Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.4% → 98.5%

Time: 10.6s

Alternatives: 8

Speedup: 0.8×

• 25.1% 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: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-20}:\\ \;\;\;\;t\_m \cdot \left(x \cdot y\_m - y\_m \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

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 4e-20)
     (* t_m (- (* x y_m) (* y_m z)))
     (* (- x z) (* t_m 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) {
	double tmp;
	if (t_m <= 4e-20) {
		tmp = t_m * ((x * y_m) - (y_m * z));
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	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 <= 4d-20) then
        tmp = t_m * ((x * y_m) - (y_m * z))
    else
        tmp = (x - z) * (t_m * y_m)
    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 <= 4e-20) {
		tmp = t_m * ((x * y_m) - (y_m * z));
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	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 <= 4e-20:
		tmp = t_m * ((x * y_m) - (y_m * z))
	else:
		tmp = (x - z) * (t_m * y_m)
	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 <= 4e-20)
		tmp = Float64(t_m * Float64(Float64(x * y_m) - Float64(y_m * z)));
	else
		tmp = Float64(Float64(x - z) * Float64(t_m * y_m));
	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 <= 4e-20)
		tmp = t_m * ((x * y_m) - (y_m * z));
	else
		tmp = (x - z) * (t_m * y_m);
	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, 4e-20], N[(t$95$m * N[(N[(x * y$95$m), $MachinePrecision] - N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.99999999999999978e-20

   1. Initial program 83.9%

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

   if 3.99999999999999978e-20 < t

   1. Initial program 92.9%

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

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-20}:\\ \;\;\;\;t \cdot \left(x \cdot y - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := t\_m \cdot \left(y\_m \cdot \left(-z\right)\right)\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \left(t\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

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 -1.55e+30) t_2 (if (<= z 2.7e-43) (* x (* t_m 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 = t_m * (y_m * -z);
	double tmp;
	if (z <= -1.55e+30) {
		tmp = t_2;
	} else if (z <= 2.7e-43) {
		tmp = x * (t_m * 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 = t_m * (y_m * -z)
    if (z <= (-1.55d+30)) then
        tmp = t_2
    else if (z <= 2.7d-43) then
        tmp = x * (t_m * 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 = t_m * (y_m * -z);
	double tmp;
	if (z <= -1.55e+30) {
		tmp = t_2;
	} else if (z <= 2.7e-43) {
		tmp = x * (t_m * 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 = t_m * (y_m * -z)
	tmp = 0
	if z <= -1.55e+30:
		tmp = t_2
	elif z <= 2.7e-43:
		tmp = x * (t_m * 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(t_m * Float64(y_m * Float64(-z)))
	tmp = 0.0
	if (z <= -1.55e+30)
		tmp = t_2;
	elseif (z <= 2.7e-43)
		tmp = Float64(x * Float64(t_m * 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 = t_m * (y_m * -z);
	tmp = 0.0;
	if (z <= -1.55e+30)
		tmp = t_2;
	elseif (z <= 2.7e-43)
		tmp = x * (t_m * 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[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[z, -1.55e+30], t$95$2, If[LessEqual[z, 2.7e-43], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5499999999999999e30 or 2.69999999999999991e-43 < z

   1. Initial program 83.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 73.8

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

   5. Simplified 73.8%

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

   if -1.5499999999999999e30 < z < 2.69999999999999991e-43

   1. Initial program 90.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 82.5

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

   5. Simplified 82.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 81.5

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

   7. Applied egg-rr 81.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+30}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(t\_m \cdot y\_m\right) \cdot \left(-z\right)\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \left(t\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

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 -2.4e+30) t_2 (if (<= z 2.3e-43) (* x (* t_m 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 = (t_m * y_m) * -z;
	double tmp;
	if (z <= -2.4e+30) {
		tmp = t_2;
	} else if (z <= 2.3e-43) {
		tmp = x * (t_m * 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 = (t_m * y_m) * -z
    if (z <= (-2.4d+30)) then
        tmp = t_2
    else if (z <= 2.3d-43) then
        tmp = x * (t_m * 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 = (t_m * y_m) * -z;
	double tmp;
	if (z <= -2.4e+30) {
		tmp = t_2;
	} else if (z <= 2.3e-43) {
		tmp = x * (t_m * 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 = (t_m * y_m) * -z
	tmp = 0
	if z <= -2.4e+30:
		tmp = t_2
	elif z <= 2.3e-43:
		tmp = x * (t_m * 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(t_m * y_m) * Float64(-z))
	tmp = 0.0
	if (z <= -2.4e+30)
		tmp = t_2;
	elseif (z <= 2.3e-43)
		tmp = Float64(x * Float64(t_m * 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 = (t_m * y_m) * -z;
	tmp = 0.0;
	if (z <= -2.4e+30)
		tmp = t_2;
	elseif (z <= 2.3e-43)
		tmp = x * (t_m * 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[(t$95$m * y$95$m), $MachinePrecision] * (-z)), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[z, -2.4e+30], t$95$2, If[LessEqual[z, 2.3e-43], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3999999999999999e30 or 2.2999999999999999e-43 < z

   1. Initial program 83.8%

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

   3. Taylor expanded in x around 0

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

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

      3. associate-*l* N/A

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

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

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

      5. lower-*.f64 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 78.6

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

   5. Simplified 78.6%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if -2.3999999999999999e30 < z < 2.2999999999999999e-43

   1. Initial program 90.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 82.5

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

   5. Simplified 82.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 81.5

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

   7. Applied egg-rr 81.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+30}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := y\_m \cdot \left(t\_m \cdot \left(-z\right)\right)\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \left(t\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

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 (* t_m (- z)))))
   (*
    y_s
    (*
     t_s
     (if (<= z -1.55e+30) t_2 (if (<= z 2.3e-43) (* x (* t_m 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 * (t_m * -z);
	double tmp;
	if (z <= -1.55e+30) {
		tmp = t_2;
	} else if (z <= 2.3e-43) {
		tmp = x * (t_m * 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 * (t_m * -z)
    if (z <= (-1.55d+30)) then
        tmp = t_2
    else if (z <= 2.3d-43) then
        tmp = x * (t_m * 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 * (t_m * -z);
	double tmp;
	if (z <= -1.55e+30) {
		tmp = t_2;
	} else if (z <= 2.3e-43) {
		tmp = x * (t_m * 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 * (t_m * -z)
	tmp = 0
	if z <= -1.55e+30:
		tmp = t_2
	elif z <= 2.3e-43:
		tmp = x * (t_m * 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(y_m * Float64(t_m * Float64(-z)))
	tmp = 0.0
	if (z <= -1.55e+30)
		tmp = t_2;
	elseif (z <= 2.3e-43)
		tmp = Float64(x * Float64(t_m * 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 * (t_m * -z);
	tmp = 0.0;
	if (z <= -1.55e+30)
		tmp = t_2;
	elseif (z <= 2.3e-43)
		tmp = x * (t_m * 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[(y$95$m * N[(t$95$m * (-z)), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[z, -1.55e+30], t$95$2, If[LessEqual[z, 2.3e-43], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5499999999999999e30 or 2.2999999999999999e-43 < z

   1. Initial program 83.8%

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

   3. Taylor expanded in x around 0

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

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

      3. associate-*l* N/A

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

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

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

      5. lower-*.f64 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 78.6

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

   5. Simplified 78.6%

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

   if -1.5499999999999999e30 < z < 2.2999999999999999e-43

   1. Initial program 90.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 82.5

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

   5. Simplified 82.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 81.5

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

   7. Applied egg-rr 81.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;y\_m \cdot \left(t\_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

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 1.2e-104) (* y_m (* t_m (- z))) (* (- x z) (* t_m 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) {
	double tmp;
	if (t_m <= 1.2e-104) {
		tmp = y_m * (t_m * -z);
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	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 <= 1.2d-104) then
        tmp = y_m * (t_m * -z)
    else
        tmp = (x - z) * (t_m * y_m)
    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 <= 1.2e-104) {
		tmp = y_m * (t_m * -z);
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	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 <= 1.2e-104:
		tmp = y_m * (t_m * -z)
	else:
		tmp = (x - z) * (t_m * y_m)
	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 <= 1.2e-104)
		tmp = Float64(y_m * Float64(t_m * Float64(-z)));
	else
		tmp = Float64(Float64(x - z) * Float64(t_m * y_m));
	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 <= 1.2e-104)
		tmp = y_m * (t_m * -z);
	else
		tmp = (x - z) * (t_m * y_m);
	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, 1.2e-104], N[(y$95$m * N[(t$95$m * (-z)), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2e-104

   1. Initial program 83.2%

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

   3. Taylor expanded in x around 0

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

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

      3. associate-*l* N/A

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

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

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

      5. lower-*.f64 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 59.9

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

   5. Simplified 59.9%

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

   if 1.2e-104 < t

   1. Initial program 92.2%

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

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.8

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

   4. Applied egg-rr 98.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 150000000:\\ \;\;\;\;t\_m \cdot \left(x \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

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 150000000.0) (* t_m (* x y_m)) (* x (* t_m 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) {
	double tmp;
	if (t_m <= 150000000.0) {
		tmp = t_m * (x * y_m);
	} else {
		tmp = x * (t_m * y_m);
	}
	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 <= 150000000.0d0) then
        tmp = t_m * (x * y_m)
    else
        tmp = x * (t_m * y_m)
    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 <= 150000000.0) {
		tmp = t_m * (x * y_m);
	} else {
		tmp = x * (t_m * y_m);
	}
	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 <= 150000000.0:
		tmp = t_m * (x * y_m)
	else:
		tmp = x * (t_m * y_m)
	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 <= 150000000.0)
		tmp = Float64(t_m * Float64(x * y_m));
	else
		tmp = Float64(x * Float64(t_m * y_m));
	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 <= 150000000.0)
		tmp = t_m * (x * y_m);
	else
		tmp = x * (t_m * y_m);
	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, 150000000.0], N[(t$95$m * N[(x * y$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e8

   1. Initial program 83.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.7

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

   5. Simplified 43.7%

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

   if 1.5e8 < t

   1. Initial program 93.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 54.8

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

   5. Simplified 54.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 58.7

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

   7. Applied egg-rr 58.7%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 150000000:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-21}:\\ \;\;\;\;y\_m \cdot \left(t\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

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-21) (* y_m (* t_m x)) (* x (* t_m 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) {
	double tmp;
	if (t_m <= 2e-21) {
		tmp = y_m * (t_m * x);
	} else {
		tmp = x * (t_m * y_m);
	}
	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-21) then
        tmp = y_m * (t_m * x)
    else
        tmp = x * (t_m * y_m)
    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-21) {
		tmp = y_m * (t_m * x);
	} else {
		tmp = x * (t_m * y_m);
	}
	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-21:
		tmp = y_m * (t_m * x)
	else:
		tmp = x * (t_m * y_m)
	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-21)
		tmp = Float64(y_m * Float64(t_m * x));
	else
		tmp = Float64(x * Float64(t_m * y_m));
	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-21)
		tmp = y_m * (t_m * x);
	else
		tmp = x * (t_m * y_m);
	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-21], N[(y$95$m * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.99999999999999982e-21

   1. Initial program 83.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 45.6

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

   5. Simplified 45.6%

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

   if 1.99999999999999982e-21 < t

   1. Initial program 93.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 52.2

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

   5. Simplified 52.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 55.8

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

   7. Applied egg-rr 55.8%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-21}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.3% accurate, 1.7× speedup

Math

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

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 (* y_m (* t_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) {
	return y_s * (t_s * (y_m * (t_m * x)));
}

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 * (y_m * (t_m * x)))
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 * (y_m * (t_m * x)));
}

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 * (y_m * (t_m * x)))

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(y_m * Float64(t_m * x))))
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 * (y_m * (t_m * x)));
end

Wolfram

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

Derivation

1. Initial program 86.6%

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

3. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 47.6

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

5. Simplified 47.6%

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

6. Final simplification 47.6%

   \[\leadsto y \cdot \left(t \cdot x\right) \]
7. 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 2024212 
(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))