Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.7% → 98.2%

Time: 9.1s

Alternatives: 9

Speedup: 1.0×

• 25.1% 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: 96.7% 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: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.0004:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot \left(x - z\right)\\ \end{array}\right) \end{array} \]

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 0.0004) (* (* (- 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 <= 0.0004) {
		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 <= 0.0004d0) 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 <= 0.0004) {
		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 <= 0.0004:
		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 <= 0.0004)
		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 <= 0.0004)
		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, 0.0004], 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 < 4.00000000000000019e-4

   1. Initial program 92.9%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

   if 4.00000000000000019e-4 < t

   1. Initial program 91.9%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 96.7

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

   4. Applied rewrites 96.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0004:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.2× speedup

Math

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

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 (pow (* y_m (- x z)) -1.0)))))

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 / pow((y_m * (x - z)), -1.0)));
}

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 / ((y_m * (x - z)) ** (-1.0d0))))
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 / Math.pow((y_m * (x - z)), -1.0)));
}

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 / math.pow((y_m * (x - z)), -1.0)))

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(y_m * Float64(x - z)) ^ -1.0))))
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 / ((y_m * (x - z)) ^ -1.0)));
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[Power[N[(y$95$m * N[(x - z), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.6%

   \[\left(x \cdot y - z \cdot y\right) \cdot t \]
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 92.6

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

   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 93.4

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

4. Applied rewrites 93.4%

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

5. Final simplification 93.4%

   \[\leadsto \frac{t}{{\left(y \cdot \left(x - z\right)\right)}^{-1}} \]
6. Add Preprocessing

Alternative 3: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.45:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+34}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

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 -0.45) t_2 (if (<= z 3.9e+34) (* (* y_m x) 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 <= -0.45) {
		tmp = t_2;
	} else if (z <= 3.9e+34) {
		tmp = (y_m * x) * 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 <= (-0.45d0)) then
        tmp = t_2
    else if (z <= 3.9d+34) then
        tmp = (y_m * x) * 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 <= -0.45) {
		tmp = t_2;
	} else if (z <= 3.9e+34) {
		tmp = (y_m * x) * 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 <= -0.45:
		tmp = t_2
	elif z <= 3.9e+34:
		tmp = (y_m * x) * 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 <= -0.45)
		tmp = t_2;
	elseif (z <= 3.9e+34)
		tmp = Float64(Float64(y_m * x) * 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 <= -0.45)
		tmp = t_2;
	elseif (z <= 3.9e+34)
		tmp = (y_m * x) * 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, -0.45], t$95$2, If[LessEqual[z, 3.9e+34], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.450000000000000011 or 3.90000000000000019e34 < z

   1. Initial program 92.5%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   if -0.450000000000000011 < z < 3.90000000000000019e34

   1. Initial program 92.8%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

Alternative 4: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.45:\\ \;\;\;\;\left(-z\right) \cdot \left(y\_m \cdot t\_m\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+34}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\_m\right) \cdot z\right) \cdot y\_m\\ \end{array}\right) \end{array} \]

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 -0.45)
     (* (- z) (* y_m t_m))
     (if (<= z 3.9e+34) (* (* y_m x) t_m) (* (* (- t_m) z) y_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 <= -0.45) {
		tmp = -z * (y_m * t_m);
	} else if (z <= 3.9e+34) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (-t_m * z) * y_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 <= (-0.45d0)) then
        tmp = -z * (y_m * t_m)
    else if (z <= 3.9d+34) then
        tmp = (y_m * x) * t_m
    else
        tmp = (-t_m * z) * y_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 <= -0.45) {
		tmp = -z * (y_m * t_m);
	} else if (z <= 3.9e+34) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (-t_m * z) * y_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 <= -0.45:
		tmp = -z * (y_m * t_m)
	elif z <= 3.9e+34:
		tmp = (y_m * x) * t_m
	else:
		tmp = (-t_m * z) * y_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 <= -0.45)
		tmp = Float64(Float64(-z) * Float64(y_m * t_m));
	elseif (z <= 3.9e+34)
		tmp = Float64(Float64(y_m * x) * t_m);
	else
		tmp = Float64(Float64(Float64(-t_m) * z) * y_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 <= -0.45)
		tmp = -z * (y_m * t_m);
	elseif (z <= 3.9e+34)
		tmp = (y_m * x) * t_m;
	else
		tmp = (-t_m * z) * y_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, -0.45], N[((-z) * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.9e+34], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[((-t$95$m) * z), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -0.450000000000000011

   1. Initial program 90.6%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if -0.450000000000000011 < z < 3.90000000000000019e34

   1. Initial program 92.8%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

   if 3.90000000000000019e34 < z

   1. Initial program 94.0%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.45:\\ \;\;\;\;\left(-z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+34}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(\left(-t\_m\right) \cdot z\right) \cdot y\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.45:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+34}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

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 (* (* (- t_m) z) y_m)))
   (*
    y_s
    (* t_s (if (<= z -0.45) t_2 (if (<= z 3.9e+34) (* (* y_m x) 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 = (-t_m * z) * y_m;
	double tmp;
	if (z <= -0.45) {
		tmp = t_2;
	} else if (z <= 3.9e+34) {
		tmp = (y_m * x) * 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 = (-t_m * z) * y_m
    if (z <= (-0.45d0)) then
        tmp = t_2
    else if (z <= 3.9d+34) then
        tmp = (y_m * x) * 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 = (-t_m * z) * y_m;
	double tmp;
	if (z <= -0.45) {
		tmp = t_2;
	} else if (z <= 3.9e+34) {
		tmp = (y_m * x) * 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 = (-t_m * z) * y_m
	tmp = 0
	if z <= -0.45:
		tmp = t_2
	elif z <= 3.9e+34:
		tmp = (y_m * x) * 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(-t_m) * z) * y_m)
	tmp = 0.0
	if (z <= -0.45)
		tmp = t_2;
	elseif (z <= 3.9e+34)
		tmp = Float64(Float64(y_m * x) * 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 = (-t_m * z) * y_m;
	tmp = 0.0;
	if (z <= -0.45)
		tmp = t_2;
	elseif (z <= 3.9e+34)
		tmp = (y_m * x) * 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[((-t$95$m) * z), $MachinePrecision] * y$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[z, -0.45], t$95$2, If[LessEqual[z, 3.9e+34], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.450000000000000011 or 3.90000000000000019e34 < z

   1. Initial program 92.5%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

   if -0.450000000000000011 < z < 3.90000000000000019e34

   1. Initial program 92.8%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

Alternative 6: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+175}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\_m\right) \cdot y\_m\\ \end{array}\right) \end{array} \]

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 -4.2e+175) (* (* (- z) y_m) t_m) (* (* (- x z) t_m) y_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 <= -4.2e+175) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = ((x - z) * t_m) * y_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 <= (-4.2d+175)) then
        tmp = (-z * y_m) * t_m
    else
        tmp = ((x - z) * t_m) * y_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 <= -4.2e+175) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = ((x - z) * t_m) * y_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 <= -4.2e+175:
		tmp = (-z * y_m) * t_m
	else:
		tmp = ((x - z) * t_m) * y_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 <= -4.2e+175)
		tmp = Float64(Float64(Float64(-z) * y_m) * t_m);
	else
		tmp = Float64(Float64(Float64(x - z) * t_m) * y_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 <= -4.2e+175)
		tmp = (-z * y_m) * t_m;
	else
		tmp = ((x - z) * t_m) * y_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, -4.2e+175], N[(N[((-z) * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.1999999999999998e175

   1. Initial program 82.4%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   if -4.1999999999999998e175 < z

   1. Initial program 93.6%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 95.3

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

   4. Applied rewrites 95.3%

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

Alternative 7: 96.7% accurate, 1.0× speedup

Math

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

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 x) (* y_m z)) 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) {
	return y_s * (t_s * (((y_m * x) - (y_m * z)) * t_m));
}

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 * x) - (y_m * z)) * t_m))
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 * x) - (y_m * z)) * t_m));
}

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 * x) - (y_m * z)) * t_m))

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(Float64(y_m * x) - Float64(y_m * z)) * t_m)))
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 * x) - (y_m * z)) * t_m));
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[(N[(y$95$m * x), $MachinePrecision] - N[(y$95$m * z), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

3. Final simplification 92.6%

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

Alternative 8: 58.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left(x \cdot t\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot x\\ \end{array}\right) \end{array} \]

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 2e-26) (* (* 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 <= 2e-26) {
		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 <= 2d-26) 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 <= 2e-26) {
		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 <= 2e-26:
		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 <= 2e-26)
		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 <= 2e-26)
		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, 2e-26], 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 < 2.0000000000000001e-26

   1. Initial program 93.1%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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. 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 95.2

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

   4. Applied rewrites 95.2%

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

   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 53.8

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

   7. Applied rewrites 53.8%

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

   if 2.0000000000000001e-26 < t

   1. Initial program 91.3%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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. 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 91.2

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

      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 94.2

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

   4. Applied rewrites 94.2%

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

   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 50.3

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

   7. Applied rewrites 50.3%

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

Alternative 9: 55.1% 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 92.6%

   \[\left(x \cdot y - z \cdot y\right) \cdot t \]
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 92.6

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

   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 93.4

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

4. Applied rewrites 93.4%

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

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 51.5

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

7. Applied rewrites 51.5%

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

Developer Target 1: 96.1% 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 2024244 
(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))