Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.5% → 98.3%

Time: 10.2s

Alternatives: 9

Speedup: 1.3×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% 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: 98.3% accurate, 0.7× speedup

Math

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

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 1e-6) (* y_m (* (- x z) t_m)) (* (- x z) (* y_m t_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 <= 1e-6) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = (x - z) * (y_m * t_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 <= 1d-6) then
        tmp = y_m * ((x - z) * t_m)
    else
        tmp = (x - z) * (y_m * t_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 <= 1e-6) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = (x - z) * (y_m * t_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 <= 1e-6:
		tmp = y_m * ((x - z) * t_m)
	else:
		tmp = (x - z) * (y_m * t_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 <= 1e-6)
		tmp = Float64(y_m * Float64(Float64(x - z) * t_m));
	else
		tmp = Float64(Float64(x - z) * Float64(y_m * t_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 <= 1e-6)
		tmp = y_m * ((x - z) * t_m);
	else
		tmp = (x - z) * (y_m * t_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, 1e-6], N[(y$95$m * N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999999955e-7

   1. Initial program 88.0%

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

      1. distribute-rgt-out-- 88.2%

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

      2. associate-*l* 91.7%

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

      3. *-commutative 91.7%

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

   3. Simplified 91.7%

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

   if 9.99999999999999955e-7 < t

   1. Initial program 93.2%

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

      1. *-commutative 93.2%

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

      2. distribute-rgt-out-- 97.0%

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

      3. associate-*r* 98.1%

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

      4. *-commutative 98.1%

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

   3. Simplified 98.1%

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

4. Final simplification 93.1%

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

Alternative 2: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+96} \lor \neg \left(z \leq 6 \cdot 10^{+210}\right):\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(\left(x - z\right) \cdot t\_m\right)\\ \end{array}\right) \end{array} \]

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 (or (<= z -7.2e+96) (not (<= z 6e+210)))
     (* t_m (* y_m (- z)))
     (* y_m (* (- x z) t_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 ((z <= -7.2e+96) || !(z <= 6e+210)) {
		tmp = t_m * (y_m * -z);
	} else {
		tmp = y_m * ((x - z) * t_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 ((z <= (-7.2d+96)) .or. (.not. (z <= 6d+210))) then
        tmp = t_m * (y_m * -z)
    else
        tmp = y_m * ((x - z) * t_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 ((z <= -7.2e+96) || !(z <= 6e+210)) {
		tmp = t_m * (y_m * -z);
	} else {
		tmp = y_m * ((x - z) * t_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 (z <= -7.2e+96) or not (z <= 6e+210):
		tmp = t_m * (y_m * -z)
	else:
		tmp = y_m * ((x - z) * t_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 ((z <= -7.2e+96) || !(z <= 6e+210))
		tmp = Float64(t_m * Float64(y_m * Float64(-z)));
	else
		tmp = Float64(y_m * Float64(Float64(x - z) * t_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 ((z <= -7.2e+96) || ~((z <= 6e+210)))
		tmp = t_m * (y_m * -z);
	else
		tmp = y_m * ((x - z) * t_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[Or[LessEqual[z, -7.2e+96], N[Not[LessEqual[z, 6e+210]], $MachinePrecision]], N[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.20000000000000026e96 or 6.00000000000000044e210 < z

   1. Initial program 82.4%

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

      1. distribute-rgt-out-- 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in x around 0 82.7%

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

      1. mul-1-neg 82.7%

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

      2. distribute-rgt-neg-out 82.7%

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

   7. Simplified 82.7%

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

   if -7.20000000000000026e96 < z < 6.00000000000000044e210

   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* 94.1%

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

      3. *-commutative 94.1%

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

   3. Simplified 94.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+96} \lor \neg \left(z \leq 6 \cdot 10^{+210}\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-7} \lor \neg \left(x \leq 35000000\right):\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(z \cdot \left(-t\_m\right)\right)\\ \end{array}\right) \end{array} \]

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 (or (<= x -1.12e-7) (not (<= x 35000000.0)))
     (* t_m (* y_m x))
     (* y_m (* z (- t_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 ((x <= -1.12e-7) || !(x <= 35000000.0)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = y_m * (z * -t_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 ((x <= (-1.12d-7)) .or. (.not. (x <= 35000000.0d0))) then
        tmp = t_m * (y_m * x)
    else
        tmp = y_m * (z * -t_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 ((x <= -1.12e-7) || !(x <= 35000000.0)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = y_m * (z * -t_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 (x <= -1.12e-7) or not (x <= 35000000.0):
		tmp = t_m * (y_m * x)
	else:
		tmp = y_m * (z * -t_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 ((x <= -1.12e-7) || !(x <= 35000000.0))
		tmp = Float64(t_m * Float64(y_m * x));
	else
		tmp = Float64(y_m * Float64(z * Float64(-t_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 ((x <= -1.12e-7) || ~((x <= 35000000.0)))
		tmp = t_m * (y_m * x);
	else
		tmp = y_m * (z * -t_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[Or[LessEqual[x, -1.12e-7], N[Not[LessEqual[x, 35000000.0]], $MachinePrecision]], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(z * (-t$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -1.12e-7 or 3.5e7 < x

   1. Initial program 85.8%

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

      1. distribute-rgt-out-- 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in x around inf 73.5%

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

      1. *-commutative 73.5%

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

   7. Simplified 73.5%

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

   if -1.12e-7 < x < 3.5e7

   1. Initial program 92.6%

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

      1. distribute-rgt-out-- 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around 0 77.4%

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

      1. mul-1-neg 77.4%

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

      2. distribute-rgt-neg-out 77.4%

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

   7. Simplified 77.4%

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

      1. distribute-rgt-neg-out 77.4%

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

      2. distribute-lft-neg-out 77.4%

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

      3. add-sqr-sqrt 42.4%

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

      4. sqrt-unprod 39.0%

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

      5. sqr-neg 39.0%

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

      6. sqrt-unprod 7.2%

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

      7. add-sqr-sqrt 13.6%

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

      8. associate-*l* 13.6%

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

      9. add-sqr-sqrt 7.2%

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

      10. sqrt-unprod 36.6%

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

      11. sqr-neg 36.6%

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

      12. sqrt-unprod 40.9%

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

      13. add-sqr-sqrt 75.1%

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

   9. Applied egg-rr 75.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-7} \lor \neg \left(x \leq 35000000\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+40} \lor \neg \left(x \leq 12500000\right):\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y\_m \cdot \left(-t\_m\right)\right)\\ \end{array}\right) \end{array} \]

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 (or (<= x -3.2e+40) (not (<= x 12500000.0)))
     (* t_m (* y_m x))
     (* z (* y_m (- t_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 ((x <= -3.2e+40) || !(x <= 12500000.0)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = z * (y_m * -t_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 ((x <= (-3.2d+40)) .or. (.not. (x <= 12500000.0d0))) then
        tmp = t_m * (y_m * x)
    else
        tmp = z * (y_m * -t_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 ((x <= -3.2e+40) || !(x <= 12500000.0)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = z * (y_m * -t_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 (x <= -3.2e+40) or not (x <= 12500000.0):
		tmp = t_m * (y_m * x)
	else:
		tmp = z * (y_m * -t_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 ((x <= -3.2e+40) || !(x <= 12500000.0))
		tmp = Float64(t_m * Float64(y_m * x));
	else
		tmp = Float64(z * Float64(y_m * Float64(-t_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 ((x <= -3.2e+40) || ~((x <= 12500000.0)))
		tmp = t_m * (y_m * x);
	else
		tmp = z * (y_m * -t_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[Or[LessEqual[x, -3.2e+40], N[Not[LessEqual[x, 12500000.0]], $MachinePrecision]], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(y$95$m * (-t$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999981e40 or 1.25e7 < x

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in x around inf 73.6%

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

      1. *-commutative 73.6%

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

   7. Simplified 73.6%

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

   if -3.19999999999999981e40 < x < 1.25e7

   1. Initial program 93.0%

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

      1. distribute-rgt-out-- 93.0%

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

      2. associate-*l* 89.7%

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

      3. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in z around inf 86.4%

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

      1. +-commutative 86.4%

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

      2. associate-*r* 86.4%

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

      3. mul-1-neg 86.4%

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

      4. cancel-sign-sub-inv 86.4%

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

      5. *-commutative 86.4%

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

      6. *-commutative 86.4%

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

      7. associate-/l* 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in x around 0 75.8%

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

      1. associate-*r* 75.8%

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

      2. neg-mul-1 75.8%

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

      3. *-commutative 75.8%

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

   10. Simplified 75.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+40} \lor \neg \left(x \leq 12500000\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+37} \lor \neg \left(x \leq 85000000\right):\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot \left(-z\right)\right)\\ \end{array}\right) \end{array} \]

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 (or (<= x -1.75e+37) (not (<= x 85000000.0)))
     (* t_m (* y_m x))
     (* t_m (* y_m (- 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 <= -1.75e+37) || !(x <= 85000000.0)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = t_m * (y_m * -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 <= (-1.75d+37)) .or. (.not. (x <= 85000000.0d0))) then
        tmp = t_m * (y_m * x)
    else
        tmp = t_m * (y_m * -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 <= -1.75e+37) || !(x <= 85000000.0)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = t_m * (y_m * -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 <= -1.75e+37) or not (x <= 85000000.0):
		tmp = t_m * (y_m * x)
	else:
		tmp = t_m * (y_m * -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 <= -1.75e+37) || !(x <= 85000000.0))
		tmp = Float64(t_m * Float64(y_m * x));
	else
		tmp = Float64(t_m * Float64(y_m * Float64(-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 <= -1.75e+37) || ~((x <= 85000000.0)))
		tmp = t_m * (y_m * x);
	else
		tmp = t_m * (y_m * -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[Or[LessEqual[x, -1.75e+37], N[Not[LessEqual[x, 85000000.0]], $MachinePrecision]], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e37 or 8.5e7 < x

   1. Initial program 84.8%

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

      1. distribute-rgt-out-- 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in x around inf 74.2%

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

      1. *-commutative 74.2%

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

   7. Simplified 74.2%

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

   if -1.75e37 < x < 8.5e7

   1. Initial program 93.0%

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

      1. distribute-rgt-out-- 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 76.6%

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

      1. mul-1-neg 76.6%

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

      2. distribute-rgt-neg-out 76.6%

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

   7. Simplified 76.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+37} \lor \neg \left(x \leq 85000000\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-18}:\\ \;\;\;\;y\_m \cdot \left(x \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \end{array}\right) \end{array} \]

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 1e-18) (* y_m (* x t_m)) (* x (* y_m t_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 <= 1e-18) {
		tmp = y_m * (x * t_m);
	} else {
		tmp = x * (y_m * t_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 <= 1d-18) then
        tmp = y_m * (x * t_m)
    else
        tmp = x * (y_m * t_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 <= 1e-18) {
		tmp = y_m * (x * t_m);
	} else {
		tmp = x * (y_m * t_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 <= 1e-18:
		tmp = y_m * (x * t_m)
	else:
		tmp = x * (y_m * t_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 <= 1e-18)
		tmp = Float64(y_m * Float64(x * t_m));
	else
		tmp = Float64(x * Float64(y_m * t_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 <= 1e-18)
		tmp = y_m * (x * t_m);
	else
		tmp = x * (y_m * t_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, 1e-18], N[(y$95$m * N[(x * t$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.0000000000000001e-18

   1. Initial program 87.9%

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

      1. distribute-rgt-out-- 88.1%

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

      2. associate-*l* 91.6%

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

      3. *-commutative 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around inf 55.6%

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

      1. associate-*r* 57.5%

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

      2. *-commutative 57.5%

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

   7. Simplified 57.5%

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

   if 1.0000000000000001e-18 < t

   1. Initial program 93.5%

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

      1. distribute-rgt-out-- 97.1%

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

      2. associate-*l* 89.7%

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

      3. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in x around inf 74.0%

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

      1. +-commutative 74.0%

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

      2. fma-define 79.4%

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

      3. mul-1-neg 79.4%

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

      4. associate-/l* 82.9%

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

      5. distribute-rgt-neg-in 82.9%

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

      6. associate-/l* 81.1%

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

      7. distribute-rgt-neg-in 81.1%

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

      8. distribute-frac-neg 81.1%

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

   7. Simplified 81.1%

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

   8. Taylor expanded in z around 0 48.6%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-18}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-20}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \end{array}\right) \end{array} \]

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 5e-20) (* t_m (* y_m x)) (* x (* y_m t_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 <= 5e-20) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = x * (y_m * t_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 <= 5d-20) then
        tmp = t_m * (y_m * x)
    else
        tmp = x * (y_m * t_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 <= 5e-20) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = x * (y_m * t_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 <= 5e-20:
		tmp = t_m * (y_m * x)
	else:
		tmp = x * (y_m * t_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 <= 5e-20)
		tmp = Float64(t_m * Float64(y_m * x));
	else
		tmp = Float64(x * Float64(y_m * t_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 <= 5e-20)
		tmp = t_m * (y_m * x);
	else
		tmp = x * (y_m * t_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, 5e-20], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.9999999999999999e-20

   1. Initial program 87.9%

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

      1. distribute-rgt-out-- 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in x around inf 55.6%

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

      1. *-commutative 55.6%

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

   7. Simplified 55.6%

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

   if 4.9999999999999999e-20 < t

   1. Initial program 93.5%

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

      1. distribute-rgt-out-- 97.1%

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

      2. associate-*l* 89.7%

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

      3. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in x around inf 74.0%

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

      1. +-commutative 74.0%

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

      2. fma-define 79.4%

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

      3. mul-1-neg 79.4%

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

      4. associate-/l* 82.9%

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

      5. distribute-rgt-neg-in 82.9%

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

      6. associate-/l* 81.1%

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

      7. distribute-rgt-neg-in 81.1%

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

      8. distribute-frac-neg 81.1%

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

   7. Simplified 81.1%

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

   8. Taylor expanded in z around 0 48.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-20}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% accurate, 1.3× speedup

Math

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

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

Derivation

1. Initial program 89.1%

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

   1. distribute-rgt-out-- 90.0%

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

3. Simplified 90.0%

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

5. Final simplification 90.0%

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

Alternative 9: 54.7% accurate, 1.8× speedup

Math

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

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 (* y_m t_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 * (y_m * t_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 * (y_m * t_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 * (y_m * t_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 * (y_m * t_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(y_m * t_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 * (y_m * t_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[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

   1. distribute-rgt-out-- 90.0%

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

   2. associate-*l* 91.2%

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

   3. *-commutative 91.2%

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

3. Simplified 91.2%

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

5. Taylor expanded in x around inf 79.5%

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

   1. +-commutative 79.5%

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

   2. fma-define 82.2%

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

   3. mul-1-neg 82.2%

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

   4. associate-/l* 78.8%

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

   5. distribute-rgt-neg-in 78.8%

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

   6. associate-/l* 78.7%

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

   7. distribute-rgt-neg-in 78.7%

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

   8. distribute-frac-neg 78.7%

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

7. Simplified 78.7%

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

8. Taylor expanded in z around 0 55.4%

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

9. Final simplification 55.4%

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

Developer target: 96.0% 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 2024066 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :alt
  (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))