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

Percentage Accurate: 88.8% → 98.0%

Time: 11.8s

Alternatives: 19

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)} \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-1}{z - y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \end{array} \end{array} \]

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 (<= (/ x_m (* (- y z) (- t z))) -1e-310)
    (* x_m (/ (/ -1.0 (- z y)) (- t z)))
    (/ (/ 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 tmp;
	if ((x_m / ((y - z) * (t - z))) <= -1e-310) {
		tmp = x_m * ((-1.0 / (z - y)) / (t - z));
	} 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) :: tmp
    if ((x_m / ((y - z) * (t - z))) <= (-1d-310)) then
        tmp = x_m * (((-1.0d0) / (z - y)) / (t - z))
    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 tmp;
	if ((x_m / ((y - z) * (t - z))) <= -1e-310) {
		tmp = x_m * ((-1.0 / (z - y)) / (t - z));
	} 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):
	tmp = 0
	if (x_m / ((y - z) * (t - z))) <= -1e-310:
		tmp = x_m * ((-1.0 / (z - y)) / (t - z))
	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)
	tmp = 0.0
	if (Float64(x_m / Float64(Float64(y - z) * Float64(t - z))) <= -1e-310)
		tmp = Float64(x_m * Float64(Float64(-1.0 / Float64(z - y)) / Float64(t - z)));
	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)
	tmp = 0.0;
	if ((x_m / ((y - z) * (t - z))) <= -1e-310)
		tmp = x_m * ((-1.0 / (z - y)) / (t - z));
	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_] := N[(x$95$s * If[LessEqual[N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-310], N[(x$95$m * N[(N[(-1.0 / N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < -9.999999999999969e-311

   1. Initial program 99.6%

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

      1. associate-/l/ 94.4%

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

      2. div-inv 94.4%

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

      3. div-inv 94.1%

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

      4. associate-*l* 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*l/ 99.6%

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

      2. *-un-lft-identity 99.6%

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

   6. Applied egg-rr 99.6%

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

   if -9.999999999999969e-311 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 85.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{z - y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-179}:\\ \;\;\;\;x\_m \cdot \frac{\frac{1}{y}}{t - z}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{z - y} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x\_m}{t - z} \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t - z}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -3.4e-37)
    (/ (/ x_m z) (- z y))
    (if (<= z 2.4e-179)
      (* x_m (/ (/ 1.0 y) (- t z)))
      (if (<= z 3.8e-70)
        (* (/ -1.0 (- z y)) (/ x_m t))
        (if (<= z 5.2e+88)
          (* (/ x_m (- t z)) (/ 1.0 y))
          (/ -1.0 (* z (/ (- t z) x_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.4e-37) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.4e-179) {
		tmp = x_m * ((1.0 / y) / (t - z));
	} else if (z <= 3.8e-70) {
		tmp = (-1.0 / (z - y)) * (x_m / t);
	} else if (z <= 5.2e+88) {
		tmp = (x_m / (t - z)) * (1.0 / y);
	} else {
		tmp = -1.0 / (z * ((t - z) / x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.4d-37)) then
        tmp = (x_m / z) / (z - y)
    else if (z <= 2.4d-179) then
        tmp = x_m * ((1.0d0 / y) / (t - z))
    else if (z <= 3.8d-70) then
        tmp = ((-1.0d0) / (z - y)) * (x_m / t)
    else if (z <= 5.2d+88) then
        tmp = (x_m / (t - z)) * (1.0d0 / y)
    else
        tmp = (-1.0d0) / (z * ((t - z) / x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -3.4e-37) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.4e-179) {
		tmp = x_m * ((1.0 / y) / (t - z));
	} else if (z <= 3.8e-70) {
		tmp = (-1.0 / (z - y)) * (x_m / t);
	} else if (z <= 5.2e+88) {
		tmp = (x_m / (t - z)) * (1.0 / y);
	} else {
		tmp = -1.0 / (z * ((t - z) / x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -3.4e-37:
		tmp = (x_m / z) / (z - y)
	elif z <= 2.4e-179:
		tmp = x_m * ((1.0 / y) / (t - z))
	elif z <= 3.8e-70:
		tmp = (-1.0 / (z - y)) * (x_m / t)
	elif z <= 5.2e+88:
		tmp = (x_m / (t - z)) * (1.0 / y)
	else:
		tmp = -1.0 / (z * ((t - z) / x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -3.4e-37)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (z <= 2.4e-179)
		tmp = Float64(x_m * Float64(Float64(1.0 / y) / Float64(t - z)));
	elseif (z <= 3.8e-70)
		tmp = Float64(Float64(-1.0 / Float64(z - y)) * Float64(x_m / t));
	elseif (z <= 5.2e+88)
		tmp = Float64(Float64(x_m / Float64(t - z)) * Float64(1.0 / y));
	else
		tmp = Float64(-1.0 / Float64(z * Float64(Float64(t - z) / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -3.4e-37)
		tmp = (x_m / z) / (z - y);
	elseif (z <= 2.4e-179)
		tmp = x_m * ((1.0 / y) / (t - z));
	elseif (z <= 3.8e-70)
		tmp = (-1.0 / (z - y)) * (x_m / t);
	elseif (z <= 5.2e+88)
		tmp = (x_m / (t - z)) * (1.0 / y);
	else
		tmp = -1.0 / (z * ((t - z) / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -3.4e-37], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.4e-179], N[(x$95$m * N[(N[(1.0 / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-70], N[(N[(-1.0 / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e+88], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(z * N[(N[(t - z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if z < -3.40000000000000018e-37

   1. Initial program 86.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 0 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

   7. Simplified 84.5%

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

   if -3.40000000000000018e-37 < z < 2.4e-179

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. associate-/l/ 92.2%

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

   5. Simplified 92.2%

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

   6. Taylor expanded in y around inf 80.1%

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

      1. div-inv 80.1%

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

      2. associate-/l* 87.2%

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

   8. Applied egg-rr 87.2%

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

   if 2.4e-179 < z < 3.7999999999999998e-70

   1. Initial program 92.3%

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

      1. associate-/l/ 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t around inf 77.3%

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

      1. div-inv 77.3%

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

   7. Applied egg-rr 77.3%

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

   if 3.7999999999999998e-70 < z < 5.2000000000000001e88

   1. Initial program 99.8%

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

      1. associate-/l/ 99.9%

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

      2. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 86.1%

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

   if 5.2000000000000001e88 < z

   1. Initial program 72.3%

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

      1. associate-/l/ 99.8%

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

      2. div-inv 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 73.3%

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

   4. Applied egg-rr 73.3%

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

   5. Taylor expanded in y around 0 69.7%

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

      1. associate-*r/ 69.7%

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

      2. *-commutative 69.7%

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

      3. frac-times 94.0%

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

      4. clear-num 94.0%

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

      5. frac-times 95.2%

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

      6. metadata-eval 95.2%

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

   7. Applied egg-rr 95.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{t - z}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{z - y} \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{t - z} \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t - z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := x\_m \cdot \frac{\frac{1}{y}}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{z - y} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \end{array} \end{array} \end{array} \]

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 (/ (/ 1.0 y) (- t z)))))
   (*
    x_s
    (if (<= z -4.8e-37)
      (/ (/ x_m z) (- z y))
      (if (<= z 2.65e-179)
        t_1
        (if (<= z 1.9e-69)
          (* (/ -1.0 (- z y)) (/ x_m t))
          (if (<= z 3.2e+88) t_1 (/ (/ x_m z) (- z t)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((1.0 / y) / (t - z));
	double tmp;
	if (z <= -4.8e-37) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.65e-179) {
		tmp = t_1;
	} else if (z <= 1.9e-69) {
		tmp = (-1.0 / (z - y)) * (x_m / t);
	} else if (z <= 3.2e+88) {
		tmp = t_1;
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * ((1.0d0 / y) / (t - z))
    if (z <= (-4.8d-37)) then
        tmp = (x_m / z) / (z - y)
    else if (z <= 2.65d-179) then
        tmp = t_1
    else if (z <= 1.9d-69) then
        tmp = ((-1.0d0) / (z - y)) * (x_m / t)
    else if (z <= 3.2d+88) then
        tmp = t_1
    else
        tmp = (x_m / z) / (z - t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((1.0 / y) / (t - z));
	double tmp;
	if (z <= -4.8e-37) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.65e-179) {
		tmp = t_1;
	} else if (z <= 1.9e-69) {
		tmp = (-1.0 / (z - y)) * (x_m / t);
	} else if (z <= 3.2e+88) {
		tmp = t_1;
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m * ((1.0 / y) / (t - z))
	tmp = 0
	if z <= -4.8e-37:
		tmp = (x_m / z) / (z - y)
	elif z <= 2.65e-179:
		tmp = t_1
	elif z <= 1.9e-69:
		tmp = (-1.0 / (z - y)) * (x_m / t)
	elif z <= 3.2e+88:
		tmp = t_1
	else:
		tmp = (x_m / z) / (z - t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(Float64(1.0 / y) / Float64(t - z)))
	tmp = 0.0
	if (z <= -4.8e-37)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (z <= 2.65e-179)
		tmp = t_1;
	elseif (z <= 1.9e-69)
		tmp = Float64(Float64(-1.0 / Float64(z - y)) * Float64(x_m / t));
	elseif (z <= 3.2e+88)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * ((1.0 / y) / (t - z));
	tmp = 0.0;
	if (z <= -4.8e-37)
		tmp = (x_m / z) / (z - y);
	elseif (z <= 2.65e-179)
		tmp = t_1;
	elseif (z <= 1.9e-69)
		tmp = (-1.0 / (z - y)) * (x_m / t);
	elseif (z <= 3.2e+88)
		tmp = t_1;
	else
		tmp = (x_m / z) / (z - t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(N[(1.0 / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -4.8e-37], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.65e-179], t$95$1, If[LessEqual[z, 1.9e-69], N[(N[(-1.0 / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e+88], t$95$1, N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -4.79999999999999982e-37

   1. Initial program 86.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 0 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

   7. Simplified 84.5%

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

   if -4.79999999999999982e-37 < z < 2.64999999999999997e-179 or 1.8999999999999999e-69 < z < 3.1999999999999999e88

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-/l/ 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in y around inf 80.3%

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

      1. div-inv 80.3%

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

      2. associate-/l* 85.9%

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

   8. Applied egg-rr 85.9%

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

   if 2.64999999999999997e-179 < z < 1.8999999999999999e-69

   1. Initial program 92.3%

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

      1. associate-/l/ 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t around inf 77.3%

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

      1. div-inv 77.3%

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

   7. Applied egg-rr 77.3%

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

   if 3.1999999999999999e88 < z

   1. Initial program 72.3%

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

   3. Taylor expanded in x around 0 72.3%

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

      1. associate-/l/ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 94.1%

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

      1. associate-*r/ 94.1%

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

      2. neg-mul-1 94.1%

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

   8. Simplified 94.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{t - z}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{z - y} \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := x\_m \cdot \frac{\frac{1}{y}}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \end{array} \end{array} \end{array} \]

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 (/ (/ 1.0 y) (- t z)))))
   (*
    x_s
    (if (<= z -7.5e-42)
      (/ (/ x_m z) (- z y))
      (if (<= z 2.6e-179)
        t_1
        (if (<= z 1.65e-62)
          (/ (/ x_m t) (- y z))
          (if (<= z 3.2e+88) t_1 (/ (/ x_m z) (- z t)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((1.0 / y) / (t - z));
	double tmp;
	if (z <= -7.5e-42) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.6e-179) {
		tmp = t_1;
	} else if (z <= 1.65e-62) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 3.2e+88) {
		tmp = t_1;
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * ((1.0d0 / y) / (t - z))
    if (z <= (-7.5d-42)) then
        tmp = (x_m / z) / (z - y)
    else if (z <= 2.6d-179) then
        tmp = t_1
    else if (z <= 1.65d-62) then
        tmp = (x_m / t) / (y - z)
    else if (z <= 3.2d+88) then
        tmp = t_1
    else
        tmp = (x_m / z) / (z - t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((1.0 / y) / (t - z));
	double tmp;
	if (z <= -7.5e-42) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.6e-179) {
		tmp = t_1;
	} else if (z <= 1.65e-62) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 3.2e+88) {
		tmp = t_1;
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m * ((1.0 / y) / (t - z))
	tmp = 0
	if z <= -7.5e-42:
		tmp = (x_m / z) / (z - y)
	elif z <= 2.6e-179:
		tmp = t_1
	elif z <= 1.65e-62:
		tmp = (x_m / t) / (y - z)
	elif z <= 3.2e+88:
		tmp = t_1
	else:
		tmp = (x_m / z) / (z - t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(Float64(1.0 / y) / Float64(t - z)))
	tmp = 0.0
	if (z <= -7.5e-42)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (z <= 2.6e-179)
		tmp = t_1;
	elseif (z <= 1.65e-62)
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	elseif (z <= 3.2e+88)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * ((1.0 / y) / (t - z));
	tmp = 0.0;
	if (z <= -7.5e-42)
		tmp = (x_m / z) / (z - y);
	elseif (z <= 2.6e-179)
		tmp = t_1;
	elseif (z <= 1.65e-62)
		tmp = (x_m / t) / (y - z);
	elseif (z <= 3.2e+88)
		tmp = t_1;
	else
		tmp = (x_m / z) / (z - t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(N[(1.0 / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -7.5e-42], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e-179], t$95$1, If[LessEqual[z, 1.65e-62], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e+88], t$95$1, N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -7.49999999999999972e-42

   1. Initial program 86.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 0 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

   7. Simplified 84.5%

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

   if -7.49999999999999972e-42 < z < 2.60000000000000005e-179 or 1.65000000000000002e-62 < z < 3.1999999999999999e88

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-/l/ 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in y around inf 80.3%

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

      1. div-inv 80.3%

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

      2. associate-/l* 85.9%

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

   8. Applied egg-rr 85.9%

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

   if 2.60000000000000005e-179 < z < 1.65000000000000002e-62

   1. Initial program 92.3%

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

      1. associate-/l/ 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t around inf 77.3%

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

   if 3.1999999999999999e88 < z

   1. Initial program 72.3%

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

   3. Taylor expanded in x around 0 72.3%

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

      1. associate-/l/ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 94.1%

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

      1. associate-*r/ 94.1%

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

      2. neg-mul-1 94.1%

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

   8. Simplified 94.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{t - z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{y \cdot \left(t - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \end{array} \end{array} \end{array} \]

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 (- t z)))))
   (*
    x_s
    (if (<= z -4.2e-40)
      (/ (/ x_m z) (- z y))
      (if (<= z 2.7e-179)
        t_1
        (if (<= z 2e-55)
          (/ (/ x_m t) (- y z))
          (if (<= z 1.8e+89) t_1 (/ (/ x_m z) (- z t)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (y * (t - z));
	double tmp;
	if (z <= -4.2e-40) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.7e-179) {
		tmp = t_1;
	} else if (z <= 2e-55) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 1.8e+89) {
		tmp = t_1;
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / (y * (t - z))
    if (z <= (-4.2d-40)) then
        tmp = (x_m / z) / (z - y)
    else if (z <= 2.7d-179) then
        tmp = t_1
    else if (z <= 2d-55) then
        tmp = (x_m / t) / (y - z)
    else if (z <= 1.8d+89) then
        tmp = t_1
    else
        tmp = (x_m / z) / (z - t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (y * (t - z));
	double tmp;
	if (z <= -4.2e-40) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 2.7e-179) {
		tmp = t_1;
	} else if (z <= 2e-55) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 1.8e+89) {
		tmp = t_1;
	} else {
		tmp = (x_m / z) / (z - t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (y * (t - z))
	tmp = 0
	if z <= -4.2e-40:
		tmp = (x_m / z) / (z - y)
	elif z <= 2.7e-179:
		tmp = t_1
	elif z <= 2e-55:
		tmp = (x_m / t) / (y - z)
	elif z <= 1.8e+89:
		tmp = t_1
	else:
		tmp = (x_m / z) / (z - t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(y * Float64(t - z)))
	tmp = 0.0
	if (z <= -4.2e-40)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (z <= 2.7e-179)
		tmp = t_1;
	elseif (z <= 2e-55)
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	elseif (z <= 1.8e+89)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (y * (t - z));
	tmp = 0.0;
	if (z <= -4.2e-40)
		tmp = (x_m / z) / (z - y);
	elseif (z <= 2.7e-179)
		tmp = t_1;
	elseif (z <= 2e-55)
		tmp = (x_m / t) / (y - z);
	elseif (z <= 1.8e+89)
		tmp = t_1;
	else
		tmp = (x_m / z) / (z - t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -4.2e-40], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e-179], t$95$1, If[LessEqual[z, 2e-55], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e+89], t$95$1, N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -4.20000000000000036e-40

   1. Initial program 86.8%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 0 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

   7. Simplified 84.5%

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

   if -4.20000000000000036e-40 < z < 2.69999999999999988e-179 or 1.99999999999999999e-55 < z < 1.8e89

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf 85.9%

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

      1. *-commutative 85.9%

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

   5. Simplified 85.9%

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

   if 2.69999999999999988e-179 < z < 1.99999999999999999e-55

   1. Initial program 92.3%

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

      1. associate-/l/ 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t around inf 77.3%

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

   if 1.8e89 < z

   1. Initial program 72.3%

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

   3. Taylor expanded in x around 0 72.3%

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

      1. associate-/l/ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 94.1%

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

      1. associate-*r/ 94.1%

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

      2. neg-mul-1 94.1%

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

   8. Simplified 94.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{y \cdot \left(t - z\right)}\\ t_2 := \frac{\frac{x\_m}{z}}{z - t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3700000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* y (- t z)))) (t_2 (/ (/ x_m z) (- z t))))
   (*
    x_s
    (if (<= z -3700000000000.0)
      t_2
      (if (<= z 2.7e-179)
        t_1
        (if (<= z 4.6e-60)
          (/ (/ x_m t) (- y z))
          (if (<= z 3.2e+88) t_1 t_2)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (y * (t - z));
	double t_2 = (x_m / z) / (z - t);
	double tmp;
	if (z <= -3700000000000.0) {
		tmp = t_2;
	} else if (z <= 2.7e-179) {
		tmp = t_1;
	} else if (z <= 4.6e-60) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 3.2e+88) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x_m / (y * (t - z))
    t_2 = (x_m / z) / (z - t)
    if (z <= (-3700000000000.0d0)) then
        tmp = t_2
    else if (z <= 2.7d-179) then
        tmp = t_1
    else if (z <= 4.6d-60) then
        tmp = (x_m / t) / (y - z)
    else if (z <= 3.2d+88) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (y * (t - z));
	double t_2 = (x_m / z) / (z - t);
	double tmp;
	if (z <= -3700000000000.0) {
		tmp = t_2;
	} else if (z <= 2.7e-179) {
		tmp = t_1;
	} else if (z <= 4.6e-60) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 3.2e+88) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (y * (t - z))
	t_2 = (x_m / z) / (z - t)
	tmp = 0
	if z <= -3700000000000.0:
		tmp = t_2
	elif z <= 2.7e-179:
		tmp = t_1
	elif z <= 4.6e-60:
		tmp = (x_m / t) / (y - z)
	elif z <= 3.2e+88:
		tmp = t_1
	else:
		tmp = t_2
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(y * Float64(t - z)))
	t_2 = Float64(Float64(x_m / z) / Float64(z - t))
	tmp = 0.0
	if (z <= -3700000000000.0)
		tmp = t_2;
	elseif (z <= 2.7e-179)
		tmp = t_1;
	elseif (z <= 4.6e-60)
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	elseif (z <= 3.2e+88)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (y * (t - z));
	t_2 = (x_m / z) / (z - t);
	tmp = 0.0;
	if (z <= -3700000000000.0)
		tmp = t_2;
	elseif (z <= 2.7e-179)
		tmp = t_1;
	elseif (z <= 4.6e-60)
		tmp = (x_m / t) / (y - z);
	elseif (z <= 3.2e+88)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3700000000000.0], t$95$2, If[LessEqual[z, 2.7e-179], t$95$1, If[LessEqual[z, 4.6e-60], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e+88], t$95$1, t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.7e12 or 3.1999999999999999e88 < z

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0 78.2%

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

      1. associate-/l/ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 93.0%

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

      1. associate-*r/ 93.0%

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

      2. neg-mul-1 93.0%

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

   8. Simplified 93.0%

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

   if -3.7e12 < z < 2.69999999999999988e-179 or 4.6000000000000003e-60 < z < 3.1999999999999999e88

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf 82.6%

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

      1. *-commutative 82.6%

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

   5. Simplified 82.6%

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

   if 2.69999999999999988e-179 < z < 4.6000000000000003e-60

   1. Initial program 92.3%

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

      1. associate-/l/ 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t around inf 77.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3700000000000:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{y \cdot \left(t - z\right)}\\ t_2 := \frac{\frac{x\_m}{z}}{z + t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -220000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y - z}\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* y (- t z)))) (t_2 (/ (/ x_m z) (+ z t))))
   (*
    x_s
    (if (<= z -220000000.0)
      t_2
      (if (<= z 2.85e-179)
        t_1
        (if (<= z 3.2e-70)
          (/ (/ x_m t) (- y z))
          (if (<= z 9.4e+89) t_1 t_2)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (y * (t - z));
	double t_2 = (x_m / z) / (z + t);
	double tmp;
	if (z <= -220000000.0) {
		tmp = t_2;
	} else if (z <= 2.85e-179) {
		tmp = t_1;
	} else if (z <= 3.2e-70) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 9.4e+89) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x_m / (y * (t - z))
    t_2 = (x_m / z) / (z + t)
    if (z <= (-220000000.0d0)) then
        tmp = t_2
    else if (z <= 2.85d-179) then
        tmp = t_1
    else if (z <= 3.2d-70) then
        tmp = (x_m / t) / (y - z)
    else if (z <= 9.4d+89) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (y * (t - z));
	double t_2 = (x_m / z) / (z + t);
	double tmp;
	if (z <= -220000000.0) {
		tmp = t_2;
	} else if (z <= 2.85e-179) {
		tmp = t_1;
	} else if (z <= 3.2e-70) {
		tmp = (x_m / t) / (y - z);
	} else if (z <= 9.4e+89) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (y * (t - z))
	t_2 = (x_m / z) / (z + t)
	tmp = 0
	if z <= -220000000.0:
		tmp = t_2
	elif z <= 2.85e-179:
		tmp = t_1
	elif z <= 3.2e-70:
		tmp = (x_m / t) / (y - z)
	elif z <= 9.4e+89:
		tmp = t_1
	else:
		tmp = t_2
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(y * Float64(t - z)))
	t_2 = Float64(Float64(x_m / z) / Float64(z + t))
	tmp = 0.0
	if (z <= -220000000.0)
		tmp = t_2;
	elseif (z <= 2.85e-179)
		tmp = t_1;
	elseif (z <= 3.2e-70)
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	elseif (z <= 9.4e+89)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (y * (t - z));
	t_2 = (x_m / z) / (z + t);
	tmp = 0.0;
	if (z <= -220000000.0)
		tmp = t_2;
	elseif (z <= 2.85e-179)
		tmp = t_1;
	elseif (z <= 3.2e-70)
		tmp = (x_m / t) / (y - z);
	elseif (z <= 9.4e+89)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m / z), $MachinePrecision] / N[(z + t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -220000000.0], t$95$2, If[LessEqual[z, 2.85e-179], t$95$1, If[LessEqual[z, 3.2e-70], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.4e+89], t$95$1, t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e8 or 9.40000000000000043e89 < z

   1. Initial program 78.2%

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

      1. associate-/l/ 99.8%

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

      2. div-inv 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 78.7%

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

   4. Applied egg-rr 78.7%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. associate-*r/ 76.1%

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

      2. *-commutative 76.1%

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

      3. neg-mul-1 76.1%

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

      4. add-sqr-sqrt 39.4%

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

      5. sqrt-unprod 65.0%

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

      6. sqr-neg 65.0%

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

      7. sqrt-unprod 32.7%

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

      8. add-sqr-sqrt 63.5%

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

   7. Applied egg-rr 63.5%

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

      1. *-un-lft-identity 63.5%

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

      2. *-commutative 63.5%

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

      3. *-commutative 63.5%

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

      4. add-sqr-sqrt 31.2%

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

      5. sqrt-unprod 70.6%

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

      6. sqr-neg 70.6%

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

      7. sqrt-unprod 39.5%

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

      8. add-sqr-sqrt 76.1%

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

      9. distribute-lft-neg-in 76.1%

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

      10. distribute-lft-neg-in 76.1%

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

      11. add-sqr-sqrt 39.5%

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

      12. sqrt-unprod 70.6%

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

      13. sqr-neg 70.6%

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

      14. sqrt-unprod 31.2%

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

      15. add-sqr-sqrt 63.5%

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

      16. sub-neg 63.5%

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

      17. distribute-lft-in 53.3%

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

      18. add-sqr-sqrt 27.2%

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

      19. sqrt-unprod 54.5%

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

      20. sqr-neg 54.5%

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

      21. sqrt-unprod 27.3%

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

      22. add-sqr-sqrt 63.5%

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

      23. +-commutative 63.5%

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

      24. distribute-lft-out 74.5%

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

   9. Applied egg-rr 74.5%

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

       1. *-lft-identity 74.5%

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

       2. associate-/r* 86.5%

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

   11. Simplified 86.5%

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

   if -2.2e8 < z < 2.85e-179 or 3.1999999999999997e-70 < z < 9.40000000000000043e89

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf 82.6%

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

      1. *-commutative 82.6%

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

   5. Simplified 82.6%

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

   if 2.85e-179 < z < 3.1999999999999997e-70

   1. Initial program 92.3%

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

      1. associate-/l/ 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t around inf 77.3%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -220000000:\\ \;\;\;\;\frac{\frac{x}{z}}{z + t}\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z + t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+160}:\\ \;\;\;\;t\_1 \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-133}:\\ \;\;\;\;t\_1 \cdot \frac{-1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z - y} \cdot \frac{x\_m}{t}\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    x_s
    (if (<= y -1.4e+160)
      (* t_1 (/ 1.0 y))
      (if (<= y -8.6e-89)
        (/ x_m (* y (- t z)))
        (if (<= y 7e-133)
          (* t_1 (/ -1.0 z))
          (* (/ -1.0 (- z y)) (/ x_m 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 t_1 = x_m / (t - z);
	double tmp;
	if (y <= -1.4e+160) {
		tmp = t_1 * (1.0 / y);
	} else if (y <= -8.6e-89) {
		tmp = x_m / (y * (t - z));
	} else if (y <= 7e-133) {
		tmp = t_1 * (-1.0 / z);
	} else {
		tmp = (-1.0 / (z - y)) * (x_m / 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) :: t_1
    real(8) :: tmp
    t_1 = x_m / (t - z)
    if (y <= (-1.4d+160)) then
        tmp = t_1 * (1.0d0 / y)
    else if (y <= (-8.6d-89)) then
        tmp = x_m / (y * (t - z))
    else if (y <= 7d-133) then
        tmp = t_1 * ((-1.0d0) / z)
    else
        tmp = ((-1.0d0) / (z - y)) * (x_m / 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 t_1 = x_m / (t - z);
	double tmp;
	if (y <= -1.4e+160) {
		tmp = t_1 * (1.0 / y);
	} else if (y <= -8.6e-89) {
		tmp = x_m / (y * (t - z));
	} else if (y <= 7e-133) {
		tmp = t_1 * (-1.0 / z);
	} else {
		tmp = (-1.0 / (z - y)) * (x_m / 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):
	t_1 = x_m / (t - z)
	tmp = 0
	if y <= -1.4e+160:
		tmp = t_1 * (1.0 / y)
	elif y <= -8.6e-89:
		tmp = x_m / (y * (t - z))
	elif y <= 7e-133:
		tmp = t_1 * (-1.0 / z)
	else:
		tmp = (-1.0 / (z - y)) * (x_m / 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)
	t_1 = Float64(x_m / Float64(t - z))
	tmp = 0.0
	if (y <= -1.4e+160)
		tmp = Float64(t_1 * Float64(1.0 / y));
	elseif (y <= -8.6e-89)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (y <= 7e-133)
		tmp = Float64(t_1 * Float64(-1.0 / z));
	else
		tmp = Float64(Float64(-1.0 / Float64(z - y)) * Float64(x_m / 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)
	t_1 = x_m / (t - z);
	tmp = 0.0;
	if (y <= -1.4e+160)
		tmp = t_1 * (1.0 / y);
	elseif (y <= -8.6e-89)
		tmp = x_m / (y * (t - z));
	elseif (y <= 7e-133)
		tmp = t_1 * (-1.0 / z);
	else
		tmp = (-1.0 / (z - y)) * (x_m / 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_] := Block[{t$95$1 = N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.4e+160], N[(t$95$1 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -8.6e-89], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e-133], N[(t$95$1 * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -1.4e160

   1. Initial program 88.9%

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

      1. associate-/l/ 99.9%

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

      2. div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf 96.5%

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

   if -1.4e160 < y < -8.59999999999999974e-89

   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 82.5%

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

      1. *-commutative 82.5%

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

   5. Simplified 82.5%

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

   if -8.59999999999999974e-89 < y < 7.00000000000000006e-133

   1. Initial program 85.2%

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

      1. associate-/l/ 97.7%

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

      2. div-inv 97.7%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in y around 0 85.5%

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

   if 7.00000000000000006e-133 < y

   1. Initial program 92.8%

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

      1. associate-/l/ 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t around inf 62.8%

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

      1. div-inv 62.8%

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

   7. Applied egg-rr 62.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{t - z} \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{t - z} \cdot \frac{-1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z - y} \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - y}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t - z}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1.36e+163)
    (/ (/ x_m z) (- z y))
    (if (<= z 7.8e+98)
      (/ x_m (* (- y z) (- t z)))
      (/ -1.0 (* z (/ (- t z) x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.36e+163) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 7.8e+98) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = -1.0 / (z * ((t - z) / x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.36d+163)) then
        tmp = (x_m / z) / (z - y)
    else if (z <= 7.8d+98) then
        tmp = x_m / ((y - z) * (t - z))
    else
        tmp = (-1.0d0) / (z * ((t - z) / x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.36e+163) {
		tmp = (x_m / z) / (z - y);
	} else if (z <= 7.8e+98) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = -1.0 / (z * ((t - z) / x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1.36e+163:
		tmp = (x_m / z) / (z - y)
	elif z <= 7.8e+98:
		tmp = x_m / ((y - z) * (t - z))
	else:
		tmp = -1.0 / (z * ((t - z) / x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1.36e+163)
		tmp = Float64(Float64(x_m / z) / Float64(z - y));
	elseif (z <= 7.8e+98)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = Float64(-1.0 / Float64(z * Float64(Float64(t - z) / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1.36e+163)
		tmp = (x_m / z) / (z - y);
	elseif (z <= 7.8e+98)
		tmp = x_m / ((y - z) * (t - z));
	else
		tmp = -1.0 / (z * ((t - z) / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1.36e+163], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e+98], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(z * N[(N[(t - z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.36000000000000001e163

   1. Initial program 79.2%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 0 91.5%

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

      1. associate-*r/ 91.5%

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

      2. neg-mul-1 91.5%

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

   7. Simplified 91.5%

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

   if -1.36000000000000001e163 < z < 7.7999999999999999e98

   1. Initial program 97.3%

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

   if 7.7999999999999999e98 < z

   1. Initial program 71.0%

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

      1. associate-/l/ 99.8%

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

      2. div-inv 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 72.0%

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

   4. Applied egg-rr 72.0%

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

   5. Taylor expanded in y around 0 69.8%

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

      1. associate-*r/ 69.9%

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

      2. *-commutative 69.9%

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

      3. frac-times 95.4%

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

      4. clear-num 95.3%

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

      5. frac-times 96.6%

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

      6. metadata-eval 96.6%

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

   7. Applied egg-rr 96.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t - z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+24} \lor \neg \left(z \leq 3.7 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

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 (or (<= z -1.75e+24) (not (<= z 3.7e+95)))
    (/ x_m (* z (+ z t)))
    (/ x_m (* (- y z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -1.75e+24) || !(z <= 3.7e+95)) {
		tmp = x_m / (z * (z + t));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.75d+24)) .or. (.not. (z <= 3.7d+95))) then
        tmp = x_m / (z * (z + t))
    else
        tmp = x_m / ((y - z) * t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -1.75e+24) || !(z <= 3.7e+95)) {
		tmp = x_m / (z * (z + t));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -1.75e+24) or not (z <= 3.7e+95):
		tmp = x_m / (z * (z + t))
	else:
		tmp = x_m / ((y - z) * t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -1.75e+24) || !(z <= 3.7e+95))
		tmp = Float64(x_m / Float64(z * Float64(z + t)));
	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 ((z <= -1.75e+24) || ~((z <= 3.7e+95)))
		tmp = x_m / (z * (z + t));
	else
		tmp = x_m / ((y - z) * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -1.75e+24], N[Not[LessEqual[z, 3.7e+95]], $MachinePrecision]], N[(x$95$m / N[(z * N[(z + t), $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 z < -1.7500000000000001e24 or 3.7000000000000001e95 < z

   1. Initial program 77.9%

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

      1. associate-/l/ 99.8%

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

      2. div-inv 99.8%

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

      3. div-inv 99.8%

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

      4. associate-*l* 78.4%

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

   4. Applied egg-rr 78.4%

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

   5. Taylor expanded in y around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. neg-mul-1 76.5%

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

      4. add-sqr-sqrt 40.0%

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

      5. sqrt-unprod 66.1%

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

      6. sqr-neg 66.1%

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

      7. sqrt-unprod 33.2%

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

      8. add-sqr-sqrt 64.4%

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

   7. Applied egg-rr 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. *-commutative 64.4%

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

      3. *-commutative 64.4%

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

      4. add-sqr-sqrt 31.6%

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

      5. sqrt-unprod 71.7%

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

      6. sqr-neg 71.7%

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

      7. sqrt-unprod 40.1%

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

      8. add-sqr-sqrt 76.5%

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

      9. distribute-lft-neg-in 76.5%

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

      10. distribute-lft-neg-in 76.5%

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

      11. add-sqr-sqrt 40.1%

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

      12. sqrt-unprod 71.7%

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

      13. sqr-neg 71.7%

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

      14. sqrt-unprod 31.6%

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

      15. add-sqr-sqrt 64.4%

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

      16. sub-neg 64.4%

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

      17. distribute-lft-in 54.1%

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

      18. add-sqr-sqrt 27.7%

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

      19. sqrt-unprod 54.5%

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

      20. sqr-neg 54.5%

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

      21. sqrt-unprod 26.9%

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

      22. add-sqr-sqrt 63.7%

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

      23. +-commutative 63.7%

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

      24. distribute-lft-out 74.9%

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

   9. Applied egg-rr 74.9%

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

       1. *-lft-identity 74.9%

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

   11. Simplified 74.9%

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

   if -1.7500000000000001e24 < z < 3.7000000000000001e95

   1. Initial program 98.3%

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

   3. Taylor expanded in t around inf 74.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+24} \lor \neg \left(z \leq 3.7 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{z \cdot \left(z + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-134} \lor \neg \left(t \leq 4.6 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(-y\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= t -6.4e-134) (not (<= t 4.6e-94)))
    (/ x_m (* (- y z) t))
    (/ x_m (* z (- y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -6.4e-134) || !(t <= 4.6e-94)) {
		tmp = x_m / ((y - z) * t);
	} else {
		tmp = x_m / (z * -y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-6.4d-134)) .or. (.not. (t <= 4.6d-94))) then
        tmp = x_m / ((y - z) * t)
    else
        tmp = x_m / (z * -y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -6.4e-134) || !(t <= 4.6e-94)) {
		tmp = x_m / ((y - z) * t);
	} else {
		tmp = x_m / (z * -y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (t <= -6.4e-134) or not (t <= 4.6e-94):
		tmp = x_m / ((y - z) * t)
	else:
		tmp = x_m / (z * -y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((t <= -6.4e-134) || !(t <= 4.6e-94))
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	else
		tmp = Float64(x_m / Float64(z * Float64(-y)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((t <= -6.4e-134) || ~((t <= 4.6e-94)))
		tmp = x_m / ((y - z) * t);
	else
		tmp = x_m / (z * -y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[t, -6.4e-134], N[Not[LessEqual[t, 4.6e-94]], $MachinePrecision]], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(z * (-y)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -6.4000000000000003e-134 or 4.5999999999999999e-94 < t

   1. Initial program 88.6%

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

   3. Taylor expanded in t around inf 76.1%

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

   if -6.4000000000000003e-134 < t < 4.5999999999999999e-94

   1. Initial program 89.0%

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

   3. Taylor expanded in x around 0 89.0%

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

      1. associate-/l/ 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in y around inf 56.1%

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

   7. Taylor expanded in t around 0 49.9%

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

      1. mul-1-neg 49.9%

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

      2. *-commutative 49.9%

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

   9. Simplified 49.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-134} \lor \neg \left(t \leq 4.6 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(-y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

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 (<= t 2.9e-209)
    (/ (/ x_m y) (- t z))
    (if (<= t 9.5e-28) (/ x_m (* z (+ z t))) (/ x_m (* (- y z) t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 2.9e-209) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 9.5e-28) {
		tmp = x_m / (z * (z + t));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.9d-209) then
        tmp = (x_m / y) / (t - z)
    else if (t <= 9.5d-28) then
        tmp = x_m / (z * (z + t))
    else
        tmp = x_m / ((y - z) * t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 2.9e-209) {
		tmp = (x_m / y) / (t - z);
	} else if (t <= 9.5e-28) {
		tmp = x_m / (z * (z + t));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 2.9e-209:
		tmp = (x_m / y) / (t - z)
	elif t <= 9.5e-28:
		tmp = x_m / (z * (z + t))
	else:
		tmp = x_m / ((y - z) * t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 2.9e-209)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (t <= 9.5e-28)
		tmp = Float64(x_m / Float64(z * Float64(z + t)));
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 2.9e-209)
		tmp = (x_m / y) / (t - z);
	elseif (t <= 9.5e-28)
		tmp = x_m / (z * (z + t));
	else
		tmp = x_m / ((y - z) * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 2.9e-209], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-28], N[(x$95$m / N[(z * N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.90000000000000026e-209

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0 86.9%

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

      1. associate-/l/ 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in y around inf 60.2%

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

   if 2.90000000000000026e-209 < t < 9.50000000000000001e-28

   1. Initial program 91.9%

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

      1. associate-/l/ 99.9%

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

      2. div-inv 99.8%

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

      3. div-inv 99.6%

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

      4. associate-*l* 91.8%

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

   4. Applied egg-rr 91.8%

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

   5. Taylor expanded in y around 0 61.9%

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

      1. associate-*r/ 61.9%

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

      2. *-commutative 61.9%

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

      3. neg-mul-1 61.9%

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

      4. add-sqr-sqrt 37.5%

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

      5. sqrt-unprod 44.3%

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

      6. sqr-neg 44.3%

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

      7. sqrt-unprod 19.9%

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

      8. add-sqr-sqrt 45.5%

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

   7. Applied egg-rr 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. *-commutative 45.5%

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

      3. *-commutative 45.5%

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

      4. add-sqr-sqrt 17.6%

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

      5. sqrt-unprod 52.6%

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

      6. sqr-neg 52.6%

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

      7. sqrt-unprod 34.9%

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

      8. add-sqr-sqrt 61.9%

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

      9. distribute-lft-neg-in 61.9%

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

      10. distribute-lft-neg-in 61.9%

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

      11. add-sqr-sqrt 34.9%

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

      12. sqrt-unprod 52.6%

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

      13. sqr-neg 52.6%

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

      14. sqrt-unprod 17.6%

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

      15. add-sqr-sqrt 45.5%

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

      16. sub-neg 45.5%

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

      17. distribute-lft-in 45.5%

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

      18. add-sqr-sqrt 27.9%

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

      19. sqrt-unprod 57.3%

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

      20. sqr-neg 57.3%

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

      21. sqrt-unprod 29.3%

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

      22. add-sqr-sqrt 64.2%

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

      23. +-commutative 64.2%

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

      24. distribute-lft-out 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. *-lft-identity 64.1%

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

   11. Simplified 64.1%

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

   if 9.50000000000000001e-28 < t

   1. Initial program 90.9%

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

   3. Taylor expanded in t around inf 85.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{z \cdot \left(z + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-202}:\\ \;\;\;\;\frac{x\_m}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

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 (<= t 4.2e-202)
    (/ x_m (* y (- t z)))
    (if (<= t 9.5e-28) (/ x_m (* z (+ z t))) (/ x_m (* (- y z) t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 4.2e-202) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 9.5e-28) {
		tmp = x_m / (z * (z + t));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 4.2d-202) then
        tmp = x_m / (y * (t - z))
    else if (t <= 9.5d-28) then
        tmp = x_m / (z * (z + t))
    else
        tmp = x_m / ((y - z) * t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 4.2e-202) {
		tmp = x_m / (y * (t - z));
	} else if (t <= 9.5e-28) {
		tmp = x_m / (z * (z + t));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 4.2e-202:
		tmp = x_m / (y * (t - z))
	elif t <= 9.5e-28:
		tmp = x_m / (z * (z + t))
	else:
		tmp = x_m / ((y - z) * t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 4.2e-202)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	elseif (t <= 9.5e-28)
		tmp = Float64(x_m / Float64(z * Float64(z + t)));
	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 <= 4.2e-202)
		tmp = x_m / (y * (t - z));
	elseif (t <= 9.5e-28)
		tmp = x_m / (z * (z + t));
	else
		tmp = x_m / ((y - z) * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 4.2e-202], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-28], N[(x$95$m / N[(z * N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.1999999999999997e-202

   1. Initial program 86.9%

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

   3. Taylor expanded in y around inf 61.4%

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

      1. *-commutative 61.4%

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

   5. Simplified 61.4%

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

   if 4.1999999999999997e-202 < t < 9.50000000000000001e-28

   1. Initial program 91.9%

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

      1. associate-/l/ 99.9%

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

      2. div-inv 99.8%

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

      3. div-inv 99.6%

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

      4. associate-*l* 91.8%

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

   4. Applied egg-rr 91.8%

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

   5. Taylor expanded in y around 0 61.9%

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

      1. associate-*r/ 61.9%

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

      2. *-commutative 61.9%

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

      3. neg-mul-1 61.9%

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

      4. add-sqr-sqrt 37.5%

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

      5. sqrt-unprod 44.3%

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

      6. sqr-neg 44.3%

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

      7. sqrt-unprod 19.9%

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

      8. add-sqr-sqrt 45.5%

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

   7. Applied egg-rr 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. *-commutative 45.5%

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

      3. *-commutative 45.5%

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

      4. add-sqr-sqrt 17.6%

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

      5. sqrt-unprod 52.6%

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

      6. sqr-neg 52.6%

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

      7. sqrt-unprod 34.9%

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

      8. add-sqr-sqrt 61.9%

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

      9. distribute-lft-neg-in 61.9%

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

      10. distribute-lft-neg-in 61.9%

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

      11. add-sqr-sqrt 34.9%

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

      12. sqrt-unprod 52.6%

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

      13. sqr-neg 52.6%

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

      14. sqrt-unprod 17.6%

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

      15. add-sqr-sqrt 45.5%

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

      16. sub-neg 45.5%

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

      17. distribute-lft-in 45.5%

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

      18. add-sqr-sqrt 27.9%

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

      19. sqrt-unprod 57.3%

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

      20. sqr-neg 57.3%

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

      21. sqrt-unprod 29.3%

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

      22. add-sqr-sqrt 64.2%

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

      23. +-commutative 64.2%

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

      24. distribute-lft-out 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. *-lft-identity 64.1%

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

   11. Simplified 64.1%

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

   if 9.50000000000000001e-28 < t

   1. Initial program 90.9%

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

   3. Taylor expanded in t around inf 85.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-202}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{z \cdot \left(z + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1400000000000 \lor \neg \left(z \leq 2.7 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{\frac{-x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \end{array} \end{array} \]

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 -1400000000000.0) (not (<= z 2.7e+91)))
    (/ (/ (- 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 <= -1400000000000.0) || !(z <= 2.7e+91)) {
		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 <= (-1400000000000.0d0)) .or. (.not. (z <= 2.7d+91))) 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 <= -1400000000000.0) || !(z <= 2.7e+91)) {
		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 <= -1400000000000.0) or not (z <= 2.7e+91):
		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 <= -1400000000000.0) || !(z <= 2.7e+91))
		tmp = Float64(Float64(Float64(-x_m) / 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 <= -1400000000000.0) || ~((z <= 2.7e+91)))
		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, -1400000000000.0], N[Not[LessEqual[z, 2.7e+91]], $MachinePrecision]], N[(N[((-x$95$m) / z), $MachinePrecision] / t), $MachinePrecision], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e12 or 2.7e91 < z

   1. Initial program 78.1%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 47.1%

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

   6. Taylor expanded in y around 0 41.1%

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

      1. associate-*r/ 41.1%

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

      2. neg-mul-1 41.1%

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

   8. Simplified 41.1%

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

      1. distribute-frac-neg 41.1%

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

      2. *-commutative 41.1%

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

      3. associate-/r* 53.5%

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

   10. Applied egg-rr 53.5%

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

   if -1.4e12 < z < 2.7e91

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0 65.4%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1400000000000 \lor \neg \left(z \leq 2.7 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-131}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-86}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t}}{y}\\ \end{array} \end{array} \]

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 (<= t -6.8e-131)
    (/ x_m (* y t))
    (if (<= t 1.55e-86) (/ x_m (* z (- y))) (/ (/ 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 (t <= -6.8e-131) {
		tmp = x_m / (y * t);
	} else if (t <= 1.55e-86) {
		tmp = x_m / (z * -y);
	} 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 (t <= (-6.8d-131)) then
        tmp = x_m / (y * t)
    else if (t <= 1.55d-86) then
        tmp = x_m / (z * -y)
    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 (t <= -6.8e-131) {
		tmp = x_m / (y * t);
	} else if (t <= 1.55e-86) {
		tmp = x_m / (z * -y);
	} 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 t <= -6.8e-131:
		tmp = x_m / (y * t)
	elif t <= 1.55e-86:
		tmp = x_m / (z * -y)
	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 (t <= -6.8e-131)
		tmp = Float64(x_m / Float64(y * t));
	elseif (t <= 1.55e-86)
		tmp = Float64(x_m / Float64(z * Float64(-y)));
	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 (t <= -6.8e-131)
		tmp = x_m / (y * t);
	elseif (t <= 1.55e-86)
		tmp = x_m / (z * -y);
	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[t, -6.8e-131], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.55e-86], N[(x$95$m / N[(z * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -6.7999999999999999e-131

   1. Initial program 88.1%

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

   3. Taylor expanded in z around 0 57.9%

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

   if -6.7999999999999999e-131 < t < 1.54999999999999994e-86

   1. Initial program 87.9%

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

   3. Taylor expanded in x around 0 87.9%

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

      1. associate-/l/ 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in y around inf 55.4%

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

   7. Taylor expanded in t around 0 49.3%

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

      1. mul-1-neg 49.3%

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

      2. *-commutative 49.3%

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

   9. Simplified 49.3%

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

   if 1.54999999999999994e-86 < t

   1. Initial program 90.2%

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

      1. associate-/l/ 97.4%

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

      2. div-inv 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. *-commutative 97.4%

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

      2. clear-num 97.3%

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

      3. un-div-inv 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in z around 0 50.3%

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

      1. associate-/r* 51.6%

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

   9. Simplified 51.6%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{z \cdot \left(-y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-50} \lor \neg \left(z \leq 10^{+99}\right):\\ \;\;\;\;\frac{x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -2.15e-50) (not (<= z 1e+99)))
    (/ x_m (* y z))
    (/ x_m (* y t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -2.15e-50) || !(z <= 1e+99)) {
		tmp = x_m / (y * z);
	} else {
		tmp = x_m / (y * t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.15d-50)) .or. (.not. (z <= 1d+99))) then
        tmp = x_m / (y * z)
    else
        tmp = x_m / (y * t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -2.15e-50) || !(z <= 1e+99)) {
		tmp = x_m / (y * z);
	} else {
		tmp = x_m / (y * t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -2.15e-50) or not (z <= 1e+99):
		tmp = x_m / (y * z)
	else:
		tmp = x_m / (y * t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -2.15e-50) || !(z <= 1e+99))
		tmp = Float64(x_m / Float64(y * z));
	else
		tmp = Float64(x_m / Float64(y * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -2.15e-50) || ~((z <= 1e+99)))
		tmp = x_m / (y * z);
	else
		tmp = x_m / (y * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -2.15e-50], N[Not[LessEqual[z, 1e+99]], $MachinePrecision]], N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.14999999999999999e-50 or 9.9999999999999997e98 < z

   1. Initial program 79.4%

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

   3. Taylor expanded in x around 0 79.4%

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

      1. associate-/l/ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf 40.1%

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

   7. Taylor expanded in t around 0 37.5%

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

      1. mul-1-neg 37.5%

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

      2. associate-/r* 37.6%

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

      3. distribute-neg-frac2 37.6%

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

   9. Simplified 37.6%

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

       1. add-sqr-sqrt 19.6%

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

       2. sqrt-unprod 53.8%

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

       3. sqr-neg 53.8%

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

       4. sqrt-unprod 17.3%

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

       5. add-sqr-sqrt 35.1%

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

       6. *-un-lft-identity 35.1%

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

       7. add-sqr-sqrt 17.3%

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

       8. sqrt-unprod 53.8%

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

       9. sqr-neg 53.8%

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

       10. sqrt-unprod 19.6%

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

       11. add-sqr-sqrt 37.6%

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

       12. associate-/l/ 37.5%

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

       13. *-commutative 37.5%

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

       14. add-sqr-sqrt 20.2%

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

       15. sqrt-unprod 56.4%

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

       16. sqr-neg 56.4%

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

       17. sqrt-unprod 17.3%

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

       18. add-sqr-sqrt 36.8%

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

   11. Applied egg-rr 36.8%

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

       1. *-lft-identity 36.8%

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

   13. Simplified 36.8%

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

   if -2.14999999999999999e-50 < z < 9.9999999999999997e98

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0 66.9%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-50} \lor \neg \left(z \leq 10^{+99}\right):\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-50}:\\ \;\;\;\;\frac{x\_m}{y \cdot z}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+113}:\\ \;\;\;\;\frac{x\_m}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot t}\\ \end{array} \end{array} \]

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 (<= z -2.15e-50)
    (/ x_m (* y z))
    (if (<= z 7.8e+113) (/ x_m (* y t)) (/ 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 (z <= -2.15e-50) {
		tmp = x_m / (y * z);
	} else if (z <= 7.8e+113) {
		tmp = x_m / (y * t);
	} 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 (z <= (-2.15d-50)) then
        tmp = x_m / (y * z)
    else if (z <= 7.8d+113) then
        tmp = x_m / (y * t)
    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 (z <= -2.15e-50) {
		tmp = x_m / (y * z);
	} else if (z <= 7.8e+113) {
		tmp = x_m / (y * t);
	} 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 z <= -2.15e-50:
		tmp = x_m / (y * z)
	elif z <= 7.8e+113:
		tmp = x_m / (y * t)
	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 (z <= -2.15e-50)
		tmp = Float64(x_m / Float64(y * z));
	elseif (z <= 7.8e+113)
		tmp = Float64(x_m / Float64(y * t));
	else
		tmp = Float64(x_m / Float64(z * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -2.15e-50)
		tmp = x_m / (y * z);
	elseif (z <= 7.8e+113)
		tmp = x_m / (y * t);
	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[z, -2.15e-50], N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e+113], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -2.14999999999999999e-50

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0 87.1%

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

      1. associate-/l/ 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf 41.2%

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

   7. Taylor expanded in t around 0 37.8%

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

      1. mul-1-neg 37.8%

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

      2. associate-/r* 36.6%

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

      3. distribute-neg-frac2 36.6%

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

   9. Simplified 36.6%

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

       1. add-sqr-sqrt 36.6%

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

       2. sqrt-unprod 52.9%

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

       3. sqr-neg 52.9%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 33.2%

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

       6. *-un-lft-identity 33.2%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 52.9%

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

       9. sqr-neg 52.9%

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

       10. sqrt-unprod 36.6%

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

       11. add-sqr-sqrt 36.6%

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

       12. associate-/l/ 37.8%

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

       13. *-commutative 37.8%

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

       14. add-sqr-sqrt 37.8%

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

       15. sqrt-unprod 53.8%

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

       16. sqr-neg 53.8%

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

       17. sqrt-unprod 0.0%

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

       18. add-sqr-sqrt 36.4%

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

   11. Applied egg-rr 36.4%

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

       1. *-lft-identity 36.4%

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

   13. Simplified 36.4%

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

   if -2.14999999999999999e-50 < z < 7.80000000000000039e113

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0 66.2%

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

   if 7.80000000000000039e113 < z

   1. Initial program 68.9%

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   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 46.9%

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

   6. Taylor expanded in y around 0 41.4%

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

      1. associate-*r/ 41.4%

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

      2. neg-mul-1 41.4%

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

   8. Simplified 41.4%

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

      1. add-sqr-sqrt 22.1%

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

      2. sqrt-unprod 45.5%

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

      3. sqr-neg 45.5%

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

      4. sqrt-unprod 19.3%

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

      5. add-sqr-sqrt 41.5%

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

      6. *-un-lft-identity 41.5%

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

      7. *-commutative 41.5%

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

      8. associate-/r* 49.5%

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

   10. Applied egg-rr 49.5%

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

       1. *-lft-identity 49.5%

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

       2. associate-/l/ 41.5%

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

       3. *-commutative 41.5%

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

   12. Simplified 41.5%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{x\_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \end{array} \end{array} \]

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 (<= x_m 5.2e-86)
    (/ x_m (* (- y z) (- t z)))
    (/ (/ 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 tmp;
	if (x_m <= 5.2e-86) {
		tmp = x_m / ((y - z) * (t - z));
	} 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) :: tmp
    if (x_m <= 5.2d-86) then
        tmp = x_m / ((y - z) * (t - z))
    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 tmp;
	if (x_m <= 5.2e-86) {
		tmp = x_m / ((y - z) * (t - z));
	} 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):
	tmp = 0
	if x_m <= 5.2e-86:
		tmp = x_m / ((y - z) * (t - z))
	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)
	tmp = 0.0
	if (x_m <= 5.2e-86)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	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)
	tmp = 0.0;
	if (x_m <= 5.2e-86)
		tmp = x_m / ((y - z) * (t - z));
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 5.2e-86], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 x < 5.2000000000000002e-86

   1. Initial program 92.8%

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

   if 5.2000000000000002e-86 < x

   1. Initial program 78.5%

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

Alternative 19: 39.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / (y * t));
}

Fortran

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

Java

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

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	return x_s * (x_m / (y * t))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / Float64(y * t)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / (y * t));
end

Wolfram

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

Derivation

1. Initial program 88.7%

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

3. Taylor expanded in z around 0 44.8%

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

4. Final simplification 44.8%

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

Developer target: 87.5% 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 2024100 
(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))))