Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.4% → 96.9%

Time: 8.3s

Alternatives: 8

Speedup: 1.4×

• 24.6% 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: 96.4% 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: 96.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * N[(t$95$m * N[(y$95$m * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 92.8

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

4. Applied rewrites 92.8%

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

5. Final simplification 92.8%

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

Alternative 2: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(y\_m \cdot x\right) \cdot t\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+204}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+226}:\\ \;\;\;\;\left(t\_m \cdot \left(x - z\right)\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* y_m x) t_m)))
   (*
    y_s
    (*
     t_s
     (if (<= x -1.2e+204)
       t_2
       (if (<= x 1.55e+226) (* (* t_m (- x z)) y_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -1.2e+204) {
		tmp = t_2;
	} else if (x <= 1.55e+226) {
		tmp = (t_m * (x - z)) * y_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (y_m * x) * t_m
    if (x <= (-1.2d+204)) then
        tmp = t_2
    else if (x <= 1.55d+226) then
        tmp = (t_m * (x - z)) * y_m
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (y_m * x) * t_m;
	double tmp;
	if (x <= -1.2e+204) {
		tmp = t_2;
	} else if (x <= 1.55e+226) {
		tmp = (t_m * (x - z)) * y_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (y_m * x) * t_m
	tmp = 0
	if x <= -1.2e+204:
		tmp = t_2
	elif x <= 1.55e+226:
		tmp = (t_m * (x - z)) * y_m
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(y_m * x) * t_m)
	tmp = 0.0
	if (x <= -1.2e+204)
		tmp = t_2;
	elseif (x <= 1.55e+226)
		tmp = Float64(Float64(t_m * Float64(x - z)) * y_m);
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (y_m * x) * t_m;
	tmp = 0.0;
	if (x <= -1.2e+204)
		tmp = t_2;
	elseif (x <= 1.55e+226)
		tmp = (t_m * (x - z)) * y_m;
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[x, -1.2e+204], t$95$2, If[LessEqual[x, 1.55e+226], N[(N[(t$95$m * N[(x - z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e204 or 1.54999999999999988e226 < x

   1. Initial program 88.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if -1.2e204 < x < 1.54999999999999988e226

   1. Initial program 92.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 96.2

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

   4. Applied rewrites 96.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+204}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+226}:\\ \;\;\;\;\left(t \cdot \left(x - z\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 10^{-30}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* (- z) y_m) t_m)))
   (*
    y_s
    (*
     t_s
     (if (<= z -1.25e-26) t_2 (if (<= z 1e-30) (* (* y_m x) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (-z * y_m) * t_m;
	double tmp;
	if (z <= -1.25e-26) {
		tmp = t_2;
	} else if (z <= 1e-30) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (-z * y_m) * t_m
    if (z <= (-1.25d-26)) then
        tmp = t_2
    else if (z <= 1d-30) then
        tmp = (y_m * x) * t_m
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (-z * y_m) * t_m;
	double tmp;
	if (z <= -1.25e-26) {
		tmp = t_2;
	} else if (z <= 1e-30) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (-z * y_m) * t_m
	tmp = 0
	if z <= -1.25e-26:
		tmp = t_2
	elif z <= 1e-30:
		tmp = (y_m * x) * t_m
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(Float64(-z) * y_m) * t_m)
	tmp = 0.0
	if (z <= -1.25e-26)
		tmp = t_2;
	elseif (z <= 1e-30)
		tmp = Float64(Float64(y_m * x) * t_m);
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (-z * y_m) * t_m;
	tmp = 0.0;
	if (z <= -1.25e-26)
		tmp = t_2;
	elseif (z <= 1e-30)
		tmp = (y_m * x) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[((-z) * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[z, -1.25e-26], t$95$2, If[LessEqual[z, 1e-30], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000005e-26 or 1e-30 < z

   1. Initial program 90.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if -1.25000000000000005e-26 < z < 1e-30

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 4: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(t\_m \cdot y\_m\right) \cdot \left(-z\right)\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 10^{-30}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* t_m y_m) (- z))))
   (*
    y_s
    (* t_s (if (<= z -6.4e-27) t_2 (if (<= z 1e-30) (* (* y_m x) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (t_m * y_m) * -z;
	double tmp;
	if (z <= -6.4e-27) {
		tmp = t_2;
	} else if (z <= 1e-30) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m * y_m) * -z
    if (z <= (-6.4d-27)) then
        tmp = t_2
    else if (z <= 1d-30) then
        tmp = (y_m * x) * t_m
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (t_m * y_m) * -z;
	double tmp;
	if (z <= -6.4e-27) {
		tmp = t_2;
	} else if (z <= 1e-30) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (t_m * y_m) * -z
	tmp = 0
	if z <= -6.4e-27:
		tmp = t_2
	elif z <= 1e-30:
		tmp = (y_m * x) * t_m
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(t_m * y_m) * Float64(-z))
	tmp = 0.0
	if (z <= -6.4e-27)
		tmp = t_2;
	elseif (z <= 1e-30)
		tmp = Float64(Float64(y_m * x) * t_m);
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (t_m * y_m) * -z;
	tmp = 0.0;
	if (z <= -6.4e-27)
		tmp = t_2;
	elseif (z <= 1e-30)
		tmp = (y_m * x) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * y$95$m), $MachinePrecision] * (-z)), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[z, -6.4e-27], t$95$2, If[LessEqual[z, 1e-30], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.39999999999999982e-27 or 1e-30 < z

   1. Initial program 90.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if -6.39999999999999982e-27 < z < 1e-30

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-27}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 10^{-30}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(\left(-t\_m\right) \cdot z\right) \cdot y\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 10^{-30}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* (- t_m) z) y_m)))
   (*
    y_s
    (* t_s (if (<= z -6.4e-27) t_2 (if (<= z 1e-30) (* (* y_m x) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (-t_m * z) * y_m;
	double tmp;
	if (z <= -6.4e-27) {
		tmp = t_2;
	} else if (z <= 1e-30) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (-t_m * z) * y_m
    if (z <= (-6.4d-27)) then
        tmp = t_2
    else if (z <= 1d-30) then
        tmp = (y_m * x) * t_m
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (-t_m * z) * y_m;
	double tmp;
	if (z <= -6.4e-27) {
		tmp = t_2;
	} else if (z <= 1e-30) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (-t_m * z) * y_m
	tmp = 0
	if z <= -6.4e-27:
		tmp = t_2
	elif z <= 1e-30:
		tmp = (y_m * x) * t_m
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(Float64(-t_m) * z) * y_m)
	tmp = 0.0
	if (z <= -6.4e-27)
		tmp = t_2;
	elseif (z <= 1e-30)
		tmp = Float64(Float64(y_m * x) * t_m);
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (-t_m * z) * y_m;
	tmp = 0.0;
	if (z <= -6.4e-27)
		tmp = t_2;
	elseif (z <= 1e-30)
		tmp = (y_m * x) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[((-t$95$m) * z), $MachinePrecision] * y$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[z, -6.4e-27], t$95$2, If[LessEqual[z, 1e-30], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.39999999999999982e-27 or 1e-30 < z

   1. Initial program 90.2%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 77.4

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

   5. Applied rewrites 77.4%

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

   if -6.39999999999999982e-27 < z < 1e-30

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 6: 57.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(t\_m \cdot y\_m\right) \cdot x\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (* y_s (* t_s (if (<= t_m 1.6e+50) (* (* y_m x) t_m) (* (* t_m y_m) x)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1.6e+50) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (t_m * y_m) * x;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.6d+50) then
        tmp = (y_m * x) * t_m
    else
        tmp = (t_m * y_m) * x
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1.6e+50) {
		tmp = (y_m * x) * t_m;
	} else {
		tmp = (t_m * y_m) * x;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 1.6e+50:
		tmp = (y_m * x) * t_m
	else:
		tmp = (t_m * y_m) * x
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 1.6e+50)
		tmp = Float64(Float64(y_m * x) * t_m);
	else
		tmp = Float64(Float64(t_m * y_m) * x);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 1.6e+50)
		tmp = (y_m * x) * t_m;
	else
		tmp = (t_m * y_m) * x;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 1.6e+50], N[(N[(y$95$m * x), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(t$95$m * y$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.59999999999999991e50

   1. Initial program 91.4%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   if 1.59999999999999991e50 < t

   1. Initial program 92.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\color{blue}{\left(x \cdot y\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      11. *-commutative N/A

          \[\leadsto \left({\color{blue}{\left(y \cdot x\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\color{blue}{\left(y \cdot x\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      13. pow2 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - \color{blue}{{\left(z \cdot y\right)}^{2}}\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      14. lower-pow.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - \color{blue}{{\left(z \cdot y\right)}^{2}}\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \color{blue}{\frac{t}{x \cdot y + z \cdot y}} \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{x \cdot y} + z \cdot y} \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{x \cdot y + \color{blue}{z \cdot y}} \]

      18. distribute-rgt-out N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{y \cdot \left(x + z\right)}} \]

      19. *-commutative N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(x + z\right) \cdot y}} \]

      20. lower-*.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(x + z\right) \cdot y}} \]

      21. +-commutative N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(z + x\right)} \cdot y} \]

      22. lower-+.f64 39.6

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(z + x\right)} \cdot y} \]

   4. Applied rewrites 39.6%

      \[\leadsto \color{blue}{\left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\left(z + x\right) \cdot y}} \]

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 64.5

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

   7. Applied rewrites 64.5%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(t\_m \cdot x\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(t\_m \cdot y\_m\right) \cdot x\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (* y_s (* t_s (if (<= t_m 2e-6) (* (* t_m x) y_m) (* (* t_m y_m) x)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2e-6) {
		tmp = (t_m * x) * y_m;
	} else {
		tmp = (t_m * y_m) * x;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2d-6) then
        tmp = (t_m * x) * y_m
    else
        tmp = (t_m * y_m) * x
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2e-6) {
		tmp = (t_m * x) * y_m;
	} else {
		tmp = (t_m * y_m) * x;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 2e-6:
		tmp = (t_m * x) * y_m
	else:
		tmp = (t_m * y_m) * x
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 2e-6)
		tmp = Float64(Float64(t_m * x) * y_m);
	else
		tmp = Float64(Float64(t_m * y_m) * x);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 2e-6)
		tmp = (t_m * x) * y_m;
	else
		tmp = (t_m * y_m) * x;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 2e-6], N[(N[(t$95$m * x), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(t$95$m * y$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.99999999999999991e-6

   1. Initial program 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\color{blue}{\left(x \cdot y\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      11. *-commutative N/A

          \[\leadsto \left({\color{blue}{\left(y \cdot x\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\color{blue}{\left(y \cdot x\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      13. pow2 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - \color{blue}{{\left(z \cdot y\right)}^{2}}\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      14. lower-pow.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - \color{blue}{{\left(z \cdot y\right)}^{2}}\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \color{blue}{\frac{t}{x \cdot y + z \cdot y}} \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{x \cdot y} + z \cdot y} \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{x \cdot y + \color{blue}{z \cdot y}} \]

      18. distribute-rgt-out N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{y \cdot \left(x + z\right)}} \]

      19. *-commutative N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(x + z\right) \cdot y}} \]

      20. lower-*.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(x + z\right) \cdot y}} \]

      21. +-commutative N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(z + x\right)} \cdot y} \]

      22. lower-+.f64 49.8

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(z + x\right)} \cdot y} \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\left(z + x\right) \cdot y}} \]

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 54.5

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

   7. Applied rewrites 54.5%

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

   9. Applied rewrites 55.3%

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

   if 1.99999999999999991e-6 < t

   1. Initial program 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\color{blue}{\left(x \cdot y\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      11. *-commutative N/A

          \[\leadsto \left({\color{blue}{\left(y \cdot x\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\color{blue}{\left(y \cdot x\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      13. pow2 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - \color{blue}{{\left(z \cdot y\right)}^{2}}\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      14. lower-pow.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - \color{blue}{{\left(z \cdot y\right)}^{2}}\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \color{blue}{\frac{t}{x \cdot y + z \cdot y}} \]

      16. lift-*.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{x \cdot y} + z \cdot y} \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{x \cdot y + \color{blue}{z \cdot y}} \]

      18. distribute-rgt-out N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{y \cdot \left(x + z\right)}} \]

      19. *-commutative N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(x + z\right) \cdot y}} \]

      20. lower-*.f64 N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(x + z\right) \cdot y}} \]

      21. +-commutative N/A

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(z + x\right)} \cdot y} \]

      22. lower-+.f64 45.2

          \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(z + x\right)} \cdot y} \]

   4. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\left(z + x\right) \cdot y}} \]

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.2

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

   7. Applied rewrites 61.2%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(t \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * N[(N[(t$95$m * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. pow2 N/A

      \[\leadsto \left(\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

   9. lower-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

   10. lift-*.f64 N/A

       \[\leadsto \left({\color{blue}{\left(x \cdot y\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

   11. *-commutative N/A

       \[\leadsto \left({\color{blue}{\left(y \cdot x\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

   12. lower-*.f64 N/A

       \[\leadsto \left({\color{blue}{\left(y \cdot x\right)}}^{2} - \left(z \cdot y\right) \cdot \left(z \cdot y\right)\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

   13. pow2 N/A

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - \color{blue}{{\left(z \cdot y\right)}^{2}}\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

   14. lower-pow.f64 N/A

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - \color{blue}{{\left(z \cdot y\right)}^{2}}\right) \cdot \frac{t}{x \cdot y + z \cdot y} \]

   15. lower-/.f64 N/A

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \color{blue}{\frac{t}{x \cdot y + z \cdot y}} \]

   16. lift-*.f64 N/A

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{x \cdot y} + z \cdot y} \]

   17. lift-*.f64 N/A

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{x \cdot y + \color{blue}{z \cdot y}} \]

   18. distribute-rgt-out N/A

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{y \cdot \left(x + z\right)}} \]

   19. *-commutative N/A

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(x + z\right) \cdot y}} \]

   20. lower-*.f64 N/A

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(x + z\right) \cdot y}} \]

   21. +-commutative N/A

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(z + x\right)} \cdot y} \]

   22. lower-+.f64 48.5

       \[\leadsto \left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\color{blue}{\left(z + x\right)} \cdot y} \]

4. Applied rewrites 48.5%

   \[\leadsto \color{blue}{\left({\left(y \cdot x\right)}^{2} - {\left(z \cdot y\right)}^{2}\right) \cdot \frac{t}{\left(z + x\right) \cdot y}} \]

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 56.5

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

7. Applied rewrites 56.5%

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

9. Applied rewrites 51.8%

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

Developer Target 1: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< t -9.231879582886777e-80)
   (* (* y t) (- x z))
   (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t < (-9.231879582886777d-80)) then
        tmp = (y * t) * (x - z)
    else if (t < 2.543067051564877d+83) then
        tmp = y * (t * (x - z))
    else
        tmp = (y * (x - z)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t < -9.231879582886777e-80:
		tmp = (y * t) * (x - z)
	elif t < 2.543067051564877e+83:
		tmp = y * (t * (x - z))
	else:
		tmp = (y * (x - z)) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t < -9.231879582886777e-80)
		tmp = Float64(Float64(y * t) * Float64(x - z));
	elseif (t < 2.543067051564877e+83)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(Float64(y * Float64(x - z)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t < -9.231879582886777e-80)
		tmp = (y * t) * (x - z);
	elseif (t < 2.543067051564877e+83)
		tmp = y * (t * (x - z));
	else
		tmp = (y * (x - z)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[t, -9.231879582886777e-80], N[(N[(y * t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], If[Less[t, 2.543067051564877e+83], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024243 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -9231879582886777/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* y t) (- x z)) (if (< t 254306705156487700000000000000000000000000000000000000000000000000000000000000000000) (* y (* t (- x z))) (* (* y (- x z)) t))))

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