Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.5% → 98.3%

Time: 8.9s

Alternatives: 7

Speedup: 1.3×

• 25.8% 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, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;y_m \cdot \left(\left(x - z\right) \cdot t_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y_m \cdot t_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 1.8e-7) (* 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 <= 1.8e-7) {
		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 <= 1.8d-7) 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 <= 1.8e-7) {
		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 <= 1.8e-7:
		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 <= 1.8e-7)
		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 <= 1.8e-7)
		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, 1.8e-7], 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 < 1.79999999999999997e-7

   1. Initial program 90.4%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 94.8%

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

      3. *-commutative 94.8%

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

   3. Simplified 94.8%

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

   if 1.79999999999999997e-7 < t

   1. Initial program 93.4%

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

      1. *-commutative 93.4%

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

      2. distribute-rgt-out-- 96.6%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 2: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+23} \lor \neg \left(x \leq 2.05 \cdot 10^{-35}\right):\\ \;\;\;\;t_m \cdot \left(y_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \left(z \cdot \left(-t_m\right)\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 -1e+23) (not (<= x 2.05e-35)))
     (* 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 <= -1e+23) || !(x <= 2.05e-35)) {
		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 <= (-1d+23)) .or. (.not. (x <= 2.05d-35))) 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 <= -1e+23) || !(x <= 2.05e-35)) {
		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 <= -1e+23) or not (x <= 2.05e-35):
		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 <= -1e+23) || !(x <= 2.05e-35))
		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 <= -1e+23) || ~((x <= 2.05e-35)))
		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, -1e+23], N[Not[LessEqual[x, 2.05e-35]], $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 < -9.9999999999999992e22 or 2.05000000000000013e-35 < x

   1. Initial program 86.1%

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

      1. distribute-rgt-out-- 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in x around inf 73.7%

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

      1. *-commutative 73.7%

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

   6. Simplified 73.7%

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

   if -9.9999999999999992e22 < x < 2.05000000000000013e-35

   1. Initial program 96.0%

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

      1. distribute-rgt-out-- 95.9%

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

      2. associate-*l* 95.2%

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

      3. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in x around 0 82.4%

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

      1. mul-1-neg 82.4%

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

      2. distribute-rgt-neg-out 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+23} \lor \neg \left(x \leq 2.05 \cdot 10^{-35}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 3: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-39} \lor \neg \left(z \leq 1.5 \cdot 10^{+31}\right):\\ \;\;\;\;t_m \cdot \left(z \cdot \left(-y_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \left(y_m \cdot x\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 -3.2e-39) (not (<= z 1.5e+31)))
     (* t_m (* z (- y_m)))
     (* t_m (* y_m x))))))

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 <= -3.2e-39) || !(z <= 1.5e+31)) {
		tmp = t_m * (z * -y_m);
	} else {
		tmp = t_m * (y_m * x);
	}
	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 <= (-3.2d-39)) .or. (.not. (z <= 1.5d+31))) then
        tmp = t_m * (z * -y_m)
    else
        tmp = t_m * (y_m * x)
    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 <= -3.2e-39) || !(z <= 1.5e+31)) {
		tmp = t_m * (z * -y_m);
	} else {
		tmp = t_m * (y_m * x);
	}
	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 <= -3.2e-39) or not (z <= 1.5e+31):
		tmp = t_m * (z * -y_m)
	else:
		tmp = t_m * (y_m * x)
	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 <= -3.2e-39) || !(z <= 1.5e+31))
		tmp = Float64(t_m * Float64(z * Float64(-y_m)));
	else
		tmp = Float64(t_m * Float64(y_m * x));
	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 <= -3.2e-39) || ~((z <= 1.5e+31)))
		tmp = t_m * (z * -y_m);
	else
		tmp = t_m * (y_m * x);
	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, -3.2e-39], N[Not[LessEqual[z, 1.5e+31]], $MachinePrecision]], N[(t$95$m * N[(z * (-y$95$m)), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.1999999999999998e-39 or 1.49999999999999995e31 < z

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in x around 0 79.2%

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

      1. mul-1-neg 79.2%

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

      2. *-commutative 79.2%

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

      3. distribute-rgt-neg-in 79.2%

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

   6. Simplified 79.2%

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

   if -3.1999999999999998e-39 < z < 1.49999999999999995e31

   1. Initial program 92.3%

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

      1. distribute-rgt-out-- 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around inf 78.5%

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

      1. *-commutative 78.5%

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

   6. Simplified 78.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-39} \lor \neg \left(z \leq 1.5 \cdot 10^{+31}\right):\\ \;\;\;\;t \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 9.1 \cdot 10^{+98}:\\ \;\;\;\;y_m \cdot \left(\left(x - z\right) \cdot t_m\right)\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \left(y_m \cdot x\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 9.1e+98) (* y_m (* (- x z) t_m)) (* t_m (* y_m x))))))

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 <= 9.1e+98) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = t_m * (y_m * x);
	}
	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 <= 9.1d+98) then
        tmp = y_m * ((x - z) * t_m)
    else
        tmp = t_m * (y_m * x)
    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 <= 9.1e+98) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = t_m * (y_m * x);
	}
	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 <= 9.1e+98:
		tmp = y_m * ((x - z) * t_m)
	else:
		tmp = t_m * (y_m * x)
	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 <= 9.1e+98)
		tmp = Float64(y_m * Float64(Float64(x - z) * t_m));
	else
		tmp = Float64(t_m * Float64(y_m * x));
	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 <= 9.1e+98)
		tmp = y_m * ((x - z) * t_m);
	else
		tmp = t_m * (y_m * x);
	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, 9.1e+98], N[(y$95$m * N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.0999999999999999e98

   1. Initial program 92.3%

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

      1. distribute-rgt-out-- 93.8%

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

      2. associate-*l* 93.4%

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

      3. *-commutative 93.4%

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

   3. Simplified 93.4%

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

   if 9.0999999999999999e98 < x

   1. Initial program 85.9%

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

      1. distribute-rgt-out-- 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around inf 82.7%

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

      1. *-commutative 82.7%

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

   6. Simplified 82.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.1 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 5: 58.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3 \cdot 10^{-13}:\\ \;\;\;\;y_m \cdot \left(x \cdot t_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y_m \cdot t_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 3e-13) (* 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 <= 3e-13) {
		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 <= 3d-13) 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 <= 3e-13) {
		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 <= 3e-13:
		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 <= 3e-13)
		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 <= 3e-13)
		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, 3e-13], 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 < 2.99999999999999984e-13

   1. Initial program 90.4%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 94.8%

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

      3. *-commutative 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in x around inf 54.7%

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

   if 2.99999999999999984e-13 < t

   1. Initial program 93.4%

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

      1. distribute-rgt-out-- 96.6%

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

      2. associate-*l* 84.8%

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

      3. *-commutative 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in x around inf 45.0%

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

      1. add-cbrt-cube 43.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(y \cdot \left(t \cdot x\right)\right) \cdot \left(y \cdot \left(t \cdot x\right)\right)\right) \cdot \left(y \cdot \left(t \cdot x\right)\right)}} \]

      2. pow3 42.9%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(y \cdot \left(t \cdot x\right)\right)}^{3}}} \]

      3. *-commutative 42.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(t \cdot x\right) \cdot y\right)}}^{3}} \]

      4. *-commutative 42.9%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(x \cdot t\right)} \cdot y\right)}^{3}} \]

      5. associate-*l* 46.1%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(x \cdot \left(t \cdot y\right)\right)}}^{3}} \]

      6. *-commutative 46.1%

         \[\leadsto \sqrt[3]{{\left(x \cdot \color{blue}{\left(y \cdot t\right)}\right)}^{3}} \]

   6. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x \cdot \left(y \cdot t\right)\right)}^{3}}} \]
   7. Step-by-step derivation

      1. rem-cbrt-cube 53.2%

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

      2. *-commutative 53.2%

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

      3. *-commutative 53.2%

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

   8. Applied egg-rr 53.2%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 6: 97.1% 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 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 91.1%

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

   1. distribute-rgt-out-- 92.7%

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

3. Simplified 92.7%

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

4. Final simplification 92.7%

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

Alternative 7: 51.2% accurate, 1.8× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 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 * 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 * 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 * 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 * 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(y_m * Float64(x * 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 * 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[(y$95$m * N[(x * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. distribute-rgt-out-- 92.7%

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

   2. associate-*l* 92.4%

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

   3. *-commutative 92.4%

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

3. Simplified 92.4%

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

4. Taylor expanded in x around inf 52.3%

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

5. Final simplification 52.3%

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

Developer target: 96.0% accurate, 0.8× 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 2023322 
(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))