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

Percentage Accurate: 89.1% → 96.6%

Time: 11.5s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.1% 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: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t - z}\\ \end{array} \end{array} \]

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 1.85e-131)
    (/ (/ x_m (- t z)) (- y z))
    (/ (/ x_m (- y z)) (- t z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 1.85e-131) {
		tmp = (x_m / (t - z)) / (y - z);
	} else {
		tmp = (x_m / (y - z)) / (t - z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 1.85d-131) then
        tmp = (x_m / (t - z)) / (y - z)
    else
        tmp = (x_m / (y - z)) / (t - z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 1.85e-131) {
		tmp = (x_m / (t - z)) / (y - z);
	} else {
		tmp = (x_m / (y - z)) / (t - z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 1.85e-131:
		tmp = (x_m / (t - z)) / (y - z)
	else:
		tmp = (x_m / (y - z)) / (t - z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 1.85e-131)
		tmp = Float64(Float64(x_m / Float64(t - z)) / Float64(y - z));
	else
		tmp = Float64(Float64(x_m / Float64(y - z)) / Float64(t - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 1.85e-131)
		tmp = (x_m / (t - z)) / (y - z);
	else
		tmp = (x_m / (y - z)) / (t - z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 1.85e-131], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001e-131

   1. Initial program 86.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

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

   if 1.8500000000000001e-131 < x

   1. Initial program 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 95.7

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

   4. Applied rewrites 95.7%

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

Alternative 2: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(t - z\right) \cdot \left(y - z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{x\_m}{-t}}{z - y}\\ \mathbf{elif}\;t\_1 \leq 10^{+300}:\\ \;\;\;\;\frac{x\_m}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{-z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (- t z) (- y z))))
   (*
    x_s
    (if (<= t_1 -5e+293)
      (/ (/ x_m (- t)) (- z y))
      (if (<= t_1 1e+300) (/ x_m t_1) (/ (/ x_m (- y z)) (- z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (t - z) * (y - z);
	double tmp;
	if (t_1 <= -5e+293) {
		tmp = (x_m / -t) / (z - y);
	} else if (t_1 <= 1e+300) {
		tmp = x_m / t_1;
	} else {
		tmp = (x_m / (y - z)) / -z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - z) * (y - z)
    if (t_1 <= (-5d+293)) then
        tmp = (x_m / -t) / (z - y)
    else if (t_1 <= 1d+300) then
        tmp = x_m / t_1
    else
        tmp = (x_m / (y - z)) / -z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (t - z) * (y - z);
	double tmp;
	if (t_1 <= -5e+293) {
		tmp = (x_m / -t) / (z - y);
	} else if (t_1 <= 1e+300) {
		tmp = x_m / t_1;
	} else {
		tmp = (x_m / (y - z)) / -z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = (t - z) * (y - z)
	tmp = 0
	if t_1 <= -5e+293:
		tmp = (x_m / -t) / (z - y)
	elif t_1 <= 1e+300:
		tmp = x_m / t_1
	else:
		tmp = (x_m / (y - z)) / -z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(t - z) * Float64(y - z))
	tmp = 0.0
	if (t_1 <= -5e+293)
		tmp = Float64(Float64(x_m / Float64(-t)) / Float64(z - y));
	elseif (t_1 <= 1e+300)
		tmp = Float64(x_m / t_1);
	else
		tmp = Float64(Float64(x_m / Float64(y - z)) / Float64(-z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (t - z) * (y - z);
	tmp = 0.0;
	if (t_1 <= -5e+293)
		tmp = (x_m / -t) / (z - y);
	elseif (t_1 <= 1e+300)
		tmp = x_m / t_1;
	else
		tmp = (x_m / (y - z)) / -z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -5e+293], N[(N[(x$95$m / (-t)), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+300], N[(x$95$m / t$95$1), $MachinePrecision], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 y z) (-.f64 t z)) < -5.00000000000000033e293

   1. Initial program 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lift--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.8

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

   9. Applied rewrites 85.8%

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

   if -5.00000000000000033e293 < (*.f64 (-.f64 y z) (-.f64 t z)) < 1.0000000000000001e300

   1. Initial program 97.8%

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

   if 1.0000000000000001e300 < (*.f64 (-.f64 y z) (-.f64 t z))

   1. Initial program 65.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.0

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

   7. Applied rewrites 78.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - z\right) \cdot \left(y - z\right) \leq -5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z - y}\\ \mathbf{elif}\;\left(t - z\right) \cdot \left(y - z\right) \leq 10^{+300}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* (- t z) (- y z)))))
   (* x_s (if (<= t_1 0.0) (/ (/ x_m (- t z)) (- y z)) t_1))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / ((t - z) * (y - z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (x_m / (t - z)) / (y - z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / ((t - z) * (y - z))
    if (t_1 <= 0.0d0) then
        tmp = (x_m / (t - z)) / (y - z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / ((t - z) * (y - z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (x_m / (t - z)) / (y - z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / ((t - z) * (y - z))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (x_m / (t - z)) / (y - z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 0.0], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < 0.0

   1. Initial program 79.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 0.0 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 95.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)} \leq 0:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-137}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{x\_m}{-z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{-t}}{z - y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -7.8e+148)
    (/ (/ x_m y) (- t z))
    (if (<= y -3e-137)
      (/ x_m (* (- t z) (- y z)))
      (if (<= y 3e-62) (/ (/ x_m (- z)) (- t z)) (/ (/ x_m (- t)) (- 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 (y <= -7.8e+148) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= -3e-137) {
		tmp = x_m / ((t - z) * (y - z));
	} else if (y <= 3e-62) {
		tmp = (x_m / -z) / (t - z);
	} else {
		tmp = (x_m / -t) / (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 (y <= (-7.8d+148)) then
        tmp = (x_m / y) / (t - z)
    else if (y <= (-3d-137)) then
        tmp = x_m / ((t - z) * (y - z))
    else if (y <= 3d-62) then
        tmp = (x_m / -z) / (t - z)
    else
        tmp = (x_m / -t) / (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 (y <= -7.8e+148) {
		tmp = (x_m / y) / (t - z);
	} else if (y <= -3e-137) {
		tmp = x_m / ((t - z) * (y - z));
	} else if (y <= 3e-62) {
		tmp = (x_m / -z) / (t - z);
	} else {
		tmp = (x_m / -t) / (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 y <= -7.8e+148:
		tmp = (x_m / y) / (t - z)
	elif y <= -3e-137:
		tmp = x_m / ((t - z) * (y - z))
	elif y <= 3e-62:
		tmp = (x_m / -z) / (t - z)
	else:
		tmp = (x_m / -t) / (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 (y <= -7.8e+148)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	elseif (y <= -3e-137)
		tmp = Float64(x_m / Float64(Float64(t - z) * Float64(y - z)));
	elseif (y <= 3e-62)
		tmp = Float64(Float64(x_m / Float64(-z)) / Float64(t - z));
	else
		tmp = Float64(Float64(x_m / Float64(-t)) / Float64(z - y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -7.8e+148)
		tmp = (x_m / y) / (t - z);
	elseif (y <= -3e-137)
		tmp = x_m / ((t - z) * (y - z));
	elseif (y <= 3e-62)
		tmp = (x_m / -z) / (t - z);
	else
		tmp = (x_m / -t) / (z - y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -7.8e+148], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3e-137], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-62], N[(N[(x$95$m / (-z)), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / (-t)), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if y < -7.80000000000000004e148

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if -7.80000000000000004e148 < y < -2.9999999999999998e-137

   1. Initial program 89.5%

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

   if -2.9999999999999998e-137 < y < 3.0000000000000001e-62

   1. Initial program 88.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 86.3

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

   7. Applied rewrites 86.3%

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

   if 3.0000000000000001e-62 < y

   1. Initial program 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lift--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 67.8

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

   9. Applied rewrites 67.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{y}}{t - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-137}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{x\_m}{-z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x_m y) (- t z))))
   (*
    x_s
    (if (<= y -7.8e+148)
      t_1
      (if (<= y -3e-137)
        (/ x_m (* (- t z) (- y z)))
        (if (<= y 2.2e+67) (/ (/ x_m (- z)) (- t z)) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / y) / (t - z);
	double tmp;
	if (y <= -7.8e+148) {
		tmp = t_1;
	} else if (y <= -3e-137) {
		tmp = x_m / ((t - z) * (y - z));
	} else if (y <= 2.2e+67) {
		tmp = (x_m / -z) / (t - z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / y) / (t - z)
    if (y <= (-7.8d+148)) then
        tmp = t_1
    else if (y <= (-3d-137)) then
        tmp = x_m / ((t - z) * (y - z))
    else if (y <= 2.2d+67) then
        tmp = (x_m / -z) / (t - z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / y) / (t - z);
	double tmp;
	if (y <= -7.8e+148) {
		tmp = t_1;
	} else if (y <= -3e-137) {
		tmp = x_m / ((t - z) * (y - z));
	} else if (y <= 2.2e+67) {
		tmp = (x_m / -z) / (t - z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / y) / (t - z)
	tmp = 0
	if y <= -7.8e+148:
		tmp = t_1
	elif y <= -3e-137:
		tmp = x_m / ((t - z) * (y - z))
	elif y <= 2.2e+67:
		tmp = (x_m / -z) / (t - z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / y) / Float64(t - z))
	tmp = 0.0
	if (y <= -7.8e+148)
		tmp = t_1;
	elseif (y <= -3e-137)
		tmp = Float64(x_m / Float64(Float64(t - z) * Float64(y - z)));
	elseif (y <= 2.2e+67)
		tmp = Float64(Float64(x_m / Float64(-z)) / Float64(t - z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / y) / (t - z);
	tmp = 0.0;
	if (y <= -7.8e+148)
		tmp = t_1;
	elseif (y <= -3e-137)
		tmp = x_m / ((t - z) * (y - z));
	elseif (y <= 2.2e+67)
		tmp = (x_m / -z) / (t - z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -7.8e+148], t$95$1, If[LessEqual[y, -3e-137], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+67], N[(N[(x$95$m / (-z)), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < -7.80000000000000004e148 or 2.2e67 < y

   1. Initial program 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 86.2

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

   7. Applied rewrites 86.2%

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

   if -7.80000000000000004e148 < y < -2.9999999999999998e-137

   1. Initial program 89.5%

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

   if -2.9999999999999998e-137 < y < 2.2e67

   1. Initial program 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.1

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

   7. Applied rewrites 83.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x\_m}{z}}{z - y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+110}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x_m z) (- z y))))
   (*
    x_s
    (if (<= z -3.3e+118)
      t_1
      (if (<= z 1.9e+110) (/ x_m (* (- t z) (- y z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / z) / (z - y);
	double tmp;
	if (z <= -3.3e+118) {
		tmp = t_1;
	} else if (z <= 1.9e+110) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m / z) / (z - y)
    if (z <= (-3.3d+118)) then
        tmp = t_1
    else if (z <= 1.9d+110) then
        tmp = x_m / ((t - z) * (y - z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / z) / (z - y);
	double tmp;
	if (z <= -3.3e+118) {
		tmp = t_1;
	} else if (z <= 1.9e+110) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / z) / (z - y)
	tmp = 0
	if z <= -3.3e+118:
		tmp = t_1
	elif z <= 1.9e+110:
		tmp = x_m / ((t - z) * (y - z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / z) / Float64(z - y))
	tmp = 0.0
	if (z <= -3.3e+118)
		tmp = t_1;
	elseif (z <= 1.9e+110)
		tmp = Float64(x_m / Float64(Float64(t - z) * Float64(y - z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / z) / (z - y);
	tmp = 0.0;
	if (z <= -3.3e+118)
		tmp = t_1;
	elseif (z <= 1.9e+110)
		tmp = x_m / ((t - z) * (y - z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / z), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3.3e+118], t$95$1, If[LessEqual[z, 1.9e+110], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e118 or 1.89999999999999994e110 < z

   1. Initial program 72.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lift--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 90.3

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

   9. Applied rewrites 90.3%

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

   if -3.3e118 < z < 1.89999999999999994e110

   1. Initial program 88.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+110}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-197}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -4.1e+14)
    (/ x_m (* (- t z) y))
    (if (<= y 1.9e-197) (/ x_m (* z (- z t))) (/ x_m (* t (- y z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -4.1e+14) {
		tmp = x_m / ((t - z) * y);
	} else if (y <= 1.9e-197) {
		tmp = x_m / (z * (z - t));
	} else {
		tmp = x_m / (t * (y - z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-4.1d+14)) then
        tmp = x_m / ((t - z) * y)
    else if (y <= 1.9d-197) then
        tmp = x_m / (z * (z - t))
    else
        tmp = x_m / (t * (y - z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -4.1e+14) {
		tmp = x_m / ((t - z) * y);
	} else if (y <= 1.9e-197) {
		tmp = x_m / (z * (z - t));
	} else {
		tmp = x_m / (t * (y - z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -4.1e+14:
		tmp = x_m / ((t - z) * y)
	elif y <= 1.9e-197:
		tmp = x_m / (z * (z - t))
	else:
		tmp = x_m / (t * (y - z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -4.1e+14)
		tmp = Float64(x_m / Float64(Float64(t - z) * y));
	elseif (y <= 1.9e-197)
		tmp = Float64(x_m / Float64(z * Float64(z - t)));
	else
		tmp = Float64(x_m / Float64(t * Float64(y - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -4.1e+14)
		tmp = x_m / ((t - z) * y);
	elseif (y <= 1.9e-197)
		tmp = x_m / (z * (z - t));
	else
		tmp = x_m / (t * (y - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -4.1e+14], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e-197], N[(x$95$m / N[(z * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.1e14

   1. Initial program 85.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -4.1e14 < y < 1.8999999999999999e-197

   1. Initial program 88.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

   if 1.8999999999999999e-197 < y

   1. Initial program 77.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

Alternative 8: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z - y\right)}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+37}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t 1.8e-70)
    (/ x_m (* z (- z y)))
    (if (<= t 1.06e+37) (/ x_m (* z (- z t))) (/ x_m (* t (- y z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 1.8e-70) {
		tmp = x_m / (z * (z - y));
	} else if (t <= 1.06e+37) {
		tmp = x_m / (z * (z - t));
	} else {
		tmp = x_m / (t * (y - z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.8d-70) then
        tmp = x_m / (z * (z - y))
    else if (t <= 1.06d+37) then
        tmp = x_m / (z * (z - t))
    else
        tmp = x_m / (t * (y - z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 1.8e-70) {
		tmp = x_m / (z * (z - y));
	} else if (t <= 1.06e+37) {
		tmp = x_m / (z * (z - t));
	} else {
		tmp = x_m / (t * (y - z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 1.8e-70:
		tmp = x_m / (z * (z - y))
	elif t <= 1.06e+37:
		tmp = x_m / (z * (z - t))
	else:
		tmp = x_m / (t * (y - z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 1.8e-70)
		tmp = Float64(x_m / Float64(z * Float64(z - y)));
	elseif (t <= 1.06e+37)
		tmp = Float64(x_m / Float64(z * Float64(z - t)));
	else
		tmp = Float64(x_m / Float64(t * Float64(y - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 1.8e-70)
		tmp = x_m / (z * (z - y));
	elseif (t <= 1.06e+37)
		tmp = x_m / (z * (z - t));
	else
		tmp = x_m / (t * (y - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 1.8e-70], N[(x$95$m / N[(z * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.06e+37], N[(x$95$m / N[(z * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.8000000000000001e-70

   1. Initial program 83.2%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 1.8000000000000001e-70 < t < 1.06e37

   1. Initial program 91.6%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if 1.06e37 < t

   1. Initial program 82.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 79.6

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

   5. Applied rewrites 79.6%

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

Alternative 9: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot \left(z - t\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-41}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z (- z t)))))
   (*
    x_s
    (if (<= z -1.4e-88) t_1 (if (<= z 8e-41) (/ x_m (* t (- y z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * (z - t));
	double tmp;
	if (z <= -1.4e-88) {
		tmp = t_1;
	} else if (z <= 8e-41) {
		tmp = x_m / (t * (y - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / (z * (z - t))
    if (z <= (-1.4d-88)) then
        tmp = t_1
    else if (z <= 8d-41) then
        tmp = x_m / (t * (y - z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * (z - t));
	double tmp;
	if (z <= -1.4e-88) {
		tmp = t_1;
	} else if (z <= 8e-41) {
		tmp = x_m / (t * (y - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * (z - t))
	tmp = 0
	if z <= -1.4e-88:
		tmp = t_1
	elif z <= 8e-41:
		tmp = x_m / (t * (y - z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * Float64(z - t)))
	tmp = 0.0
	if (z <= -1.4e-88)
		tmp = t_1;
	elseif (z <= 8e-41)
		tmp = Float64(x_m / Float64(t * Float64(y - z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * (z - t));
	tmp = 0.0;
	if (z <= -1.4e-88)
		tmp = t_1;
	elseif (z <= 8e-41)
		tmp = x_m / (t * (y - z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.4e-88], t$95$1, If[LessEqual[z, 8e-41], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999988e-88 or 8.00000000000000005e-41 < z

   1. Initial program 80.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -1.39999999999999988e-88 < z < 8.00000000000000005e-41

   1. Initial program 89.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 76.3

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

   5. Applied rewrites 76.3%

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

Alternative 10: 69.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -10000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 490000000000:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z z))))
   (*
    x_s
    (if (<= z -10000.0)
      t_1
      (if (<= z 490000000000.0) (/ x_m (* t (- y z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -10000.0) {
		tmp = t_1;
	} else if (z <= 490000000000.0) {
		tmp = x_m / (t * (y - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / (z * z)
    if (z <= (-10000.0d0)) then
        tmp = t_1
    else if (z <= 490000000000.0d0) then
        tmp = x_m / (t * (y - z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -10000.0) {
		tmp = t_1;
	} else if (z <= 490000000000.0) {
		tmp = x_m / (t * (y - z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -10000.0:
		tmp = t_1
	elif z <= 490000000000.0:
		tmp = x_m / (t * (y - z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -10000.0)
		tmp = t_1;
	elseif (z <= 490000000000.0)
		tmp = Float64(x_m / Float64(t * Float64(y - z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -10000.0)
		tmp = t_1;
	elseif (z <= 490000000000.0)
		tmp = x_m / (t * (y - z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -10000.0], t$95$1, If[LessEqual[z, 490000000000.0], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1e4 or 4.9e11 < z

   1. Initial program 77.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if -1e4 < z < 4.9e11

   1. Initial program 90.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 68.3

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

   5. Applied rewrites 68.3%

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

Alternative 11: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -7.8e+148) (/ (/ x_m 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 (y <= -7.8e+148) {
		tmp = (x_m / 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 (y <= (-7.8d+148)) then
        tmp = (x_m / 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 (y <= -7.8e+148) {
		tmp = (x_m / 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 y <= -7.8e+148:
		tmp = (x_m / 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 (y <= -7.8e+148)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	else
		tmp = Float64(x_m / Float64(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 (y <= -7.8e+148)
		tmp = (x_m / 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[y, -7.8e+148], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -7.80000000000000004e148

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if -7.80000000000000004e148 < y

   1. Initial program 84.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10500000000:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z z))))
   (*
    x_s
    (if (<= z -1.4e-88) t_1 (if (<= z 10500000000.0) (/ x_m (* t y)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -1.4e-88) {
		tmp = t_1;
	} else if (z <= 10500000000.0) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m / (z * z)
    if (z <= (-1.4d-88)) then
        tmp = t_1
    else if (z <= 10500000000.0d0) then
        tmp = x_m / (t * y)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * z);
	double tmp;
	if (z <= -1.4e-88) {
		tmp = t_1;
	} else if (z <= 10500000000.0) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -1.4e-88:
		tmp = t_1
	elif z <= 10500000000.0:
		tmp = x_m / (t * y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -1.4e-88)
		tmp = t_1;
	elseif (z <= 10500000000.0)
		tmp = Float64(x_m / Float64(t * y));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -1.4e-88)
		tmp = t_1;
	elseif (z <= 10500000000.0)
		tmp = x_m / (t * y);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.4e-88], t$95$1, If[LessEqual[z, 10500000000.0], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999988e-88 or 1.05e10 < z

   1. Initial program 79.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if -1.39999999999999988e-88 < z < 1.05e10

   1. Initial program 89.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

Alternative 13: 89.1% accurate, 1.0× speedup

Math

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

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 (* (- 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) {
	return x_s * (x_m / ((t - z) * (y - z)));
}

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 / ((t - z) * (y - z)))
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 / ((t - z) * (y - z)));
}

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

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(Float64(t - z) * Float64(y - z))))
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 / ((t - z) * (y - z)));
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[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

3. Final simplification 83.8%

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

Alternative 14: 39.5% accurate, 1.4× speedup

Math

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

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 (* 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) {
	return x_s * (x_m / (t * y));
}

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 / (t * y))
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 / (t * y));
}

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 / (t * y))

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(t * y)))
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 / (t * y));
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[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in z around 0

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

   1. lower-*.f64 31.8

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

5. Applied rewrites 31.8%

   \[\leadsto \frac{x}{\color{blue}{t \cdot y}} \]
6. Add Preprocessing

Developer Target 1: 87.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

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

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