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

Percentage Accurate: 89.6% → 97.9%

Time: 11.9s

Alternatives: 17

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: 89.6% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-299}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* (- y z) (- t z)))))
   (* x_s (if (<= t_1 -2e-299) t_1 (/ (/ x_m (- t z)) (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / ((y - z) * (t - z));
	double tmp;
	if (t_1 <= -2e-299) {
		tmp = t_1;
	} else {
		tmp = (x_m / (t - z)) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / ((y - z) * (t - z))
    if (t_1 <= (-2d-299)) then
        tmp = t_1
    else
        tmp = (x_m / (t - z)) / (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / ((y - z) * (t - z));
	double tmp;
	if (t_1 <= -2e-299) {
		tmp = t_1;
	} else {
		tmp = (x_m / (t - z)) / (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / ((y - z) * (t - z))
	tmp = 0
	if t_1 <= -2e-299:
		tmp = t_1
	else:
		tmp = (x_m / (t - z)) / (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)))
	tmp = 0.0
	if (t_1 <= -2e-299)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / ((y - z) * (t - z));
	tmp = 0.0;
	if (t_1 <= -2e-299)
		tmp = t_1;
	else
		tmp = (x_m / (t - z)) / (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -2e-299], t$95$1, N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < -1.99999999999999998e-299

   1. Initial program 96.9%

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

   if -1.99999999999999998e-299 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 89.6%

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

      1. associate-/l/ 97.1%

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

   3. Simplified 97.1%

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

Alternative 2: 82.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{t - z}}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{+204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \end{array} \]

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 (- t z)) y)))
   (*
    x_s
    (if (<= y -2.25e+204)
      t_1
      (if (<= y -2e-16)
        (/ x_m (* y (- t z)))
        (if (<= y -5.6e-81)
          t_1
          (if (<= y 6e-135) (/ (/ 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 t_1 = (x_m / (t - z)) / y;
	double tmp;
	if (y <= -2.25e+204) {
		tmp = t_1;
	} else if (y <= -2e-16) {
		tmp = x_m / (y * (t - z));
	} else if (y <= -5.6e-81) {
		tmp = t_1;
	} else if (y <= 6e-135) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (x_m / (t - z)) / y
    if (y <= (-2.25d+204)) then
        tmp = t_1
    else if (y <= (-2d-16)) then
        tmp = x_m / (y * (t - z))
    else if (y <= (-5.6d-81)) then
        tmp = t_1
    else if (y <= 6d-135) 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 t_1 = (x_m / (t - z)) / y;
	double tmp;
	if (y <= -2.25e+204) {
		tmp = t_1;
	} else if (y <= -2e-16) {
		tmp = x_m / (y * (t - z));
	} else if (y <= -5.6e-81) {
		tmp = t_1;
	} else if (y <= 6e-135) {
		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):
	t_1 = (x_m / (t - z)) / y
	tmp = 0
	if y <= -2.25e+204:
		tmp = t_1
	elif y <= -2e-16:
		tmp = x_m / (y * (t - z))
	elif y <= -5.6e-81:
		tmp = t_1
	elif y <= 6e-135:
		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)
	t_1 = Float64(Float64(x_m / Float64(t - z)) / y)
	tmp = 0.0
	if (y <= -2.25e+204)
		tmp = t_1;
	elseif (y <= -2e-16)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (y <= -5.6e-81)
		tmp = t_1;
	elseif (y <= 6e-135)
		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)
	t_1 = (x_m / (t - z)) / y;
	tmp = 0.0;
	if (y <= -2.25e+204)
		tmp = t_1;
	elseif (y <= -2e-16)
		tmp = x_m / (y * (t - z));
	elseif (y <= -5.6e-81)
		tmp = t_1;
	elseif (y <= 6e-135)
		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_] := Block[{t$95$1 = N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -2.25e+204], t$95$1, If[LessEqual[y, -2e-16], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.6e-81], t$95$1, If[LessEqual[y, 6e-135], 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 4 regimes

2. if y < -2.25000000000000001e204 or -2e-16 < y < -5.5999999999999998e-81

   1. Initial program 92.4%

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

      1. associate-/l/ 99.9%

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

      2. clear-num 99.8%

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

      3. inv-pow 99.8%

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

      4. div-inv 99.8%

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

      5. clear-num 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 81.0%

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

      1. *-lft-identity 81.0%

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

      2. times-frac 88.6%

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

      3. associate-*l/ 88.6%

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

      4. *-lft-identity 88.6%

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

   9. Simplified 88.6%

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

   if -2.25000000000000001e204 < y < -2e-16

   1. Initial program 88.3%

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

   3. Taylor expanded in y around inf 82.4%

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

      1. *-commutative 82.4%

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

   5. Simplified 82.4%

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

   if -5.5999999999999998e-81 < y < 6.00000000000000024e-135

   1. Initial program 93.7%

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

   3. Taylor expanded in x around 0 93.7%

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

      1. associate-/l/ 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in y around 0 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   8. Simplified 87.4%

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

   if 6.00000000000000024e-135 < y

   1. Initial program 91.0%

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

      1. associate-/l/ 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in t around inf 70.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{t - z}}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -6.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \end{array} \]

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 (- t z)) y)))
   (*
    x_s
    (if (<= y -1.55e+203)
      t_1
      (if (<= y -6.9e-18)
        (/ x_m (* y (- t z)))
        (if (<= y -1.7e-86)
          t_1
          (if (<= y 8.6e-137)
            (/ 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 t_1 = (x_m / (t - z)) / y;
	double tmp;
	if (y <= -1.55e+203) {
		tmp = t_1;
	} else if (y <= -6.9e-18) {
		tmp = x_m / (y * (t - z));
	} else if (y <= -1.7e-86) {
		tmp = t_1;
	} else if (y <= 8.6e-137) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (x_m / (t - z)) / y
    if (y <= (-1.55d+203)) then
        tmp = t_1
    else if (y <= (-6.9d-18)) then
        tmp = x_m / (y * (t - z))
    else if (y <= (-1.7d-86)) then
        tmp = t_1
    else if (y <= 8.6d-137) 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 t_1 = (x_m / (t - z)) / y;
	double tmp;
	if (y <= -1.55e+203) {
		tmp = t_1;
	} else if (y <= -6.9e-18) {
		tmp = x_m / (y * (t - z));
	} else if (y <= -1.7e-86) {
		tmp = t_1;
	} else if (y <= 8.6e-137) {
		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):
	t_1 = (x_m / (t - z)) / y
	tmp = 0
	if y <= -1.55e+203:
		tmp = t_1
	elif y <= -6.9e-18:
		tmp = x_m / (y * (t - z))
	elif y <= -1.7e-86:
		tmp = t_1
	elif y <= 8.6e-137:
		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)
	t_1 = Float64(Float64(x_m / Float64(t - z)) / y)
	tmp = 0.0
	if (y <= -1.55e+203)
		tmp = t_1;
	elseif (y <= -6.9e-18)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (y <= -1.7e-86)
		tmp = t_1;
	elseif (y <= 8.6e-137)
		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)
	t_1 = (x_m / (t - z)) / y;
	tmp = 0.0;
	if (y <= -1.55e+203)
		tmp = t_1;
	elseif (y <= -6.9e-18)
		tmp = x_m / (y * (t - z));
	elseif (y <= -1.7e-86)
		tmp = t_1;
	elseif (y <= 8.6e-137)
		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_] := Block[{t$95$1 = N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.55e+203], t$95$1, If[LessEqual[y, -6.9e-18], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.7e-86], t$95$1, If[LessEqual[y, 8.6e-137], 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 4 regimes

2. if y < -1.55e203 or -6.9000000000000003e-18 < y < -1.7e-86

   1. Initial program 92.4%

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

      1. associate-/l/ 99.9%

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

      2. clear-num 99.8%

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

      3. inv-pow 99.8%

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

      4. div-inv 99.8%

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

      5. clear-num 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 81.0%

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

      1. *-lft-identity 81.0%

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

      2. times-frac 88.6%

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

      3. associate-*l/ 88.6%

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

      4. *-lft-identity 88.6%

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

   9. Simplified 88.6%

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

   if -1.55e203 < y < -6.9000000000000003e-18

   1. Initial program 88.3%

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

   3. Taylor expanded in y around inf 82.4%

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

      1. *-commutative 82.4%

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

   5. Simplified 82.4%

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

   if -1.7e-86 < y < 8.59999999999999959e-137

   1. Initial program 93.6%

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

   3. Taylor expanded in y around 0 86.3%

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

      1. associate-*r/ 86.3%

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

      2. neg-mul-1 86.3%

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

   5. Simplified 86.3%

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

   if 8.59999999999999959e-137 < y

   1. Initial program 91.1%

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

      1. associate-/l/ 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in t around inf 71.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y}\\ \mathbf{elif}\;y \leq -6.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{t \cdot \frac{y}{x\_m}}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{+57}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ 1.0 (* t (/ y x_m)))))
   (*
    x_s
    (if (<= y -2.6e+123)
      t_1
      (if (<= y -1.9e+57)
        (/ x_m (* z (- y)))
        (if (<= y -1.55e-86)
          (/ (/ x_m t) y)
          (if (<= y 1.05e-141) (/ (/ x_m 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 = 1.0 / (t * (y / x_m));
	double tmp;
	if (y <= -2.6e+123) {
		tmp = t_1;
	} else if (y <= -1.9e+57) {
		tmp = x_m / (z * -y);
	} else if (y <= -1.55e-86) {
		tmp = (x_m / t) / y;
	} else if (y <= 1.05e-141) {
		tmp = (x_m / 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 = 1.0d0 / (t * (y / x_m))
    if (y <= (-2.6d+123)) then
        tmp = t_1
    else if (y <= (-1.9d+57)) then
        tmp = x_m / (z * -y)
    else if (y <= (-1.55d-86)) then
        tmp = (x_m / t) / y
    else if (y <= 1.05d-141) then
        tmp = (x_m / 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 = 1.0 / (t * (y / x_m));
	double tmp;
	if (y <= -2.6e+123) {
		tmp = t_1;
	} else if (y <= -1.9e+57) {
		tmp = x_m / (z * -y);
	} else if (y <= -1.55e-86) {
		tmp = (x_m / t) / y;
	} else if (y <= 1.05e-141) {
		tmp = (x_m / 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 = 1.0 / (t * (y / x_m))
	tmp = 0
	if y <= -2.6e+123:
		tmp = t_1
	elif y <= -1.9e+57:
		tmp = x_m / (z * -y)
	elif y <= -1.55e-86:
		tmp = (x_m / t) / y
	elif y <= 1.05e-141:
		tmp = (x_m / 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(1.0 / Float64(t * Float64(y / x_m)))
	tmp = 0.0
	if (y <= -2.6e+123)
		tmp = t_1;
	elseif (y <= -1.9e+57)
		tmp = Float64(x_m / Float64(z * Float64(-y)));
	elseif (y <= -1.55e-86)
		tmp = Float64(Float64(x_m / t) / y);
	elseif (y <= 1.05e-141)
		tmp = Float64(Float64(x_m / z) / Float64(-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 = 1.0 / (t * (y / x_m));
	tmp = 0.0;
	if (y <= -2.6e+123)
		tmp = t_1;
	elseif (y <= -1.9e+57)
		tmp = x_m / (z * -y);
	elseif (y <= -1.55e-86)
		tmp = (x_m / t) / y;
	elseif (y <= 1.05e-141)
		tmp = (x_m / 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[(1.0 / N[(t * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -2.6e+123], t$95$1, If[LessEqual[y, -1.9e+57], N[(x$95$m / N[(z * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.55e-86], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.05e-141], N[(N[(x$95$m / z), $MachinePrecision] / (-t)), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -2.59999999999999985e123 or 1.05e-141 < y

   1. Initial program 89.7%

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

   3. Taylor expanded in z around 0 58.8%

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

      1. clear-num 58.8%

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

      2. inv-pow 58.8%

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

      3. *-commutative 58.8%

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

      4. associate-/l* 65.2%

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

   5. Applied egg-rr 65.2%

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

      1. unpow-1 65.2%

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

      2. associate-*r/ 58.8%

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

      3. *-commutative 58.8%

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

      4. associate-/l* 69.1%

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

   7. Simplified 69.1%

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

   if -2.59999999999999985e123 < y < -1.8999999999999999e57

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf 94.5%

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

      1. *-commutative 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in t around 0 53.6%

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

      1. mul-1-neg 53.6%

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

      2. distribute-neg-frac2 53.6%

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

      3. distribute-rgt-neg-out 53.6%

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

   8. Simplified 53.6%

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

   if -1.8999999999999999e57 < y < -1.54999999999999994e-86

   1. Initial program 91.3%

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

      1. associate-/l/ 99.7%

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

      2. clear-num 99.4%

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

      3. inv-pow 99.4%

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

      4. div-inv 99.5%

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

      5. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 95.0%

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

      3. inv-pow 95.0%

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

      4. associate-/r/ 95.2%

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

      5. *-commutative 95.2%

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

      6. clear-num 94.9%

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

      7. associate-*l/ 95.4%

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

      8. *-un-lft-identity 95.4%

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

   6. Applied egg-rr 95.4%

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

      1. clear-num 95.1%

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

      2. associate-/r/ 95.1%

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

   8. Applied egg-rr 95.1%

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

   9. Taylor expanded in z around 0 55.0%

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

       1. associate-/r* 54.9%

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

   11. Simplified 54.9%

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

   if -1.54999999999999994e-86 < y < 1.05e-141

   1. Initial program 93.6%

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

      1. associate-/l/ 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in t around inf 51.5%

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

   6. Taylor expanded in y around 0 50.3%

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

      1. associate-*r/ 50.3%

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

      2. neg-mul-1 50.3%

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

      3. *-commutative 50.3%

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

   8. Simplified 50.3%

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

   9. Taylor expanded in x around 0 50.3%

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

       1. mul-1-neg 50.3%

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

       2. associate-/l/ 54.9%

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

       3. distribute-neg-frac2 54.9%

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

   11. Simplified 54.9%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{t \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{z \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{t}}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+94}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+170}:\\ \;\;\;\;\frac{x\_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 t) y)))
   (*
    x_s
    (if (<= t -5.5e-65)
      t_1
      (if (<= t 2.1e-109)
        (/ x_m (* z (- y)))
        (if (<= t 2.15e+94)
          (/ x_m (* y t))
          (if (<= t 1.2e+170) (/ x_m (* 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 / t) / y;
	double tmp;
	if (t <= -5.5e-65) {
		tmp = t_1;
	} else if (t <= 2.1e-109) {
		tmp = x_m / (z * -y);
	} else if (t <= 2.15e+94) {
		tmp = x_m / (y * t);
	} else if (t <= 1.2e+170) {
		tmp = x_m / (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 / t) / y
    if (t <= (-5.5d-65)) then
        tmp = t_1
    else if (t <= 2.1d-109) then
        tmp = x_m / (z * -y)
    else if (t <= 2.15d+94) then
        tmp = x_m / (y * t)
    else if (t <= 1.2d+170) then
        tmp = x_m / (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 / t) / y;
	double tmp;
	if (t <= -5.5e-65) {
		tmp = t_1;
	} else if (t <= 2.1e-109) {
		tmp = x_m / (z * -y);
	} else if (t <= 2.15e+94) {
		tmp = x_m / (y * t);
	} else if (t <= 1.2e+170) {
		tmp = x_m / (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 / t) / y
	tmp = 0
	if t <= -5.5e-65:
		tmp = t_1
	elif t <= 2.1e-109:
		tmp = x_m / (z * -y)
	elif t <= 2.15e+94:
		tmp = x_m / (y * t)
	elif t <= 1.2e+170:
		tmp = x_m / (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 / t) / y)
	tmp = 0.0
	if (t <= -5.5e-65)
		tmp = t_1;
	elseif (t <= 2.1e-109)
		tmp = Float64(x_m / Float64(z * Float64(-y)));
	elseif (t <= 2.15e+94)
		tmp = Float64(x_m / Float64(y * t));
	elseif (t <= 1.2e+170)
		tmp = Float64(x_m / 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 / t) / y;
	tmp = 0.0;
	if (t <= -5.5e-65)
		tmp = t_1;
	elseif (t <= 2.1e-109)
		tmp = x_m / (z * -y);
	elseif (t <= 2.15e+94)
		tmp = x_m / (y * t);
	elseif (t <= 1.2e+170)
		tmp = x_m / (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 / t), $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -5.5e-65], t$95$1, If[LessEqual[t, 2.1e-109], N[(x$95$m / N[(z * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.15e+94], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e+170], N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < -5.4999999999999999e-65 or 1.2e170 < t

   1. Initial program 90.6%

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

      1. associate-/l/ 95.0%

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

      2. clear-num 94.8%

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

      3. inv-pow 94.8%

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

      4. div-inv 94.7%

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

      5. clear-num 94.8%

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

   4. Applied egg-rr 94.8%

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

      1. *-commutative 94.8%

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

      2. associate-/r/ 95.4%

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

      3. inv-pow 95.4%

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

      4. associate-/r/ 95.8%

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

      5. *-commutative 95.8%

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

      6. clear-num 95.7%

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

      7. associate-*l/ 95.9%

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

      8. *-un-lft-identity 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. clear-num 95.8%

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

      2. associate-/r/ 95.7%

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

   8. Applied egg-rr 95.7%

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

   9. Taylor expanded in z around 0 53.1%

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

       1. associate-/r* 64.1%

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

   11. Simplified 64.1%

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

   if -5.4999999999999999e-65 < t < 2.09999999999999996e-109

   1. Initial program 92.4%

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

   3. Taylor expanded in y around inf 54.1%

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

      1. *-commutative 54.1%

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

   5. Simplified 54.1%

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

   6. Taylor expanded in t around 0 41.1%

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

      1. mul-1-neg 41.1%

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

      2. distribute-neg-frac2 41.1%

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

      3. distribute-rgt-neg-out 41.1%

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

   8. Simplified 41.1%

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

   if 2.09999999999999996e-109 < t < 2.15e94

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0 54.3%

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

   if 2.15e94 < t < 1.2e170

   1. Initial program 3.9%

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

      1. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 6.6%

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

   6. Taylor expanded in y around 0 6.6%

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

      1. associate-*r/ 6.6%

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

      2. neg-mul-1 6.6%

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

      3. *-commutative 6.6%

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

   8. Simplified 6.6%

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

      1. add-sqr-sqrt 6.6%

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

      2. sqrt-unprod 2.6%

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

      3. sqr-neg 2.6%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 2.0%

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

      6. *-un-lft-identity 2.0%

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

      7. associate-/r* 2.0%

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

   10. Applied egg-rr 2.0%

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

       1. *-lft-identity 2.0%

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

       2. associate-/l/ 2.0%

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

       3. *-commutative 2.0%

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

   12. Simplified 2.0%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{z \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+94}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+170}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.35 \cdot 10^{+46}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-86} \lor \neg \left(y \leq 4.1 \cdot 10^{-141}\right):\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{-t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -4.35e+46)
    (/ x_m (* z (- y)))
    (if (or (<= y -1.6e-86) (not (<= y 4.1e-141)))
      (/ (/ x_m t) y)
      (/ (/ x_m z) (- t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -4.35e+46) {
		tmp = x_m / (z * -y);
	} else if ((y <= -1.6e-86) || !(y <= 4.1e-141)) {
		tmp = (x_m / t) / y;
	} else {
		tmp = (x_m / z) / -t;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-4.35d+46)) then
        tmp = x_m / (z * -y)
    else if ((y <= (-1.6d-86)) .or. (.not. (y <= 4.1d-141))) then
        tmp = (x_m / t) / y
    else
        tmp = (x_m / z) / -t
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -4.35e+46) {
		tmp = x_m / (z * -y);
	} else if ((y <= -1.6e-86) || !(y <= 4.1e-141)) {
		tmp = (x_m / t) / y;
	} else {
		tmp = (x_m / z) / -t;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -4.35e+46:
		tmp = x_m / (z * -y)
	elif (y <= -1.6e-86) or not (y <= 4.1e-141):
		tmp = (x_m / t) / y
	else:
		tmp = (x_m / z) / -t
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -4.35e+46)
		tmp = Float64(x_m / Float64(z * Float64(-y)));
	elseif ((y <= -1.6e-86) || !(y <= 4.1e-141))
		tmp = Float64(Float64(x_m / t) / y);
	else
		tmp = Float64(Float64(x_m / z) / Float64(-t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -4.35e+46)
		tmp = x_m / (z * -y);
	elseif ((y <= -1.6e-86) || ~((y <= 4.1e-141)))
		tmp = (x_m / t) / y;
	else
		tmp = (x_m / z) / -t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -4.35e+46], N[(x$95$m / N[(z * (-y)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -1.6e-86], N[Not[LessEqual[y, 4.1e-141]], $MachinePrecision]], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] / (-t)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.34999999999999981e46

   1. Initial program 87.7%

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

   3. Taylor expanded in y around inf 87.6%

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

      1. *-commutative 87.6%

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

   5. Simplified 87.6%

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

   6. Taylor expanded in t around 0 54.2%

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

      1. mul-1-neg 54.2%

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

      2. distribute-neg-frac2 54.2%

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

      3. distribute-rgt-neg-out 54.2%

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

   8. Simplified 54.2%

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

   if -4.34999999999999981e46 < y < -1.60000000000000003e-86 or 4.10000000000000002e-141 < y

   1. Initial program 91.8%

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

      1. associate-/l/ 98.6%

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

      2. clear-num 98.1%

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

      3. inv-pow 98.1%

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

      4. div-inv 98.0%

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

      5. clear-num 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-/r/ 95.7%

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

      3. inv-pow 95.7%

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

      4. associate-/r/ 96.3%

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

      5. *-commutative 96.3%

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

      6. clear-num 96.0%

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

      7. associate-*l/ 96.2%

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

      8. *-un-lft-identity 96.2%

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

   6. Applied egg-rr 96.2%

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

      1. clear-num 96.0%

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

      2. associate-/r/ 96.0%

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

   8. Applied egg-rr 96.0%

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

   9. Taylor expanded in z around 0 57.4%

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

       1. associate-/r* 62.4%

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

   11. Simplified 62.4%

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

   if -1.60000000000000003e-86 < y < 4.10000000000000002e-141

   1. Initial program 93.6%

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

      1. associate-/l/ 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in t around inf 51.5%

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

   6. Taylor expanded in y around 0 50.3%

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

      1. associate-*r/ 50.3%

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

      2. neg-mul-1 50.3%

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

      3. *-commutative 50.3%

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

   8. Simplified 50.3%

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

   9. Taylor expanded in x around 0 50.3%

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

       1. mul-1-neg 50.3%

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

       2. associate-/l/ 54.9%

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

       3. distribute-neg-frac2 54.9%

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

   11. Simplified 54.9%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.35 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{z \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-86} \lor \neg \left(y \leq 4.1 \cdot 10^{-141}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-85} \lor \neg \left(y \leq 2.75 \cdot 10^{-139}\right):\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{-z \cdot t}\\ \end{array} \end{array} \]

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 (<= y -7.8e+46)
    (/ x_m (* z (- y)))
    (if (or (<= y -6e-85) (not (<= y 2.75e-139)))
      (/ (/ x_m t) y)
      (/ x_m (- (* z t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -7.8e+46) {
		tmp = x_m / (z * -y);
	} else if ((y <= -6e-85) || !(y <= 2.75e-139)) {
		tmp = (x_m / t) / y;
	} else {
		tmp = x_m / -(z * t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-7.8d+46)) then
        tmp = x_m / (z * -y)
    else if ((y <= (-6d-85)) .or. (.not. (y <= 2.75d-139))) then
        tmp = (x_m / t) / y
    else
        tmp = x_m / -(z * t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -7.8e+46) {
		tmp = x_m / (z * -y);
	} else if ((y <= -6e-85) || !(y <= 2.75e-139)) {
		tmp = (x_m / t) / y;
	} else {
		tmp = x_m / -(z * t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -7.8e+46:
		tmp = x_m / (z * -y)
	elif (y <= -6e-85) or not (y <= 2.75e-139):
		tmp = (x_m / t) / y
	else:
		tmp = x_m / -(z * t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -7.8e+46)
		tmp = Float64(x_m / Float64(z * Float64(-y)));
	elseif ((y <= -6e-85) || !(y <= 2.75e-139))
		tmp = Float64(Float64(x_m / t) / y);
	else
		tmp = Float64(x_m / Float64(-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 (y <= -7.8e+46)
		tmp = x_m / (z * -y);
	elseif ((y <= -6e-85) || ~((y <= 2.75e-139)))
		tmp = (x_m / t) / y;
	else
		tmp = x_m / -(z * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -7.8e+46], N[(x$95$m / N[(z * (-y)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -6e-85], N[Not[LessEqual[y, 2.75e-139]], $MachinePrecision]], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision], N[(x$95$m / (-N[(z * t), $MachinePrecision])), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999999e46

   1. Initial program 87.7%

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

   3. Taylor expanded in y around inf 87.6%

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

      1. *-commutative 87.6%

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

   5. Simplified 87.6%

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

   6. Taylor expanded in t around 0 54.2%

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

      1. mul-1-neg 54.2%

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

      2. distribute-neg-frac2 54.2%

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

      3. distribute-rgt-neg-out 54.2%

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

   8. Simplified 54.2%

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

   if -7.7999999999999999e46 < y < -6.00000000000000044e-85 or 2.7499999999999998e-139 < y

   1. Initial program 91.8%

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

      1. associate-/l/ 98.6%

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

      2. clear-num 98.1%

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

      3. inv-pow 98.1%

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

      4. div-inv 98.0%

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

      5. clear-num 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-/r/ 95.7%

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

      3. inv-pow 95.7%

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

      4. associate-/r/ 96.3%

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

      5. *-commutative 96.3%

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

      6. clear-num 96.0%

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

      7. associate-*l/ 96.2%

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

      8. *-un-lft-identity 96.2%

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

   6. Applied egg-rr 96.2%

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

      1. clear-num 96.0%

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

      2. associate-/r/ 96.0%

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

   8. Applied egg-rr 96.0%

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

   9. Taylor expanded in z around 0 57.4%

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

       1. associate-/r* 62.4%

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

   11. Simplified 62.4%

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

   if -6.00000000000000044e-85 < y < 2.7499999999999998e-139

   1. Initial program 93.6%

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

      1. associate-/l/ 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in t around inf 51.5%

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

   6. Taylor expanded in y around 0 50.3%

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

      1. associate-*r/ 50.3%

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

      2. neg-mul-1 50.3%

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

      3. *-commutative 50.3%

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

   8. Simplified 50.3%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{z \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-85} \lor \neg \left(y \leq 2.75 \cdot 10^{-139}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-81}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-1}{z - t}}{y}\\ \mathbf{elif}\;y \leq 2.06 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

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 -3.1e-81)
    (* x_m (/ (/ -1.0 (- z t)) y))
    (if (<= y 2.06e-135) (/ (/ 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 <= -3.1e-81) {
		tmp = x_m * ((-1.0 / (z - t)) / y);
	} else if (y <= 2.06e-135) {
		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 <= (-3.1d-81)) then
        tmp = x_m * (((-1.0d0) / (z - t)) / y)
    else if (y <= 2.06d-135) 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 <= -3.1e-81) {
		tmp = x_m * ((-1.0 / (z - t)) / y);
	} else if (y <= 2.06e-135) {
		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 <= -3.1e-81:
		tmp = x_m * ((-1.0 / (z - t)) / y)
	elif y <= 2.06e-135:
		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 <= -3.1e-81)
		tmp = Float64(x_m * Float64(Float64(-1.0 / Float64(z - t)) / y));
	elseif (y <= 2.06e-135)
		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 <= -3.1e-81)
		tmp = x_m * ((-1.0 / (z - t)) / y);
	elseif (y <= 2.06e-135)
		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, -3.1e-81], N[(x$95$m * N[(N[(-1.0 / N[(z - t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.06e-135], 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 < -3.09999999999999988e-81

   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/ 96.0%

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

      2. clear-num 95.8%

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

      3. inv-pow 95.8%

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

      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. Applied egg-rr 95.9%

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

   7. Taylor expanded in y around inf 82.0%

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

      1. *-lft-identity 82.0%

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

      2. times-frac 81.3%

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

      3. associate-*l/ 81.3%

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

      4. *-lft-identity 81.3%

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

   9. Simplified 81.3%

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

       1. div-inv 81.2%

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

       2. associate-/l* 83.6%

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

   11. Applied egg-rr 83.6%

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

   if -3.09999999999999988e-81 < y < 2.0599999999999999e-135

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0 93.6%

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

      1. associate-/l/ 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in y around 0 88.4%

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

      1. associate-*r/ 88.4%

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

      2. neg-mul-1 88.4%

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

   8. Simplified 88.4%

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

   if 2.0599999999999999e-135 < y

   1. Initial program 91.1%

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

      1. associate-/l/ 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in t around inf 71.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{z - t}}{y}\\ \mathbf{elif}\;y \leq 2.06 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1750:\\ \;\;\;\;\frac{1}{t \cdot \frac{y}{x\_m}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{-x\_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -1750.0)
    (/ 1.0 (* t (/ y x_m)))
    (if (<= t 3.8e-113) (/ (/ (- x_m) z) y) (/ x_m (* (- y z) t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -1750.0) {
		tmp = 1.0 / (t * (y / x_m));
	} else if (t <= 3.8e-113) {
		tmp = (-x_m / z) / y;
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1750.0d0)) then
        tmp = 1.0d0 / (t * (y / x_m))
    else if (t <= 3.8d-113) then
        tmp = (-x_m / z) / y
    else
        tmp = x_m / ((y - z) * t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -1750.0) {
		tmp = 1.0 / (t * (y / x_m));
	} else if (t <= 3.8e-113) {
		tmp = (-x_m / z) / y;
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -1750.0:
		tmp = 1.0 / (t * (y / x_m))
	elif t <= 3.8e-113:
		tmp = (-x_m / z) / y
	else:
		tmp = x_m / ((y - z) * t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -1750.0)
		tmp = Float64(1.0 / Float64(t * Float64(y / x_m)));
	elseif (t <= 3.8e-113)
		tmp = Float64(Float64(Float64(-x_m) / z) / y);
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -1750.0)
		tmp = 1.0 / (t * (y / x_m));
	elseif (t <= 3.8e-113)
		tmp = (-x_m / z) / y;
	else
		tmp = x_m / ((y - z) * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -1750.0], N[(1.0 / N[(t * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e-113], N[(N[((-x$95$m) / z), $MachinePrecision] / y), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1750

   1. Initial program 93.3%

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

   3. Taylor expanded in z around 0 51.6%

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

      1. clear-num 51.5%

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

      2. inv-pow 51.5%

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

      3. *-commutative 51.5%

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

      4. associate-/l* 65.3%

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

   5. Applied egg-rr 65.3%

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

      1. unpow-1 65.3%

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

      2. associate-*r/ 51.5%

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

      3. *-commutative 51.5%

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

      4. associate-/l* 55.3%

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

   7. Simplified 55.3%

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

   if -1750 < t < 3.79999999999999983e-113

   1. Initial program 93.3%

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

      1. associate-/l/ 97.7%

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

      2. clear-num 97.2%

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

      3. inv-pow 97.2%

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

      4. div-inv 97.2%

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

      5. clear-num 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. unpow-1 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in y around inf 58.0%

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

      1. *-lft-identity 58.0%

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

      2. times-frac 61.8%

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

      3. associate-*l/ 61.8%

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

      4. *-lft-identity 61.8%

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

   9. Simplified 61.8%

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

   10. Taylor expanded in t around 0 43.7%

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

       1. associate-*r/ 43.7%

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

       2. neg-mul-1 43.7%

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

   12. Simplified 43.7%

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

   if 3.79999999999999983e-113 < t

   1. Initial program 88.7%

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

   3. Taylor expanded in t around inf 75.4%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1750:\\ \;\;\;\;\frac{1}{t \cdot \frac{y}{x}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{-x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq 4.65 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{-x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \end{array} \end{array} \]

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 (<= y -4.8e-82)
    (* (/ 1.0 t) (/ x_m y))
    (if (<= y 4.65e-141) (/ (/ (- x_m) z) t) (/ (/ 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 (y <= -4.8e-82) {
		tmp = (1.0 / t) * (x_m / y);
	} else if (y <= 4.65e-141) {
		tmp = (-x_m / z) / t;
	} 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 (y <= (-4.8d-82)) then
        tmp = (1.0d0 / t) * (x_m / y)
    else if (y <= 4.65d-141) then
        tmp = (-x_m / z) / t
    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 (y <= -4.8e-82) {
		tmp = (1.0 / t) * (x_m / y);
	} else if (y <= 4.65e-141) {
		tmp = (-x_m / z) / t;
	} 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 y <= -4.8e-82:
		tmp = (1.0 / t) * (x_m / y)
	elif y <= 4.65e-141:
		tmp = (-x_m / z) / t
	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 (y <= -4.8e-82)
		tmp = Float64(Float64(1.0 / t) * Float64(x_m / y));
	elseif (y <= 4.65e-141)
		tmp = Float64(Float64(Float64(-x_m) / z) / t);
	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 (y <= -4.8e-82)
		tmp = (1.0 / t) * (x_m / y);
	elseif (y <= 4.65e-141)
		tmp = (-x_m / z) / t;
	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[LessEqual[y, -4.8e-82], N[(N[(1.0 / t), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.65e-141], N[(N[((-x$95$m) / z), $MachinePrecision] / t), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.80000000000000017e-82

   1. Initial program 89.7%

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

   3. Taylor expanded in z around 0 58.6%

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

      1. *-un-lft-identity 58.6%

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

      2. times-frac 65.4%

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

   5. Applied egg-rr 65.4%

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

   if -4.80000000000000017e-82 < y < 4.6500000000000001e-141

   1. Initial program 93.6%

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

      1. associate-/l/ 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in t around inf 51.5%

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

   6. Taylor expanded in y around 0 50.3%

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

      1. associate-*r/ 50.3%

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

      2. neg-mul-1 50.3%

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

      3. *-commutative 50.3%

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

   8. Simplified 50.3%

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

   9. Taylor expanded in x around 0 50.3%

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

       1. mul-1-neg 50.3%

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

       2. associate-/l/ 54.9%

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

       3. distribute-neg-frac2 54.9%

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

   11. Simplified 54.9%

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

   if 4.6500000000000001e-141 < y

   1. Initial program 91.1%

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

      1. associate-/l/ 98.4%

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

      2. clear-num 97.8%

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

      3. inv-pow 97.8%

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

      4. div-inv 97.7%

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

      5. clear-num 97.9%

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

   4. Applied egg-rr 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-/r/ 96.0%

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

      3. inv-pow 96.0%

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

      4. associate-/r/ 96.6%

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

      5. *-commutative 96.6%

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

      6. clear-num 96.3%

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

      7. associate-*l/ 96.4%

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

      8. *-un-lft-identity 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. clear-num 96.3%

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

      2. associate-/r/ 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in z around 0 57.7%

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

       1. associate-/r* 64.0%

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

   11. Simplified 64.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 4.65 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+65} \lor \neg \left(z \leq 1.4 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x\_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \end{array} \end{array} \]

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 -1.8e+65) (not (<= z 1.4e+67)))
    (/ 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 <= -1.8e+65) || !(z <= 1.4e+67)) {
		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 <= (-1.8d+65)) .or. (.not. (z <= 1.4d+67))) 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 <= -1.8e+65) || !(z <= 1.4e+67)) {
		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 <= -1.8e+65) or not (z <= 1.4e+67):
		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 <= -1.8e+65) || !(z <= 1.4e+67))
		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 <= -1.8e+65) || ~((z <= 1.4e+67)))
		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, -1.8e+65], N[Not[LessEqual[z, 1.4e+67]], $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 < -1.79999999999999989e65 or 1.3999999999999999e67 < z

   1. Initial program 86.0%

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

      1. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 45.1%

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

   6. Taylor expanded in y around 0 42.1%

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

      1. associate-*r/ 42.1%

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

      2. neg-mul-1 42.1%

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

      3. *-commutative 42.1%

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

   8. Simplified 42.1%

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

      1. add-sqr-sqrt 25.0%

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

      2. sqrt-unprod 46.2%

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

      3. sqr-neg 46.2%

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

      4. sqrt-unprod 17.1%

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

      5. add-sqr-sqrt 41.0%

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

      6. *-un-lft-identity 41.0%

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

      7. associate-/r* 51.2%

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

   10. Applied egg-rr 51.2%

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

       1. *-lft-identity 51.2%

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

       2. associate-/l/ 41.0%

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

       3. *-commutative 41.0%

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

   12. Simplified 41.0%

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

   if -1.79999999999999989e65 < z < 1.3999999999999999e67

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0 52.9%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+65} \lor \neg \left(z \leq 1.4 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array} \end{array} \]

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 (<= z -8.5e+149) (/ (/ x_m z) (- z t)) (/ x_m (* (- y z) (- t z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -8.5e+149) {
		tmp = (x_m / z) / (z - t);
	} else {
		tmp = x_m / ((y - z) * (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 <= (-8.5d+149)) then
        tmp = (x_m / z) / (z - t)
    else
        tmp = x_m / ((y - z) * (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 <= -8.5e+149) {
		tmp = (x_m / z) / (z - t);
	} else {
		tmp = x_m / ((y - z) * (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 <= -8.5e+149:
		tmp = (x_m / z) / (z - t)
	else:
		tmp = x_m / ((y - z) * (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 <= -8.5e+149)
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * 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 <= -8.5e+149)
		tmp = (x_m / z) / (z - t);
	else
		tmp = x_m / ((y - z) * (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[LessEqual[z, -8.5e+149], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -8.49999999999999956e149

   1. Initial program 79.3%

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

   3. Taylor expanded in x around 0 79.3%

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

      1. associate-/l/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 99.6%

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

      1. associate-*r/ 99.6%

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

      2. neg-mul-1 99.6%

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

   8. Simplified 99.6%

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

   if -8.49999999999999956e149 < z

   1. Initial program 92.9%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= t 2.45e+27) (/ (/ x_m (- t z)) y) (/ (/ x_m t) (- y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 2.45e+27) {
		tmp = (x_m / (t - z)) / y;
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.45d+27) then
        tmp = (x_m / (t - z)) / y
    else
        tmp = (x_m / t) / (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 2.45e+27) {
		tmp = (x_m / (t - z)) / y;
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 2.45e+27:
		tmp = (x_m / (t - z)) / y
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 2.45e+27)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 2.45e+27)
		tmp = (x_m / (t - z)) / y;
	else
		tmp = (x_m / t) / (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 2.45e+27], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.45000000000000007e27

   1. Initial program 92.9%

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

      1. associate-/l/ 95.3%

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

      2. clear-num 95.0%

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

      3. inv-pow 95.0%

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

      4. div-inv 94.9%

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

      5. clear-num 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. unpow-1 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in y around inf 58.7%

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

      1. *-lft-identity 58.7%

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

      2. times-frac 63.3%

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

      3. associate-*l/ 63.3%

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

      4. *-lft-identity 63.3%

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

   9. Simplified 63.3%

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

   if 2.45000000000000007e27 < t

   1. Initial program 87.5%

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

      1. associate-/l/ 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t around inf 92.7%

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

Alternative 14: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{-17}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= y -1.16e-17) (/ x_m (* y (- t z))) (/ (/ x_m t) (- y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.16e-17) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.16d-17)) then
        tmp = x_m / (y * (t - z))
    else
        tmp = (x_m / t) / (y - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.16e-17) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -1.16e-17:
		tmp = x_m / (y * (t - z))
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -1.16e-17)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -1.16e-17)
		tmp = x_m / (y * (t - z));
	else
		tmp = (x_m / t) / (y - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -1.16e-17], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.16e-17

   1. Initial program 88.4%

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

   3. Taylor expanded in y around inf 84.1%

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

      1. *-commutative 84.1%

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

   5. Simplified 84.1%

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

   if -1.16e-17 < y

   1. Initial program 92.7%

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

      1. associate-/l/ 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in t around inf 61.0%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.3% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= y -3.5e-10) (/ x_m (* y (- t z))) (/ x_m (* (- y z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.5e-10) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3.5d-10)) then
        tmp = x_m / (y * (t - z))
    else
        tmp = x_m / ((y - z) * t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.5e-10) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -3.5e-10:
		tmp = x_m / (y * (t - z))
	else:
		tmp = x_m / ((y - z) * t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -3.5e-10)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -3.5e-10)
		tmp = x_m / (y * (t - z));
	else
		tmp = x_m / ((y - z) * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -3.5e-10], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -3.4999999999999998e-10

   1. Initial program 88.1%

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

   3. Taylor expanded in y around inf 85.1%

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

      1. *-commutative 85.1%

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

   5. Simplified 85.1%

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

   if -3.4999999999999998e-10 < y

   1. Initial program 92.8%

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

   3. Taylor expanded in t around inf 60.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+118}:\\ \;\;\;\;\frac{x\_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \end{array} \end{array} \]

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 (<= z -1.3e+118) (/ x_m (* z t)) (/ (/ 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 <= -1.3e+118) {
		tmp = x_m / (z * t);
	} 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 <= (-1.3d+118)) then
        tmp = x_m / (z * t)
    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 <= -1.3e+118) {
		tmp = x_m / (z * t);
	} 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 <= -1.3e+118:
		tmp = x_m / (z * t)
	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 <= -1.3e+118)
		tmp = Float64(x_m / Float64(z * t));
	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 <= -1.3e+118)
		tmp = x_m / (z * t);
	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[LessEqual[z, -1.3e+118], N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.30000000000000008e118

   1. Initial program 84.9%

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

      1. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 33.0%

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

   6. Taylor expanded in y around 0 33.0%

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

      1. associate-*r/ 33.0%

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

      2. neg-mul-1 33.0%

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

      3. *-commutative 33.0%

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

   8. Simplified 33.0%

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

      1. add-sqr-sqrt 22.4%

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

      2. sqrt-unprod 35.1%

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

      3. sqr-neg 35.1%

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

      4. sqrt-unprod 10.5%

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

      5. add-sqr-sqrt 32.8%

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

      6. *-un-lft-identity 32.8%

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

      7. associate-/r* 41.5%

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

   10. Applied egg-rr 41.5%

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

       1. *-lft-identity 41.5%

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

       2. associate-/l/ 32.8%

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

       3. *-commutative 32.8%

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

   12. Simplified 32.8%

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

   if -1.30000000000000008e118 < z

   1. Initial program 92.6%

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

      1. associate-/l/ 95.4%

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

      2. clear-num 95.2%

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

      3. inv-pow 95.2%

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

      4. div-inv 95.1%

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

      5. clear-num 95.3%

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

   4. Applied egg-rr 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-/r/ 95.7%

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

      3. inv-pow 95.7%

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

      4. associate-/r/ 95.9%

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

      5. *-commutative 95.9%

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

      6. clear-num 95.8%

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

      7. associate-*l/ 95.9%

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

      8. *-un-lft-identity 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. clear-num 95.8%

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

      2. associate-/r/ 95.7%

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

   8. Applied egg-rr 95.7%

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

   9. Taylor expanded in z around 0 46.2%

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

       1. associate-/r* 51.6%

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

   11. Simplified 51.6%

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

Alternative 17: 39.1% 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 91.6%

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

3. Taylor expanded in z around 0 42.5%

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

4. Final simplification 42.5%

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

Developer target: 88.1% 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 2024086 
(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))))