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

   5. Simplified 95.8%

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

Alternative 8: 78.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-19} \lor \neg \left(z \leq 1.65 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -2.8e-19) (not (<= z 1.65e-77)))
    (/ (/ x_m z) (- z t))
    (/ x_m (* y (- t z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -2.8e-19) || !(z <= 1.65e-77)) {
		tmp = (x_m / z) / (z - t);
	} else {
		tmp = x_m / (y * (t - 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 ((z <= (-2.8d-19)) .or. (.not. (z <= 1.65d-77))) then
        tmp = (x_m / z) / (z - t)
    else
        tmp = x_m / (y * (t - 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 ((z <= -2.8e-19) || !(z <= 1.65e-77)) {
		tmp = (x_m / z) / (z - t);
	} else {
		tmp = x_m / (y * (t - 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 (z <= -2.8e-19) or not (z <= 1.65e-77):
		tmp = (x_m / z) / (z - t)
	else:
		tmp = x_m / (y * (t - 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 ((z <= -2.8e-19) || !(z <= 1.65e-77))
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	else
		tmp = Float64(x_m / Float64(y * Float64(t - 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 ((z <= -2.8e-19) || ~((z <= 1.65e-77)))
		tmp = (x_m / z) / (z - t);
	else
		tmp = x_m / (y * (t - 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[Or[LessEqual[z, -2.8e-19], N[Not[LessEqual[z, 1.65e-77]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.80000000000000003e-19 or 1.64999999999999996e-77 < z

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0 70.2%

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

      1. mul-1-neg 70.2%

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

      2. associate-/r* 81.5%

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

      3. distribute-neg-frac2 81.5%

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

      4. sub-neg 81.5%

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

      5. +-commutative 81.5%

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

      6. distribute-neg-in 81.5%

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

      7. remove-double-neg 81.5%

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

      8. unsub-neg 81.5%

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

   5. Simplified 81.5%

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

   if -2.80000000000000003e-19 < z < 1.64999999999999996e-77

   1. Initial program 97.1%

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

   3. Taylor expanded in y around inf 84.0%

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

      1. *-commutative 84.0%

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

   5. Simplified 84.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-19} \lor \neg \left(z \leq 1.65 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -33000000 \lor \neg \left(z \leq 3 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \end{array} \end{array} \]

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 -33000000.0) (not (<= z 3e-73)))
    (/ (/ 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 <= -33000000.0) || !(z <= 3e-73)) {
		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 <= (-33000000.0d0)) .or. (.not. (z <= 3d-73))) 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 <= -33000000.0) || !(z <= 3e-73)) {
		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 <= -33000000.0) or not (z <= 3e-73):
		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 <= -33000000.0) || !(z <= 3e-73))
		tmp = Float64(Float64(x_m / 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 <= -33000000.0) || ~((z <= 3e-73)))
		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, -33000000.0], N[Not[LessEqual[z, 3e-73]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e7 or 3e-73 < z

   1. Initial program 83.1%

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

   3. Taylor expanded in t around 0 71.8%

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

      1. mul-1-neg 71.8%

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

      2. associate-/r* 82.2%

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

      3. distribute-neg-frac2 82.2%

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

      4. neg-sub0 82.2%

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

      5. sub-neg 82.2%

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

      6. +-commutative 82.2%

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

      7. associate--r+ 82.2%

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

      8. neg-sub0 82.2%

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

      9. remove-double-neg 82.2%

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

   5. Simplified 82.2%

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

   6. Taylor expanded in z around inf 68.6%

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

   if -3.3e7 < z < 3e-73

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0 73.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -33000000 \lor \neg \left(z \leq 3 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5500000 \lor \neg \left(z \leq 2.7 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \end{array} \end{array} \]

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 -5500000.0) (not (<= z 2.7e-73)))
    (/ 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 <= -5500000.0) || !(z <= 2.7e-73)) {
		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 <= (-5500000.0d0)) .or. (.not. (z <= 2.7d-73))) 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 <= -5500000.0) || !(z <= 2.7e-73)) {
		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 <= -5500000.0) or not (z <= 2.7e-73):
		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 <= -5500000.0) || !(z <= 2.7e-73))
		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 <= -5500000.0) || ~((z <= 2.7e-73)))
		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, -5500000.0], N[Not[LessEqual[z, 2.7e-73]], $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 < -5.5e6 or 2.69999999999999994e-73 < z

   1. Initial program 83.1%

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

   3. Taylor expanded in y around 0 71.0%

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

      1. mul-1-neg 71.0%

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

      2. distribute-rgt-neg-in 71.0%

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

      3. sub-neg 71.0%

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

      4. +-commutative 71.0%

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

      5. distribute-neg-in 71.0%

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

      6. remove-double-neg 71.0%

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

      7. unsub-neg 71.0%

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

   5. Simplified 71.0%

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

   6. Taylor expanded in z around inf 59.6%

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

   if -5.5e6 < z < 2.69999999999999994e-73

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0 73.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5500000 \lor \neg \left(z \leq 2.7 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6800000 \lor \neg \left(z \leq 4 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x\_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \end{array} \end{array} \]

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 -6800000.0) (not (<= z 4e-63)))
    (/ x_m (* z t))
    (/ 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 <= -6800000.0) || !(z <= 4e-63)) {
		tmp = x_m / (z * t);
	} 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 <= (-6800000.0d0)) .or. (.not. (z <= 4d-63))) then
        tmp = x_m / (z * t)
    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 <= -6800000.0) || !(z <= 4e-63)) {
		tmp = x_m / (z * t);
	} 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 <= -6800000.0) or not (z <= 4e-63):
		tmp = x_m / (z * t)
	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 <= -6800000.0) || !(z <= 4e-63))
		tmp = Float64(x_m / Float64(z * t));
	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 <= -6800000.0) || ~((z <= 4e-63)))
		tmp = x_m / (z * t);
	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, -6800000.0], N[Not[LessEqual[z, 4e-63]], $MachinePrecision]], N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -6.8e6 or 4.00000000000000027e-63 < z

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. distribute-rgt-neg-in 70.6%

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

      3. sub-neg 70.6%

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

      4. +-commutative 70.6%

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

      5. distribute-neg-in 70.6%

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

      6. remove-double-neg 70.6%

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

      7. unsub-neg 70.6%

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

   5. Simplified 70.6%

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

   6. Taylor expanded in z around 0 38.0%

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

      1. associate-*r/ 38.0%

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

      2. neg-mul-1 38.0%

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

      3. *-commutative 38.0%

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

   8. Simplified 38.0%

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

      1. add-sqr-sqrt 21.4%

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

      2. sqrt-unprod 34.0%

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

      3. sqr-neg 34.0%

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

      4. sqrt-unprod 12.2%

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

      5. add-sqr-sqrt 30.0%

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

      6. *-un-lft-identity 30.0%

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

   10. Applied egg-rr 30.0%

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

       1. *-lft-identity 30.0%

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

   12. Simplified 30.0%

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

   if -6.8e6 < z < 4.00000000000000027e-63

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0 72.0%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6800000 \lor \neg \left(z \leq 4 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.4% 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 89.5%

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

3. Taylor expanded in z around 0 45.0%

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

4. Final simplification 45.0%

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

Developer Target 1: 88.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - z) * (t - z)
    if ((x / t_1) < 0.0d0) then
        tmp = (x / (y - z)) / (t - z)
    else
        tmp = x * (1.0d0 / t_1)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024157 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

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

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