Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.2% → 98.0%

Time: 8.0s

Alternatives: 10

Speedup: 0.9×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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 * y)) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * y)) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.18 \cdot 10^{-51}:\\ \;\;\;\;\left(x \cdot y\_m - z \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot \left(x - z\right)\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (*
   t_s
   (if (<= t_m 1.18e-51)
     (* (- (* x y_m) (* z y_m)) t_m)
     (* (* y_m t_m) (- x z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1.18e-51) {
		tmp = ((x * y_m) - (z * y_m)) * t_m;
	} else {
		tmp = (y_m * t_m) * (x - z);
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.18d-51) then
        tmp = ((x * y_m) - (z * y_m)) * t_m
    else
        tmp = (y_m * t_m) * (x - z)
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1.18e-51) {
		tmp = ((x * y_m) - (z * y_m)) * t_m;
	} else {
		tmp = (y_m * t_m) * (x - z);
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 1.18e-51:
		tmp = ((x * y_m) - (z * y_m)) * t_m
	else:
		tmp = (y_m * t_m) * (x - z)
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 1.18e-51)
		tmp = Float64(Float64(Float64(x * y_m) - Float64(z * y_m)) * t_m);
	else
		tmp = Float64(Float64(y_m * t_m) * Float64(x - z));
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 1.18e-51)
		tmp = ((x * y_m) - (z * y_m)) * t_m;
	else
		tmp = (y_m * t_m) * (x - z);
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 1.18e-51], N[(N[(N[(x * y$95$m), $MachinePrecision] - N[(z * y$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(y$95$m * t$95$m), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.18000000000000004e-51

   1. Initial program 90.8%

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

   if 1.18000000000000004e-51 < t

   1. Initial program 93.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{-51}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.5× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (* y_s (* t_s (/ t_m (/ (/ -1.0 y_m) (- z x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	return y_s * (t_s * (t_m / ((-1.0 / y_m) / (z - x))));
}

Fortran

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

Java

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

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	return y_s * (t_s * (t_m / ((-1.0 / y_m) / (z - x))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	return Float64(y_s * Float64(t_s * Float64(t_m / Float64(Float64(-1.0 / y_m) / Float64(z - x)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip-- N/A

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

   10. lift--.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 91.7

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. distribute-rgt-out-- N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower--.f64 94.1

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

4. Applied rewrites 94.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. unpow-1 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. unpow-1 N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 94.1

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

6. Applied rewrites 94.1%

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

7. Final simplification 94.1%

   \[\leadsto \frac{t}{\frac{\frac{-1}{y}}{z - x}} \]
8. Add Preprocessing

Alternative 3: 55.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y\_m - z \cdot y\_m \leq 4 \cdot 10^{-217}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\_m\right) \cdot t\_m\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (*
   t_s
   (if (<= (- (* x y_m) (* z y_m)) 4e-217)
     (* (* y_m t_m) x)
     (* (* x y_m) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (((x * y_m) - (z * y_m)) <= 4e-217) {
		tmp = (y_m * t_m) * x;
	} else {
		tmp = (x * y_m) * t_m;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (((x * y_m) - (z * y_m)) <= 4d-217) then
        tmp = (y_m * t_m) * x
    else
        tmp = (x * y_m) * t_m
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (((x * y_m) - (z * y_m)) <= 4e-217) {
		tmp = (y_m * t_m) * x;
	} else {
		tmp = (x * y_m) * t_m;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if ((x * y_m) - (z * y_m)) <= 4e-217:
		tmp = (y_m * t_m) * x
	else:
		tmp = (x * y_m) * t_m
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (Float64(Float64(x * y_m) - Float64(z * y_m)) <= 4e-217)
		tmp = Float64(Float64(y_m * t_m) * x);
	else
		tmp = Float64(Float64(x * y_m) * t_m);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (((x * y_m) - (z * y_m)) <= 4e-217)
		tmp = (y_m * t_m) * x;
	else
		tmp = (x * y_m) * t_m;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[N[(N[(x * y$95$m), $MachinePrecision] - N[(z * y$95$m), $MachinePrecision]), $MachinePrecision], 4e-217], N[(N[(y$95$m * t$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < 4.00000000000000033e-217

   1. Initial program 94.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 94.6

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 54.5

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

   7. Applied rewrites 54.5%

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

   if 4.00000000000000033e-217 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 88.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \leq 4 \cdot 10^{-217}:\\ \;\;\;\;\left(y \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+31}:\\ \;\;\;\;\left(x \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* (- z) y_m) t_m)))
   (*
    y_s
    (*
     t_s
     (if (<= z -2.6e-33) t_2 (if (<= z 2.9e+31) (* (* x y_m) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (-z * y_m) * t_m;
	double tmp;
	if (z <= -2.6e-33) {
		tmp = t_2;
	} else if (z <= 2.9e+31) {
		tmp = (x * y_m) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (-z * y_m) * t_m
    if (z <= (-2.6d-33)) then
        tmp = t_2
    else if (z <= 2.9d+31) then
        tmp = (x * y_m) * t_m
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (-z * y_m) * t_m;
	double tmp;
	if (z <= -2.6e-33) {
		tmp = t_2;
	} else if (z <= 2.9e+31) {
		tmp = (x * y_m) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (-z * y_m) * t_m
	tmp = 0
	if z <= -2.6e-33:
		tmp = t_2
	elif z <= 2.9e+31:
		tmp = (x * y_m) * t_m
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(Float64(-z) * y_m) * t_m)
	tmp = 0.0
	if (z <= -2.6e-33)
		tmp = t_2;
	elseif (z <= 2.9e+31)
		tmp = Float64(Float64(x * y_m) * t_m);
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (-z * y_m) * t_m;
	tmp = 0.0;
	if (z <= -2.6e-33)
		tmp = t_2;
	elseif (z <= 2.9e+31)
		tmp = (x * y_m) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[((-z) * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[z, -2.6e-33], t$95$2, If[LessEqual[z, 2.9e+31], N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.59999999999999994e-33 or 2.9e31 < z

   1. Initial program 87.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -2.59999999999999994e-33 < z < 2.9e31

   1. Initial program 96.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-33}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\right) \cdot t\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+31}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-33}:\\ \;\;\;\;\left(\left(-t\_m\right) \cdot z\right) \cdot y\_m\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+31}:\\ \;\;\;\;\left(x \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot \left(-z\right)\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (*
   t_s
   (if (<= z -2.6e-33)
     (* (* (- t_m) z) y_m)
     (if (<= z 2.9e+31) (* (* x y_m) t_m) (* (* y_m t_m) (- z)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (z <= -2.6e-33) {
		tmp = (-t_m * z) * y_m;
	} else if (z <= 2.9e+31) {
		tmp = (x * y_m) * t_m;
	} else {
		tmp = (y_m * t_m) * -z;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (z <= (-2.6d-33)) then
        tmp = (-t_m * z) * y_m
    else if (z <= 2.9d+31) then
        tmp = (x * y_m) * t_m
    else
        tmp = (y_m * t_m) * -z
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (z <= -2.6e-33) {
		tmp = (-t_m * z) * y_m;
	} else if (z <= 2.9e+31) {
		tmp = (x * y_m) * t_m;
	} else {
		tmp = (y_m * t_m) * -z;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if z <= -2.6e-33:
		tmp = (-t_m * z) * y_m
	elif z <= 2.9e+31:
		tmp = (x * y_m) * t_m
	else:
		tmp = (y_m * t_m) * -z
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (z <= -2.6e-33)
		tmp = Float64(Float64(Float64(-t_m) * z) * y_m);
	elseif (z <= 2.9e+31)
		tmp = Float64(Float64(x * y_m) * t_m);
	else
		tmp = Float64(Float64(y_m * t_m) * Float64(-z));
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (z <= -2.6e-33)
		tmp = (-t_m * z) * y_m;
	elseif (z <= 2.9e+31)
		tmp = (x * y_m) * t_m;
	else
		tmp = (y_m * t_m) * -z;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[z, -2.6e-33], N[(N[((-t$95$m) * z), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[z, 2.9e+31], N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(y$95$m * t$95$m), $MachinePrecision] * (-z)), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -2.59999999999999994e-33

   1. Initial program 90.5%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -2.59999999999999994e-33 < z < 2.9e31

   1. Initial program 96.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if 2.9e31 < z

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 82.4

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.4

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

   6. Applied rewrites 92.4%

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

   7. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.3

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

   9. Applied rewrites 76.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-33}:\\ \;\;\;\;\left(\left(-t\right) \cdot z\right) \cdot y\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+31}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(x \cdot y\_m\right) \cdot t\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(-t\_m\right) \cdot z\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* x y_m) t_m)))
   (*
    y_s
    (*
     t_s
     (if (<= x -1.7e-31) t_2 (if (<= x 4.8e-38) (* (* (- t_m) z) y_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) * t_m;
	double tmp;
	if (x <= -1.7e-31) {
		tmp = t_2;
	} else if (x <= 4.8e-38) {
		tmp = (-t_m * z) * y_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x * y_m) * t_m
    if (x <= (-1.7d-31)) then
        tmp = t_2
    else if (x <= 4.8d-38) then
        tmp = (-t_m * z) * y_m
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) * t_m;
	double tmp;
	if (x <= -1.7e-31) {
		tmp = t_2;
	} else if (x <= 4.8e-38) {
		tmp = (-t_m * z) * y_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (x * y_m) * t_m
	tmp = 0
	if x <= -1.7e-31:
		tmp = t_2
	elif x <= 4.8e-38:
		tmp = (-t_m * z) * y_m
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(x * y_m) * t_m)
	tmp = 0.0
	if (x <= -1.7e-31)
		tmp = t_2;
	elseif (x <= 4.8e-38)
		tmp = Float64(Float64(Float64(-t_m) * z) * y_m);
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (x * y_m) * t_m;
	tmp = 0.0;
	if (x <= -1.7e-31)
		tmp = t_2;
	elseif (x <= 4.8e-38)
		tmp = (-t_m * z) * y_m;
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[x, -1.7e-31], t$95$2, If[LessEqual[x, 4.8e-38], N[(N[((-t$95$m) * z), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7000000000000001e-31 or 4.80000000000000044e-38 < x

   1. Initial program 88.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if -1.7000000000000001e-31 < x < 4.80000000000000044e-38

   1. Initial program 96.2%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 78.5

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

   5. Applied rewrites 78.5%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-31}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(-t\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot \left(x - z\right)\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (* t_s (if (<= t_m 2e+16) (* (* (- x z) t_m) y_m) (* (* y_m t_m) (- x z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2e+16) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = (y_m * t_m) * (x - z);
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2d+16) then
        tmp = ((x - z) * t_m) * y_m
    else
        tmp = (y_m * t_m) * (x - z)
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2e+16) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = (y_m * t_m) * (x - z);
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 2e+16:
		tmp = ((x - z) * t_m) * y_m
	else:
		tmp = (y_m * t_m) * (x - z)
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 2e+16)
		tmp = Float64(Float64(Float64(x - z) * t_m) * y_m);
	else
		tmp = Float64(Float64(y_m * t_m) * Float64(x - z));
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 2e+16)
		tmp = ((x - z) * t_m) * y_m;
	else
		tmp = (y_m * t_m) * (x - z);
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 2e+16], N[(N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(y$95$m * t$95$m), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2e16

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 90.3

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

   4. Applied rewrites 90.3%

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

   if 2e16 < t

   1. Initial program 95.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

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

4. Final simplification 92.4%

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

Alternative 8: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{+96}:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (* t_s (if (<= z 8.5e+96) (* (* (- x z) t_m) y_m) (* (* (- z) y_m) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (z <= 8.5e+96) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = (-z * y_m) * t_m;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (z <= 8.5d+96) then
        tmp = ((x - z) * t_m) * y_m
    else
        tmp = (-z * y_m) * t_m
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (z <= 8.5e+96) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = (-z * y_m) * t_m;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if z <= 8.5e+96:
		tmp = ((x - z) * t_m) * y_m
	else:
		tmp = (-z * y_m) * t_m
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (z <= 8.5e+96)
		tmp = Float64(Float64(Float64(x - z) * t_m) * y_m);
	else
		tmp = Float64(Float64(Float64(-z) * y_m) * t_m);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (z <= 8.5e+96)
		tmp = ((x - z) * t_m) * y_m;
	else
		tmp = (-z * y_m) * t_m;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[z, 8.5e+96], N[(N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[((-z) * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.50000000000000025e96

   1. Initial program 94.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 90.3

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

   4. Applied rewrites 90.3%

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

   if 8.50000000000000025e96 < z

   1. Initial program 80.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 79.0

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

   5. Applied rewrites 79.0%

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

Alternative 9: 57.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{+31}:\\ \;\;\;\;\left(x \cdot t\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot x\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (* y_s (* t_s (if (<= t_m 1.35e+31) (* (* x t_m) y_m) (* (* y_m t_m) x)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1.35e+31) {
		tmp = (x * t_m) * y_m;
	} else {
		tmp = (y_m * t_m) * x;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.35d+31) then
        tmp = (x * t_m) * y_m
    else
        tmp = (y_m * t_m) * x
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1.35e+31) {
		tmp = (x * t_m) * y_m;
	} else {
		tmp = (y_m * t_m) * x;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 1.35e+31:
		tmp = (x * t_m) * y_m
	else:
		tmp = (y_m * t_m) * x
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 1.35e+31)
		tmp = Float64(Float64(x * t_m) * y_m);
	else
		tmp = Float64(Float64(y_m * t_m) * x);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 1.35e+31)
		tmp = (x * t_m) * y_m;
	else
		tmp = (y_m * t_m) * x;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 1.35e+31], N[(N[(x * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(y$95$m * t$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.34999999999999993e31

   1. Initial program 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 90.5

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.1

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

   7. Applied rewrites 52.1%

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

   if 1.34999999999999993e31 < t

   1. Initial program 95.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 95.1

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out-- N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 54.9

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

   7. Applied rewrites 54.9%

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

Alternative 10: 54.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	return y_s * (t_s * ((y_m * t_m) * x));
}

Fortran

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

Java

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

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	return y_s * (t_s * ((y_m * t_m) * x))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	return Float64(y_s * Float64(t_s * Float64(Float64(y_m * t_m) * x)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp = code(y_s, t_s, x, y_m, z, t_m)
	tmp = y_s * (t_s * ((y_m * t_m) * x));
end

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip-- N/A

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

   10. lift--.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 91.7

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. distribute-rgt-out-- N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower--.f64 94.1

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

4. Applied rewrites 94.1%

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

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 54.0

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

7. Applied rewrites 54.0%

   \[\leadsto \color{blue}{\left(y \cdot t\right) \cdot x} \]
8. Add Preprocessing

Developer Target 1: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< t -9.231879582886777e-80)
   (* (* y t) (- x z))
   (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	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) :: tmp
    if (t < (-9.231879582886777d-80)) then
        tmp = (y * t) * (x - z)
    else if (t < 2.543067051564877d+83) then
        tmp = y * (t * (x - z))
    else
        tmp = (y * (x - z)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t < -9.231879582886777e-80:
		tmp = (y * t) * (x - z)
	elif t < 2.543067051564877e+83:
		tmp = y * (t * (x - z))
	else:
		tmp = (y * (x - z)) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t < -9.231879582886777e-80)
		tmp = Float64(Float64(y * t) * Float64(x - z));
	elseif (t < 2.543067051564877e+83)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(Float64(y * Float64(x - z)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t < -9.231879582886777e-80)
		tmp = (y * t) * (x - z);
	elseif (t < 2.543067051564877e+83)
		tmp = y * (t * (x - z));
	else
		tmp = (y * (x - z)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[t, -9.231879582886777e-80], N[(N[(y * t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], If[Less[t, 2.543067051564877e+83], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024267 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -9231879582886777/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* y t) (- x z)) (if (< t 254306705156487700000000000000000000000000000000000000000000000000000000000000000000) (* y (* t (- x z))) (* (* y (- x z)) t))))

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