Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 94.0% → 97.3%

Time: 11.2s

Alternatives: 5

Speedup: 1.3×

• 25.2% 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: 94.0% 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: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y, z, t_m] = \mathsf{sort}([x, y, z, t_m])\\ \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\_m\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s x y z t_m)
 :precision binary64
 (* t_s (if (<= t_m 1.8e-42) (* y (* (- x z) t_m)) (* (- x z) (* y t_m)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y && y < z && z < t_m);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 1.8e-42) {
		tmp = y * ((x - z) * t_m);
	} else {
		tmp = (x - z) * (y * t_m);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.8d-42) then
        tmp = y * ((x - z) * t_m)
    else
        tmp = (x - z) * (y * t_m)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y && y < z && z < t_m;
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 1.8e-42) {
		tmp = y * ((x - z) * t_m);
	} else {
		tmp = (x - z) * (y * t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y, z, t_m] = sort([x, y, z, t_m])
def code(t_s, x, y, z, t_m):
	tmp = 0
	if t_m <= 1.8e-42:
		tmp = y * ((x - z) * t_m)
	else:
		tmp = (x - z) * (y * t_m)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
x, y, z, t_m = sort([x, y, z, t_m])
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 1.8e-42)
		tmp = Float64(y * Float64(Float64(x - z) * t_m));
	else
		tmp = Float64(Float64(x - z) * Float64(y * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y, z, t_m = num2cell(sort([x, y, z, t_m])){:}
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 1.8e-42)
		tmp = y * ((x - z) * t_m);
	else
		tmp = (x - z) * (y * t_m);
	end
	tmp_2 = 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]
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-42], N[(y * N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(y * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8000000000000001e-42

   1. Initial program 86.0%

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

      1. distribute-rgt-out-- 88.3%

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

      2. associate-*l* 96.1%

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

      3. *-commutative 96.1%

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

   3. Simplified 96.1%

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

   if 1.8000000000000001e-42 < t

   1. Initial program 94.7%

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

      1. *-commutative 94.7%

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

      2. distribute-rgt-out-- 94.7%

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

      3. associate-*r* 98.5%

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

      4. *-commutative 98.5%

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

   3. Simplified 98.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y, z, t_m] = \mathsf{sort}([x, y, z, t_m])\\ \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+184}:\\ \;\;\;\;t\_m \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-129} \lor \neg \left(x \leq 5.8 \cdot 10^{-164}\right):\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= x -2.35e+184)
    (* t_m (* y x))
    (if (or (<= x -2.4e-129) (not (<= x 5.8e-164)))
      (* y (* (- x z) t_m))
      (* t_m (* y (- z)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y && y < z && z < t_m);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (x <= -2.35e+184) {
		tmp = t_m * (y * x);
	} else if ((x <= -2.4e-129) || !(x <= 5.8e-164)) {
		tmp = y * ((x - z) * t_m);
	} else {
		tmp = t_m * (y * -z);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (x <= (-2.35d+184)) then
        tmp = t_m * (y * x)
    else if ((x <= (-2.4d-129)) .or. (.not. (x <= 5.8d-164))) then
        tmp = y * ((x - z) * t_m)
    else
        tmp = t_m * (y * -z)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y && y < z && z < t_m;
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (x <= -2.35e+184) {
		tmp = t_m * (y * x);
	} else if ((x <= -2.4e-129) || !(x <= 5.8e-164)) {
		tmp = y * ((x - z) * t_m);
	} else {
		tmp = t_m * (y * -z);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y, z, t_m] = sort([x, y, z, t_m])
def code(t_s, x, y, z, t_m):
	tmp = 0
	if x <= -2.35e+184:
		tmp = t_m * (y * x)
	elif (x <= -2.4e-129) or not (x <= 5.8e-164):
		tmp = y * ((x - z) * t_m)
	else:
		tmp = t_m * (y * -z)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
x, y, z, t_m = sort([x, y, z, t_m])
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (x <= -2.35e+184)
		tmp = Float64(t_m * Float64(y * x));
	elseif ((x <= -2.4e-129) || !(x <= 5.8e-164))
		tmp = Float64(y * Float64(Float64(x - z) * t_m));
	else
		tmp = Float64(t_m * Float64(y * Float64(-z)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y, z, t_m = num2cell(sort([x, y, z, t_m])){:}
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (x <= -2.35e+184)
		tmp = t_m * (y * x);
	elseif ((x <= -2.4e-129) || ~((x <= 5.8e-164)))
		tmp = y * ((x - z) * t_m);
	else
		tmp = t_m * (y * -z);
	end
	tmp_2 = 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]
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[x, -2.35e+184], N[(t$95$m * N[(y * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -2.4e-129], N[Not[LessEqual[x, 5.8e-164]], $MachinePrecision]], N[(y * N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y * (-z)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < -2.3500000000000002e184

   1. Initial program 56.1%

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

      1. distribute-rgt-out-- 61.2%

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

      2. associate-*l* 95.3%

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

      3. *-commutative 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in x around inf 56.7%

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

      1. *-commutative 56.7%

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

   7. Simplified 56.7%

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

   if -2.3500000000000002e184 < x < -2.39999999999999989e-129 or 5.8e-164 < x

   1. Initial program 91.6%

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

      1. distribute-rgt-out-- 93.5%

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

      2. associate-*l* 95.1%

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

      3. *-commutative 95.1%

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

   3. Simplified 95.1%

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

   if -2.39999999999999989e-129 < x < 5.8e-164

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 91.5%

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

      2. associate-*l* 92.9%

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

      3. *-commutative 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around 0 89.1%

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

      1. mul-1-neg 89.1%

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

      2. distribute-rgt-neg-in 89.1%

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

      3. distribute-rgt-neg-in 89.1%

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

   7. Simplified 89.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+184}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-129} \lor \neg \left(x \leq 5.8 \cdot 10^{-164}\right):\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y, z, t_m] = \mathsf{sort}([x, y, z, t_m])\\ \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+56} \lor \neg \left(z \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;t\_m \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (or (<= z -3.8e+56) (not (<= z 5.4e-9)))
    (* t_m (* y (- z)))
    (* t_m (* y x)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y && y < z && z < t_m);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if ((z <= -3.8e+56) || !(z <= 5.4e-9)) {
		tmp = t_m * (y * -z);
	} else {
		tmp = t_m * (y * x);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((z <= (-3.8d+56)) .or. (.not. (z <= 5.4d-9))) then
        tmp = t_m * (y * -z)
    else
        tmp = t_m * (y * x)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y && y < z && z < t_m;
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if ((z <= -3.8e+56) || !(z <= 5.4e-9)) {
		tmp = t_m * (y * -z);
	} else {
		tmp = t_m * (y * x);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y, z, t_m] = sort([x, y, z, t_m])
def code(t_s, x, y, z, t_m):
	tmp = 0
	if (z <= -3.8e+56) or not (z <= 5.4e-9):
		tmp = t_m * (y * -z)
	else:
		tmp = t_m * (y * x)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
x, y, z, t_m = sort([x, y, z, t_m])
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if ((z <= -3.8e+56) || !(z <= 5.4e-9))
		tmp = Float64(t_m * Float64(y * Float64(-z)));
	else
		tmp = Float64(t_m * Float64(y * x));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y, z, t_m = num2cell(sort([x, y, z, t_m])){:}
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if ((z <= -3.8e+56) || ~((z <= 5.4e-9)))
		tmp = t_m * (y * -z);
	else
		tmp = t_m * (y * x);
	end
	tmp_2 = 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]
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[Or[LessEqual[z, -3.8e+56], N[Not[LessEqual[z, 5.4e-9]], $MachinePrecision]], N[(t$95$m * N[(y * (-z)), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.79999999999999996e56 or 5.4000000000000004e-9 < z

   1. Initial program 85.8%

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

      1. distribute-rgt-out-- 89.4%

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

      2. associate-*l* 92.6%

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

      3. *-commutative 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around 0 73.0%

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

      1. mul-1-neg 73.0%

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

      2. distribute-rgt-neg-in 73.0%

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

      3. distribute-rgt-neg-in 73.0%

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

   7. Simplified 73.0%

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

   if -3.79999999999999996e56 < z < 5.4000000000000004e-9

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 90.9%

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

      2. associate-*l* 96.2%

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

      3. *-commutative 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in x around inf 73.4%

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

      1. *-commutative 73.4%

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

   7. Simplified 73.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+56} \lor \neg \left(z \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y, z, t_m = num2cell(sort([x, y, z, t_m])){:}
function tmp = code(t_s, x, y, z, t_m)
	tmp = t_s * ((y * (x - 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]
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.5%

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

   1. distribute-rgt-out-- 90.2%

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

3. Simplified 90.2%

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

5. Final simplification 90.2%

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

Alternative 5: 54.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y, z, t_m = num2cell(sort([x, y, z, t_m])){:}
function tmp = code(t_s, x, y, z, t_m)
	tmp = t_s * (t_m * (y * 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]
NOTE: x, y, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * N[(t$95$m * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.5%

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

   1. distribute-rgt-out-- 90.2%

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

   2. associate-*l* 94.5%

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

   3. *-commutative 94.5%

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

3. Simplified 94.5%

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

5. Taylor expanded in x around inf 52.0%

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

   1. *-commutative 52.0%

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

7. Simplified 52.0%

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

8. Final simplification 52.0%

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

Developer target: 96.2% 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 2024034 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

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