Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 97.2% → 98.4%

Time: 11.9s

Alternatives: 8

Speedup: 1.3×

• 25.9% 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: 97.2% 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: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 50000000:\\ \;\;\;\;t\_m \cdot \left(\left(x - z\right) \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (*
   y_s
   (if (<= t_m 50000000.0) (* t_m (* (- x z) y_m)) (* (- x z) (* t_m y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 50000000.0) {
		tmp = t_m * ((x - z) * y_m);
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_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 <= 50000000.0d0) then
        tmp = t_m * ((x - z) * y_m)
    else
        tmp = (x - z) * (t_m * y_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 50000000.0) {
		tmp = t_m * ((x - z) * y_m);
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 50000000.0:
		tmp = t_m * ((x - z) * y_m)
	else:
		tmp = (x - z) * (t_m * y_m)
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 50000000.0)
		tmp = Float64(t_m * Float64(Float64(x - z) * y_m));
	else
		tmp = Float64(Float64(x - z) * Float64(t_m * y_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 50000000.0)
		tmp = t_m * ((x - z) * y_m);
	else
		tmp = (x - z) * (t_m * y_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[LessEqual[t$95$m, 50000000.0], N[(t$95$m * N[(N[(x - z), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5e7

   1. Initial program 86.4%

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

      1. *-commutative N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

      8. *-lowering-*.f64 90.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

   3. Simplified 90.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y - z \cdot y\right), \color{blue}{t}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(x - z\right)\right), t\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x - z\right) \cdot y\right), t\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x - z\right), y\right), t\right) \]

      8. --lowering--.f64 89.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), y\right), t\right) \]

   6. Applied egg-rr 89.0%

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

   if 5e7 < t

   1. Initial program 96.4%

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

      1. *-commutative N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

      8. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

   3. Simplified 99.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 50000000:\\ \;\;\;\;t \cdot \left(\left(x - z\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := 0 - t\_m \cdot \left(z \cdot y\_m\right)\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-54}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 0.0175:\\ \;\;\;\;t\_m \cdot \left(x \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (- 0.0 (* t_m (* z y_m)))))
   (*
    t_s
    (*
     y_s
     (if (<= z -8.5e-54) t_2 (if (<= z 0.0175) (* t_m (* x y_m)) t_2))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = 0.0 - (t_m * (z * y_m));
	double tmp;
	if (z <= -8.5e-54) {
		tmp = t_2;
	} else if (z <= 0.0175) {
		tmp = t_m * (x * y_m);
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_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 = 0.0d0 - (t_m * (z * y_m))
    if (z <= (-8.5d-54)) then
        tmp = t_2
    else if (z <= 0.0175d0) then
        tmp = t_m * (x * y_m)
    else
        tmp = t_2
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = 0.0 - (t_m * (z * y_m));
	double tmp;
	if (z <= -8.5e-54) {
		tmp = t_2;
	} else if (z <= 0.0175) {
		tmp = t_m * (x * y_m);
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	t_2 = 0.0 - (t_m * (z * y_m))
	tmp = 0
	if z <= -8.5e-54:
		tmp = t_2
	elif z <= 0.0175:
		tmp = t_m * (x * y_m)
	else:
		tmp = t_2
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	t_2 = Float64(0.0 - Float64(t_m * Float64(z * y_m)))
	tmp = 0.0
	if (z <= -8.5e-54)
		tmp = t_2;
	elseif (z <= 0.0175)
		tmp = Float64(t_m * Float64(x * y_m));
	else
		tmp = t_2;
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	t_2 = 0.0 - (t_m * (z * y_m));
	tmp = 0.0;
	if (z <= -8.5e-54)
		tmp = t_2;
	elseif (z <= 0.0175)
		tmp = t_m * (x * y_m);
	else
		tmp = t_2;
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(0.0 - N[(t$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[z, -8.5e-54], t$95$2, If[LessEqual[z, 0.0175], N[(t$95$m * N[(x * y$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e-54 or 0.017500000000000002 < z

   1. Initial program 87.4%

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

      1. *-commutative N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

      8. *-lowering-*.f64 91.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

   3. Simplified 91.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. flip-- N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(t, \color{blue}{\left(\frac{x \cdot y + z \cdot y}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) - \left(z \cdot y\right) \cdot \left(z \cdot y\right)}\right)}\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(t, \left(\frac{1}{\color{blue}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) - \left(z \cdot y\right) \cdot \left(z \cdot y\right)}{x \cdot y + z \cdot y}}}\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(t, \left(\frac{1}{x \cdot y - \color{blue}{z \cdot y}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot y - z \cdot y\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \left(y \cdot \color{blue}{\left(x - z\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \left(\left(x - z\right) \cdot \color{blue}{y}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{y}\right)\right)\right) \]

      15. --lowering--.f64 91.6%

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), y\right)\right)\right) \]

   6. Applied egg-rr 91.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(t, \color{blue}{\left(\frac{-1}{y \cdot z}\right)}\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(-1, \color{blue}{\left(y \cdot z\right)}\right)\right) \]

      2. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   9. Simplified 75.4%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. associate-/r* N/A

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

       4. div-inv N/A

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

       5. neg-mul-1 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(\frac{1}{y}\right)}{z}} \cdot t \]

       6. clear-num N/A

          \[\leadsto \frac{z}{\mathsf{neg}\left(\frac{1}{y}\right)} \cdot t \]

       7. distribute-frac-neg2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{z}{\frac{1}{y}}\right)\right) \cdot t \]

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

          \[\leadsto \mathsf{neg}\left(\frac{z}{\frac{1}{y}} \cdot t\right) \]

       9. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{z}{\frac{1}{y}} \cdot t\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{z}{\frac{1}{y}}\right), t\right)\right) \]

       11. div-inv N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(z \cdot \frac{1}{\frac{1}{y}}\right), t\right)\right) \]

       12. remove-double-div N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(z \cdot y\right), t\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(y \cdot z\right), t\right)\right) \]

       14. *-lowering-*.f64 75.5%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), t\right)\right) \]

   11. Applied egg-rr 75.5%

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

   if -8.5e-54 < z < 0.017500000000000002

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(x \cdot y\right)}, t\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot x\right), t\right) \]

      2. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, x\right), t\right) \]

   5. Simplified 76.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-54}:\\ \;\;\;\;0 - t \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 0.0175:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := t\_m \cdot \left(x \cdot y\_m\right)\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{+49}:\\ \;\;\;\;0 - z \cdot \left(t\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (* x y_m))))
   (*
    t_s
    (*
     y_s
     (if (<= x -4.2e+62)
       t_2
       (if (<= x 1.08e+49) (- 0.0 (* z (* t_m y_m))) t_2))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = t_m * (x * y_m);
	double tmp;
	if (x <= -4.2e+62) {
		tmp = t_2;
	} else if (x <= 1.08e+49) {
		tmp = 0.0 - (z * (t_m * y_m));
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_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 * (x * y_m)
    if (x <= (-4.2d+62)) then
        tmp = t_2
    else if (x <= 1.08d+49) then
        tmp = 0.0d0 - (z * (t_m * y_m))
    else
        tmp = t_2
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = t_m * (x * y_m);
	double tmp;
	if (x <= -4.2e+62) {
		tmp = t_2;
	} else if (x <= 1.08e+49) {
		tmp = 0.0 - (z * (t_m * y_m));
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	t_2 = t_m * (x * y_m)
	tmp = 0
	if x <= -4.2e+62:
		tmp = t_2
	elif x <= 1.08e+49:
		tmp = 0.0 - (z * (t_m * y_m))
	else:
		tmp = t_2
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	t_2 = Float64(t_m * Float64(x * y_m))
	tmp = 0.0
	if (x <= -4.2e+62)
		tmp = t_2;
	elseif (x <= 1.08e+49)
		tmp = Float64(0.0 - Float64(z * Float64(t_m * y_m)));
	else
		tmp = t_2;
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	t_2 = t_m * (x * y_m);
	tmp = 0.0;
	if (x <= -4.2e+62)
		tmp = t_2;
	elseif (x <= 1.08e+49)
		tmp = 0.0 - (z * (t_m * y_m));
	else
		tmp = t_2;
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -4.2e+62], t$95$2, If[LessEqual[x, 1.08e+49], N[(0.0 - N[(z * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e62 or 1.08000000000000001e49 < x

   1. Initial program 82.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(x \cdot y\right)}, t\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot x\right), t\right) \]

      2. *-lowering-*.f64 72.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, x\right), t\right) \]

   5. Simplified 72.1%

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

   if -4.2e62 < x < 1.08000000000000001e49

   1. Initial program 93.3%

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

      1. *-commutative N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

      8. *-lowering-*.f64 92.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

   3. Simplified 92.8%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. flip-- N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(t, \color{blue}{\left(\frac{x \cdot y + z \cdot y}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) - \left(z \cdot y\right) \cdot \left(z \cdot y\right)}\right)}\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(t, \left(\frac{1}{\color{blue}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) - \left(z \cdot y\right) \cdot \left(z \cdot y\right)}{x \cdot y + z \cdot y}}}\right)\right) \]

      10. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(t, \left(\frac{1}{x \cdot y - \color{blue}{z \cdot y}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot y - z \cdot y\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \left(y \cdot \color{blue}{\left(x - z\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \left(\left(x - z\right) \cdot \color{blue}{y}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{y}\right)\right)\right) \]

      15. --lowering--.f64 93.8%

          \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), y\right)\right)\right) \]

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(t, \color{blue}{\left(\frac{-1}{y \cdot z}\right)}\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(-1, \color{blue}{\left(y \cdot z\right)}\right)\right) \]

      2. *-lowering-*.f64 74.0%

         \[\leadsto \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   9. Simplified 74.0%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. associate-/r* N/A

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

       4. div-inv N/A

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

       5. neg-mul-1 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(\frac{1}{y}\right)}{z}} \cdot t \]

       6. clear-num N/A

          \[\leadsto \frac{z}{\mathsf{neg}\left(\frac{1}{y}\right)} \cdot t \]

       7. distribute-frac-neg2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{z}{\frac{1}{y}}\right)\right) \cdot t \]

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

          \[\leadsto \mathsf{neg}\left(\frac{z}{\frac{1}{y}} \cdot t\right) \]

       9. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{z}{\frac{1}{y}} \cdot t\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{z}{\frac{1}{y}}\right), t\right)\right) \]

       11. div-inv N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(z \cdot \frac{1}{\frac{1}{y}}\right), t\right)\right) \]

       12. remove-double-div N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(z \cdot y\right), t\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(y \cdot z\right), t\right)\right) \]

       14. *-lowering-*.f64 74.1%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), t\right)\right) \]

   11. Applied egg-rr 74.1%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(t \cdot \left(y \cdot z\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(t \cdot y\right) \cdot z\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(t \cdot y\right), z\right)\right) \]

       4. *-lowering-*.f64 74.7%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, y\right), z\right)\right) \]

   13. Applied egg-rr 74.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{+49}:\\ \;\;\;\;0 - z \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+142}:\\ \;\;\;\;t\_m \cdot \left(x \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(t\_m \cdot \left(x - z\right)\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (* y_s (if (<= x -1.6e+142) (* t_m (* x y_m)) (* y_m (* t_m (- x z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (x <= -1.6e+142) {
		tmp = t_m * (x * y_m);
	} else {
		tmp = y_m * (t_m * (x - z));
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_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 <= (-1.6d+142)) then
        tmp = t_m * (x * y_m)
    else
        tmp = y_m * (t_m * (x - z))
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (x <= -1.6e+142) {
		tmp = t_m * (x * y_m);
	} else {
		tmp = y_m * (t_m * (x - z));
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if x <= -1.6e+142:
		tmp = t_m * (x * y_m)
	else:
		tmp = y_m * (t_m * (x - z))
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (x <= -1.6e+142)
		tmp = Float64(t_m * Float64(x * y_m));
	else
		tmp = Float64(y_m * Float64(t_m * Float64(x - z)));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (x <= -1.6e+142)
		tmp = t_m * (x * y_m);
	else
		tmp = y_m * (t_m * (x - z));
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[LessEqual[x, -1.6e+142], N[(t$95$m * N[(x * y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(t$95$m * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000003e142

   1. Initial program 77.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(x \cdot y\right)}, t\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot x\right), t\right) \]

      2. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, x\right), t\right) \]

   5. Simplified 75.0%

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

   if -1.60000000000000003e142 < x

   1. Initial program 90.4%

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

      1. *-commutative N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

      8. *-lowering-*.f64 92.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

   3. Simplified 92.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x - z\right) \cdot t\right), \color{blue}{y}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x - z\right), t\right), y\right) \]

      5. --lowering--.f64 94.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), t\right), y\right) \]

   6. Applied egg-rr 94.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+142}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{+31}:\\ \;\;\;\;t\_m \cdot \left(x \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (* t_s (* y_s (if (<= t_m 6e+31) (* t_m (* x y_m)) (* x (* t_m y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 6e+31) {
		tmp = t_m * (x * y_m);
	} else {
		tmp = x * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_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 <= 6d+31) then
        tmp = t_m * (x * y_m)
    else
        tmp = x * (t_m * y_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 6e+31) {
		tmp = t_m * (x * y_m);
	} else {
		tmp = x * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 6e+31:
		tmp = t_m * (x * y_m)
	else:
		tmp = x * (t_m * y_m)
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 6e+31)
		tmp = Float64(t_m * Float64(x * y_m));
	else
		tmp = Float64(x * Float64(t_m * y_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 6e+31)
		tmp = t_m * (x * y_m);
	else
		tmp = x * (t_m * y_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[LessEqual[t$95$m, 6e+31], N[(t$95$m * N[(x * y$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.99999999999999978e31

   1. Initial program 86.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(x \cdot y\right)}, t\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot x\right), t\right) \]

      2. *-lowering-*.f64 49.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, x\right), t\right) \]

   5. Simplified 49.3%

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

   if 5.99999999999999978e31 < t

   1. Initial program 96.2%

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

      1. *-commutative N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

      8. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(y, t\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 56.3%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+31}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 15000000:\\ \;\;\;\;y\_m \cdot \left(t\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (* t_s (* y_s (if (<= t_m 15000000.0) (* y_m (* t_m x)) (* x (* t_m y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 15000000.0) {
		tmp = y_m * (t_m * x);
	} else {
		tmp = x * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_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 <= 15000000.0d0) then
        tmp = y_m * (t_m * x)
    else
        tmp = x * (t_m * y_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 15000000.0) {
		tmp = y_m * (t_m * x);
	} else {
		tmp = x * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 15000000.0:
		tmp = y_m * (t_m * x)
	else:
		tmp = x * (t_m * y_m)
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 15000000.0)
		tmp = Float64(y_m * Float64(t_m * x));
	else
		tmp = Float64(x * Float64(t_m * y_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 15000000.0)
		tmp = y_m * (t_m * x);
	else
		tmp = x * (t_m * y_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[LessEqual[t$95$m, 15000000.0], N[(y$95$m * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e7

   1. Initial program 86.4%

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

      1. *-commutative N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

      8. *-lowering-*.f64 90.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

   3. Simplified 90.6%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(t \cdot x\right)}\right) \]

      4. *-lowering-*.f64 54.9%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(t, \color{blue}{x}\right)\right) \]

   7. Simplified 54.9%

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

   if 1.5e7 < t

   1. Initial program 96.4%

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

      1. *-commutative N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

      8. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(y, t\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 54.2%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 15000000:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * N[(t$95$m * N[(N[(x - z), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

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

   1. *-commutative N/A

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

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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

   8. *-lowering-*.f64 92.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

3. Simplified 92.7%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y - z \cdot y\right), \color{blue}{t}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(x - z\right)\right), t\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(x - z\right) \cdot y\right), t\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x - z\right), y\right), t\right) \]

   8. --lowering--.f64 91.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), y\right), t\right) \]

6. Applied egg-rr 91.1%

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

7. Final simplification 91.1%

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

Alternative 8: 54.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

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

   1. *-commutative N/A

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

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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x - z\right), \color{blue}{\left(t \cdot y\right)}\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(\color{blue}{t} \cdot y\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \left(y \cdot \color{blue}{t}\right)\right) \]

   8. *-lowering-*.f64 92.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, z\right), \mathsf{*.f64}\left(y, \color{blue}{t}\right)\right) \]

3. Simplified 92.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(y, t\right)\right) \]
6. Step-by-step derivation

7. Simplified 52.9%

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

8. Final simplification 52.9%

   \[\leadsto x \cdot \left(t \cdot y\right) \]
9. Add Preprocessing

Developer Target 1: 96.3% accurate, 0.5× 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 2024158 
(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))