Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 93.5% → 97.7%

Time: 23.2s

Alternatives: 9

Speedup: 1.3×

• 25.7% 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: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t] = \mathsf{sort}([x, y_m, z, t])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 10^{+16}:\\ \;\;\;\;y\_m \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y\_m \cdot t\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z t)
 :precision binary64
 (* y_s (if (<= t 1e+16) (* y_m (* (- x z) t)) (* (- x z) (* y_m t)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t);
double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if (t <= 1e+16) {
		tmp = y_m * ((x - z) * t);
	} else {
		tmp = (x - z) * (y_m * t);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z, t)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1d+16) then
        tmp = y_m * ((x - z) * t)
    else
        tmp = (x - z) * (y_m * t)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t;
public static double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if (t <= 1e+16) {
		tmp = y_m * ((x - z) * t);
	} else {
		tmp = (x - z) * (y_m * t);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t] = sort([x, y_m, z, t])
def code(y_s, x, y_m, z, t):
	tmp = 0
	if t <= 1e+16:
		tmp = y_m * ((x - z) * t)
	else:
		tmp = (x - z) * (y_m * t)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t = sort([x, y_m, z, t])
function code(y_s, x, y_m, z, t)
	tmp = 0.0
	if (t <= 1e+16)
		tmp = Float64(y_m * Float64(Float64(x - z) * t));
	else
		tmp = Float64(Float64(x - z) * Float64(y_m * t));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t = num2cell(sort([x, y_m, z, t])){:}
function tmp_2 = code(y_s, x, y_m, z, t)
	tmp = 0.0;
	if (t <= 1e+16)
		tmp = y_m * ((x - z) * t);
	else
		tmp = (x - z) * (y_m * t);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_, t_] := N[(y$95$s * If[LessEqual[t, 1e+16], N[(y$95$m * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(y$95$m * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1e16

   1. Initial program 94.0%

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

      1. distribute-rgt-out-- 94.5%

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

      2. associate-*l* 93.5%

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

      3. *-commutative 93.5%

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

   3. Simplified 93.5%

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

   if 1e16 < t

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt-out-- 99.8%

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

      3. associate-*r* 98.1%

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

      4. *-commutative 98.1%

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

   3. Simplified 98.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{+16}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t] = \mathsf{sort}([x, y_m, z, t])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+181} \lor \neg \left(z \leq 1.5 \cdot 10^{+171}\right):\\ \;\;\;\;t \cdot \left(y\_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z t)
 :precision binary64
 (*
  y_s
  (if (or (<= z -3.4e+181) (not (<= z 1.5e+171)))
    (* t (* y_m (- z)))
    (* y_m (* (- x z) t)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t);
double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if ((z <= -3.4e+181) || !(z <= 1.5e+171)) {
		tmp = t * (y_m * -z);
	} else {
		tmp = y_m * ((x - z) * t);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z, t)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-3.4d+181)) .or. (.not. (z <= 1.5d+171))) then
        tmp = t * (y_m * -z)
    else
        tmp = y_m * ((x - z) * t)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t;
public static double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if ((z <= -3.4e+181) || !(z <= 1.5e+171)) {
		tmp = t * (y_m * -z);
	} else {
		tmp = y_m * ((x - z) * t);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t] = sort([x, y_m, z, t])
def code(y_s, x, y_m, z, t):
	tmp = 0
	if (z <= -3.4e+181) or not (z <= 1.5e+171):
		tmp = t * (y_m * -z)
	else:
		tmp = y_m * ((x - z) * t)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t = sort([x, y_m, z, t])
function code(y_s, x, y_m, z, t)
	tmp = 0.0
	if ((z <= -3.4e+181) || !(z <= 1.5e+171))
		tmp = Float64(t * Float64(y_m * Float64(-z)));
	else
		tmp = Float64(y_m * Float64(Float64(x - z) * t));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t = num2cell(sort([x, y_m, z, t])){:}
function tmp_2 = code(y_s, x, y_m, z, t)
	tmp = 0.0;
	if ((z <= -3.4e+181) || ~((z <= 1.5e+171)))
		tmp = t * (y_m * -z);
	else
		tmp = y_m * ((x - z) * t);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_, t_] := N[(y$95$s * If[Or[LessEqual[z, -3.4e+181], N[Not[LessEqual[z, 1.5e+171]], $MachinePrecision]], N[(t * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.40000000000000031e181 or 1.5e171 < z

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around 0 88.3%

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

      1. neg-mul-1 88.3%

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

   7. Simplified 88.3%

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

   if -3.40000000000000031e181 < z < 1.5e171

   1. Initial program 96.4%

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

      1. distribute-rgt-out-- 96.9%

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

      2. associate-*l* 93.6%

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

      3. *-commutative 93.6%

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

   3. Simplified 93.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+181} \lor \neg \left(z \leq 1.5 \cdot 10^{+171}\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t] = \mathsf{sort}([x, y_m, z, t])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-74} \lor \neg \left(z \leq 5.8 \cdot 10^{+29}\right):\\ \;\;\;\;t \cdot \left(y\_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z t)
 :precision binary64
 (*
  y_s
  (if (or (<= z -1.4e-74) (not (<= z 5.8e+29)))
    (* t (* y_m (- z)))
    (* x (* y_m t)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t);
double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if ((z <= -1.4e-74) || !(z <= 5.8e+29)) {
		tmp = t * (y_m * -z);
	} else {
		tmp = x * (y_m * t);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z, t)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.4d-74)) .or. (.not. (z <= 5.8d+29))) then
        tmp = t * (y_m * -z)
    else
        tmp = x * (y_m * t)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t;
public static double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if ((z <= -1.4e-74) || !(z <= 5.8e+29)) {
		tmp = t * (y_m * -z);
	} else {
		tmp = x * (y_m * t);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t] = sort([x, y_m, z, t])
def code(y_s, x, y_m, z, t):
	tmp = 0
	if (z <= -1.4e-74) or not (z <= 5.8e+29):
		tmp = t * (y_m * -z)
	else:
		tmp = x * (y_m * t)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t = sort([x, y_m, z, t])
function code(y_s, x, y_m, z, t)
	tmp = 0.0
	if ((z <= -1.4e-74) || !(z <= 5.8e+29))
		tmp = Float64(t * Float64(y_m * Float64(-z)));
	else
		tmp = Float64(x * Float64(y_m * t));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t = num2cell(sort([x, y_m, z, t])){:}
function tmp_2 = code(y_s, x, y_m, z, t)
	tmp = 0.0;
	if ((z <= -1.4e-74) || ~((z <= 5.8e+29)))
		tmp = t * (y_m * -z);
	else
		tmp = x * (y_m * t);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_, t_] := N[(y$95$s * If[Or[LessEqual[z, -1.4e-74], N[Not[LessEqual[z, 5.8e+29]], $MachinePrecision]], N[(t * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999994e-74 or 5.7999999999999999e29 < z

   1. Initial program 93.8%

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

      1. distribute-rgt-out-- 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in x around 0 78.7%

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

      1. neg-mul-1 78.7%

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

   7. Simplified 78.7%

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

   if -1.39999999999999994e-74 < z < 5.7999999999999999e29

   1. Initial program 97.2%

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

      1. *-commutative 97.2%

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

      2. distribute-rgt-out-- 97.2%

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

      3. associate-*r* 95.3%

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

      4. *-commutative 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in x around inf 87.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-74} \lor \neg \left(z \leq 5.8 \cdot 10^{+29}\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t] = \mathsf{sort}([x, y_m, z, t])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+37} \lor \neg \left(x \leq 6.7 \cdot 10^{-16}\right):\\ \;\;\;\;t \cdot \left(y\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\right) \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z t)
 :precision binary64
 (*
  y_s
  (if (or (<= x -2.6e+37) (not (<= x 6.7e-16)))
    (* t (* y_m x))
    (* (* y_m t) (- z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t);
double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if ((x <= -2.6e+37) || !(x <= 6.7e-16)) {
		tmp = t * (y_m * x);
	} else {
		tmp = (y_m * t) * -z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z, t)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.6d+37)) .or. (.not. (x <= 6.7d-16))) then
        tmp = t * (y_m * x)
    else
        tmp = (y_m * t) * -z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t;
public static double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if ((x <= -2.6e+37) || !(x <= 6.7e-16)) {
		tmp = t * (y_m * x);
	} else {
		tmp = (y_m * t) * -z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t] = sort([x, y_m, z, t])
def code(y_s, x, y_m, z, t):
	tmp = 0
	if (x <= -2.6e+37) or not (x <= 6.7e-16):
		tmp = t * (y_m * x)
	else:
		tmp = (y_m * t) * -z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t = sort([x, y_m, z, t])
function code(y_s, x, y_m, z, t)
	tmp = 0.0
	if ((x <= -2.6e+37) || !(x <= 6.7e-16))
		tmp = Float64(t * Float64(y_m * x));
	else
		tmp = Float64(Float64(y_m * t) * Float64(-z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t = num2cell(sort([x, y_m, z, t])){:}
function tmp_2 = code(y_s, x, y_m, z, t)
	tmp = 0.0;
	if ((x <= -2.6e+37) || ~((x <= 6.7e-16)))
		tmp = t * (y_m * x);
	else
		tmp = (y_m * t) * -z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_, t_] := N[(y$95$s * If[Or[LessEqual[x, -2.6e+37], N[Not[LessEqual[x, 6.7e-16]], $MachinePrecision]], N[(t * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * t), $MachinePrecision] * (-z)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -2.5999999999999999e37 or 6.7000000000000004e-16 < x

   1. Initial program 92.2%

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

      1. distribute-rgt-out-- 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in x around inf 77.7%

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

   if -2.5999999999999999e37 < x < 6.7000000000000004e-16

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. distribute-rgt-out-- 97.7%

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

      3. associate-*r* 93.5%

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

      4. *-commutative 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in x around 0 79.9%

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

      1. neg-mul-1 80.9%

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

   7. Simplified 79.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+37} \lor \neg \left(x \leq 6.7 \cdot 10^{-16}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t] = \mathsf{sort}([x, y_m, z, t])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-29} \lor \neg \left(x \leq 2.45 \cdot 10^{-14}\right):\\ \;\;\;\;t \cdot \left(y\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z t)
 :precision binary64
 (*
  y_s
  (if (or (<= x -4.6e-29) (not (<= x 2.45e-14)))
    (* t (* y_m x))
    (* y_m (* z (- t))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t);
double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if ((x <= -4.6e-29) || !(x <= 2.45e-14)) {
		tmp = t * (y_m * x);
	} else {
		tmp = y_m * (z * -t);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z, t)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-4.6d-29)) .or. (.not. (x <= 2.45d-14))) then
        tmp = t * (y_m * x)
    else
        tmp = y_m * (z * -t)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t;
public static double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if ((x <= -4.6e-29) || !(x <= 2.45e-14)) {
		tmp = t * (y_m * x);
	} else {
		tmp = y_m * (z * -t);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t] = sort([x, y_m, z, t])
def code(y_s, x, y_m, z, t):
	tmp = 0
	if (x <= -4.6e-29) or not (x <= 2.45e-14):
		tmp = t * (y_m * x)
	else:
		tmp = y_m * (z * -t)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t = sort([x, y_m, z, t])
function code(y_s, x, y_m, z, t)
	tmp = 0.0
	if ((x <= -4.6e-29) || !(x <= 2.45e-14))
		tmp = Float64(t * Float64(y_m * x));
	else
		tmp = Float64(y_m * Float64(z * Float64(-t)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t = num2cell(sort([x, y_m, z, t])){:}
function tmp_2 = code(y_s, x, y_m, z, t)
	tmp = 0.0;
	if ((x <= -4.6e-29) || ~((x <= 2.45e-14)))
		tmp = t * (y_m * x);
	else
		tmp = y_m * (z * -t);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_, t_] := N[(y$95$s * If[Or[LessEqual[x, -4.6e-29], N[Not[LessEqual[x, 2.45e-14]], $MachinePrecision]], N[(t * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(z * (-t)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -4.59999999999999982e-29 or 2.44999999999999997e-14 < x

   1. Initial program 93.3%

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

      1. distribute-rgt-out-- 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in x around inf 75.9%

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

   if -4.59999999999999982e-29 < x < 2.44999999999999997e-14

   1. Initial program 97.4%

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

      1. distribute-rgt-out-- 97.4%

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

      2. associate-*l* 92.6%

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

      3. *-commutative 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around 0 81.0%

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

      1. mul-1-neg 81.0%

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

      2. *-commutative 81.0%

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

      3. distribute-rgt-neg-in 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-29} \lor \neg \left(x \leq 2.45 \cdot 10^{-14}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t] = \mathsf{sort}([x, y_m, z, t])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{+186}:\\ \;\;\;\;t \cdot \left(y\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z t)
 :precision binary64
 (* y_s (if (<= t 2.4e+186) (* t (* y_m x)) (* x (* y_m t)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t);
double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if (t <= 2.4e+186) {
		tmp = t * (y_m * x);
	} else {
		tmp = x * (y_m * t);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z, t)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 2.4d+186) then
        tmp = t * (y_m * x)
    else
        tmp = x * (y_m * t)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t;
public static double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if (t <= 2.4e+186) {
		tmp = t * (y_m * x);
	} else {
		tmp = x * (y_m * t);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t] = sort([x, y_m, z, t])
def code(y_s, x, y_m, z, t):
	tmp = 0
	if t <= 2.4e+186:
		tmp = t * (y_m * x)
	else:
		tmp = x * (y_m * t)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t = sort([x, y_m, z, t])
function code(y_s, x, y_m, z, t)
	tmp = 0.0
	if (t <= 2.4e+186)
		tmp = Float64(t * Float64(y_m * x));
	else
		tmp = Float64(x * Float64(y_m * t));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t = num2cell(sort([x, y_m, z, t])){:}
function tmp_2 = code(y_s, x, y_m, z, t)
	tmp = 0.0;
	if (t <= 2.4e+186)
		tmp = t * (y_m * x);
	else
		tmp = x * (y_m * t);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_, t_] := N[(y$95$s * If[LessEqual[t, 2.4e+186], N[(t * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.39999999999999995e186

   1. Initial program 94.7%

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

      1. distribute-rgt-out-- 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around inf 52.4%

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

   if 2.39999999999999995e186 < t

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt-out-- 99.8%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 74.4%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{+186}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t] = \mathsf{sort}([x, y_m, z, t])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 100000000000:\\ \;\;\;\;y\_m \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z t)
 :precision binary64
 (* y_s (if (<= t 100000000000.0) (* y_m (* x t)) (* x (* y_m t)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t);
double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if (t <= 100000000000.0) {
		tmp = y_m * (x * t);
	} else {
		tmp = x * (y_m * t);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z, t)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 100000000000.0d0) then
        tmp = y_m * (x * t)
    else
        tmp = x * (y_m * t)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t;
public static double code(double y_s, double x, double y_m, double z, double t) {
	double tmp;
	if (t <= 100000000000.0) {
		tmp = y_m * (x * t);
	} else {
		tmp = x * (y_m * t);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t] = sort([x, y_m, z, t])
def code(y_s, x, y_m, z, t):
	tmp = 0
	if t <= 100000000000.0:
		tmp = y_m * (x * t)
	else:
		tmp = x * (y_m * t)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t = sort([x, y_m, z, t])
function code(y_s, x, y_m, z, t)
	tmp = 0.0
	if (t <= 100000000000.0)
		tmp = Float64(y_m * Float64(x * t));
	else
		tmp = Float64(x * Float64(y_m * t));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t = num2cell(sort([x, y_m, z, t])){:}
function tmp_2 = code(y_s, x, y_m, z, t)
	tmp = 0.0;
	if (t <= 100000000000.0)
		tmp = y_m * (x * t);
	else
		tmp = x * (y_m * t);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_, t_] := N[(y$95$s * If[LessEqual[t, 100000000000.0], N[(y$95$m * N[(x * t), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1e11

   1. Initial program 93.9%

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

      1. distribute-rgt-out-- 94.5%

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

      2. associate-*l* 93.5%

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

      3. *-commutative 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in x around inf 54.5%

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

   if 1e11 < t

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-rgt-out-- 99.8%

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

      3. associate-*r* 98.1%

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

      4. *-commutative 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in x around inf 58.9%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 100000000000:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. distribute-rgt-out-- 95.7%

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

3. Simplified 95.7%

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

Alternative 9: 52.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. distribute-rgt-out-- 95.7%

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

   2. associate-*l* 90.9%

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

   3. *-commutative 90.9%

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

3. Simplified 90.9%

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

5. Taylor expanded in x around inf 53.4%

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

6. Final simplification 53.4%

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

Developer Target 1: 96.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024143 
(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))