Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.4% → 98.0%

Time: 8.4s

Alternatives: 6

Speedup: 1.3×

• 25.3% 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: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
y_s = copysign(1.0, y);
t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1e-6) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = (x - z) * (y_m * t_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1d-6) then
        tmp = y_m * ((x - z) * t_m)
    else
        tmp = (x - z) * (y_m * t_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1e-6) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = (x - z) * (y_m * t_m);
	}
	return t_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 1e-6:
		tmp = y_m * ((x - z) * t_m)
	else:
		tmp = (x - z) * (y_m * t_m)
	return t_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
t_m = abs(t)
t_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 1e-6)
		tmp = Float64(y_m * Float64(Float64(x - z) * t_m));
	else
		tmp = Float64(Float64(x - z) * Float64(y_m * t_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 1e-6)
		tmp = y_m * ((x - z) * t_m);
	else
		tmp = (x - z) * (y_m * t_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999999955e-7

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 89.6%

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

      2. associate-*l* 90.1%

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

      3. *-commutative 90.1%

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

   3. Simplified 90.1%

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

   if 9.99999999999999955e-7 < t

   1. Initial program 95.7%

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

      1. *-commutative 95.7%

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

      2. distribute-rgt-out-- 99.9%

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

      3. associate-*r* 98.6%

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

      4. *-commutative 98.6%

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

   3. Simplified 98.6%

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

4. Final simplification 92.5%

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

Alternative 2: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := t_m \cdot \left(y_m \cdot \left(-z\right)\right)\\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-74}:\\ \;\;\;\;t_m \cdot \left(y_m \cdot x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-8} \lor \neg \left(z \leq 1.12 \cdot 10^{+49}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y_m \cdot t_m\right)\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (* y_m (- z)))))
   (*
    t_s
    (*
     y_s
     (if (<= z -2.7e+50)
       t_2
       (if (<= z 6.5e-74)
         (* t_m (* y_m x))
         (if (or (<= z 7.5e-8) (not (<= z 1.12e+49)))
           t_2
           (* x (* y_m t_m)))))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = t_m * (y_m * -z);
	double tmp;
	if (z <= -2.7e+50) {
		tmp = t_2;
	} else if (z <= 6.5e-74) {
		tmp = t_m * (y_m * x);
	} else if ((z <= 7.5e-8) || !(z <= 1.12e+49)) {
		tmp = t_2;
	} else {
		tmp = x * (y_m * t_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (y_m * -z)
    if (z <= (-2.7d+50)) then
        tmp = t_2
    else if (z <= 6.5d-74) then
        tmp = t_m * (y_m * x)
    else if ((z <= 7.5d-8) .or. (.not. (z <= 1.12d+49))) then
        tmp = t_2
    else
        tmp = x * (y_m * t_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = t_m * (y_m * -z);
	double tmp;
	if (z <= -2.7e+50) {
		tmp = t_2;
	} else if (z <= 6.5e-74) {
		tmp = t_m * (y_m * x);
	} else if ((z <= 7.5e-8) || !(z <= 1.12e+49)) {
		tmp = t_2;
	} else {
		tmp = x * (y_m * t_m);
	}
	return t_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	t_2 = t_m * (y_m * -z)
	tmp = 0
	if z <= -2.7e+50:
		tmp = t_2
	elif z <= 6.5e-74:
		tmp = t_m * (y_m * x)
	elif (z <= 7.5e-8) or not (z <= 1.12e+49):
		tmp = t_2
	else:
		tmp = x * (y_m * t_m)
	return t_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
t_m = abs(t)
t_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	t_2 = Float64(t_m * Float64(y_m * Float64(-z)))
	tmp = 0.0
	if (z <= -2.7e+50)
		tmp = t_2;
	elseif (z <= 6.5e-74)
		tmp = Float64(t_m * Float64(y_m * x));
	elseif ((z <= 7.5e-8) || !(z <= 1.12e+49))
		tmp = t_2;
	else
		tmp = Float64(x * Float64(y_m * t_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	t_2 = t_m * (y_m * -z);
	tmp = 0.0;
	if (z <= -2.7e+50)
		tmp = t_2;
	elseif (z <= 6.5e-74)
		tmp = t_m * (y_m * x);
	elseif ((z <= 7.5e-8) || ~((z <= 1.12e+49)))
		tmp = t_2;
	else
		tmp = x * (y_m * t_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[z, -2.7e+50], t$95$2, If[LessEqual[z, 6.5e-74], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 7.5e-8], N[Not[LessEqual[z, 1.12e+49]], $MachinePrecision]], t$95$2, N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7e50 or 6.5000000000000002e-74 < z < 7.4999999999999997e-8 or 1.12000000000000005e49 < z

   1. Initial program 86.4%

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

      1. distribute-rgt-out-- 89.9%

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

      2. associate-*l* 84.4%

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

      3. *-commutative 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in x around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. distribute-rgt-neg-in 75.4%

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

      3. distribute-rgt-neg-in 75.4%

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

   7. Simplified 75.4%

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

   if -2.7e50 < z < 6.5000000000000002e-74

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 94.1%

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

      2. associate-*l* 95.4%

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

      3. *-commutative 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in x around inf 81.8%

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

      1. *-commutative 81.8%

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

   7. Simplified 81.8%

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

   if 7.4999999999999997e-8 < z < 1.12000000000000005e49

   1. Initial program 99.7%

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

      1. distribute-rgt-out-- 99.7%

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

      2. associate-*l* 76.0%

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

      3. *-commutative 76.0%

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

   3. Simplified 76.0%

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

      1. associate-*r* 100.0%

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

      2. flip-- 68.4%

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

      3. associate-*r/ 60.5%

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

      4. pow2 60.5%

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

      5. pow2 60.5%

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

   6. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{\frac{\left(y \cdot t\right) \cdot \left({x}^{2} - {z}^{2}\right)}{x + z}} \]
   7. Step-by-step derivation

      1. *-commutative 60.5%

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

      2. associate-/l* 67.3%

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

      3. div-sub 59.0%

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

      4. associate-/l/ 67.2%

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

      5. associate-/r/ 59.8%

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

      6. associate-/l/ 59.8%

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

      7. associate-/r/ 59.7%

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

      8. distribute-rgt-out-- 59.7%

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

      9. div-sub 59.7%

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

      10. associate-/l* 52.7%

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

      11. *-commutative 52.7%

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

      12. associate-/l* 60.5%

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

      13. unpow2 60.5%

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

      14. unpow2 60.5%

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

      15. difference-of-squares 60.5%

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

      16. associate-/r* 76.1%

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

   8. Simplified 76.1%

      \[\leadsto \color{blue}{y \cdot \frac{t}{\frac{1}{x - z}}} \]

   9. Taylor expanded in x around inf 91.5%

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

       1. *-commutative 91.5%

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

       2. associate-*r* 91.8%

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

   11. Simplified 91.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+50}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-74}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-8} \lor \neg \left(z \leq 1.12 \cdot 10^{+49}\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := t_m \cdot \left(y_m \cdot \left(-z\right)\right)\\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-74}:\\ \;\;\;\;t_m \cdot \left(y_m \cdot x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;y_m \cdot \left(z \cdot \left(-t_m\right)\right)\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(y_m \cdot t_m\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (* y_m (- z)))))
   (*
    t_s
    (*
     y_s
     (if (<= z -6e+48)
       t_2
       (if (<= z 6.5e-74)
         (* t_m (* y_m x))
         (if (<= z 8.5e-9)
           (* y_m (* z (- t_m)))
           (if (<= z 2.25e+50) (* x (* y_m t_m)) t_2))))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = t_m * (y_m * -z);
	double tmp;
	if (z <= -6e+48) {
		tmp = t_2;
	} else if (z <= 6.5e-74) {
		tmp = t_m * (y_m * x);
	} else if (z <= 8.5e-9) {
		tmp = y_m * (z * -t_m);
	} else if (z <= 2.25e+50) {
		tmp = x * (y_m * t_m);
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (y_m * -z)
    if (z <= (-6d+48)) then
        tmp = t_2
    else if (z <= 6.5d-74) then
        tmp = t_m * (y_m * x)
    else if (z <= 8.5d-9) then
        tmp = y_m * (z * -t_m)
    else if (z <= 2.25d+50) then
        tmp = x * (y_m * t_m)
    else
        tmp = t_2
    end if
    code = t_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = t_m * (y_m * -z);
	double tmp;
	if (z <= -6e+48) {
		tmp = t_2;
	} else if (z <= 6.5e-74) {
		tmp = t_m * (y_m * x);
	} else if (z <= 8.5e-9) {
		tmp = y_m * (z * -t_m);
	} else if (z <= 2.25e+50) {
		tmp = x * (y_m * t_m);
	} else {
		tmp = t_2;
	}
	return t_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	t_2 = t_m * (y_m * -z)
	tmp = 0
	if z <= -6e+48:
		tmp = t_2
	elif z <= 6.5e-74:
		tmp = t_m * (y_m * x)
	elif z <= 8.5e-9:
		tmp = y_m * (z * -t_m)
	elif z <= 2.25e+50:
		tmp = x * (y_m * t_m)
	else:
		tmp = t_2
	return t_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
t_m = abs(t)
t_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	t_2 = Float64(t_m * Float64(y_m * Float64(-z)))
	tmp = 0.0
	if (z <= -6e+48)
		tmp = t_2;
	elseif (z <= 6.5e-74)
		tmp = Float64(t_m * Float64(y_m * x));
	elseif (z <= 8.5e-9)
		tmp = Float64(y_m * Float64(z * Float64(-t_m)));
	elseif (z <= 2.25e+50)
		tmp = Float64(x * Float64(y_m * t_m));
	else
		tmp = t_2;
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	t_2 = t_m * (y_m * -z);
	tmp = 0.0;
	if (z <= -6e+48)
		tmp = t_2;
	elseif (z <= 6.5e-74)
		tmp = t_m * (y_m * x);
	elseif (z <= 8.5e-9)
		tmp = y_m * (z * -t_m);
	elseif (z <= 2.25e+50)
		tmp = x * (y_m * t_m);
	else
		tmp = t_2;
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[z, -6e+48], t$95$2, If[LessEqual[z, 6.5e-74], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e-9], N[(y$95$m * N[(z * (-t$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.25e+50], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -5.9999999999999999e48 or 2.25000000000000007e50 < z

   1. Initial program 84.7%

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

      1. distribute-rgt-out-- 88.8%

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

      2. associate-*l* 82.2%

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

      3. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in x around 0 74.9%

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

      1. mul-1-neg 74.9%

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

      2. distribute-rgt-neg-in 74.9%

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

      3. distribute-rgt-neg-in 74.9%

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

   7. Simplified 74.9%

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

   if -5.9999999999999999e48 < z < 6.5000000000000002e-74

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 94.1%

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

      2. associate-*l* 95.4%

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

      3. *-commutative 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in x around inf 81.8%

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

      1. *-commutative 81.8%

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

   7. Simplified 81.8%

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

   if 6.5000000000000002e-74 < z < 8.5e-9

   1. Initial program 97.8%

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

      1. distribute-rgt-out-- 97.8%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 78.7%

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

      1. mul-1-neg 78.7%

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

      2. associate-*r* 74.6%

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

      3. *-commutative 74.6%

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

      4. distribute-rgt-neg-out 74.6%

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

      5. associate-*l* 80.7%

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

   7. Simplified 80.7%

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

   if 8.5e-9 < z < 2.25000000000000007e50

   1. Initial program 99.7%

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

      1. distribute-rgt-out-- 99.7%

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

      2. associate-*l* 76.0%

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

      3. *-commutative 76.0%

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

   3. Simplified 76.0%

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

      1. associate-*r* 100.0%

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

      2. flip-- 68.4%

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

      3. associate-*r/ 60.5%

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

      4. pow2 60.5%

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

      5. pow2 60.5%

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

   6. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{\frac{\left(y \cdot t\right) \cdot \left({x}^{2} - {z}^{2}\right)}{x + z}} \]
   7. Step-by-step derivation

      1. *-commutative 60.5%

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

      2. associate-/l* 67.3%

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

      3. div-sub 59.0%

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

      4. associate-/l/ 67.2%

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

      5. associate-/r/ 59.8%

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

      6. associate-/l/ 59.8%

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

      7. associate-/r/ 59.7%

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

      8. distribute-rgt-out-- 59.7%

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

      9. div-sub 59.7%

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

      10. associate-/l* 52.7%

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

      11. *-commutative 52.7%

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

      12. associate-/l* 60.5%

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

      13. unpow2 60.5%

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

      14. unpow2 60.5%

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

      15. difference-of-squares 60.5%

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

      16. associate-/r* 76.1%

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

   8. Simplified 76.1%

      \[\leadsto \color{blue}{y \cdot \frac{t}{\frac{1}{x - z}}} \]

   9. Taylor expanded in x around inf 91.5%

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

       1. *-commutative 91.5%

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

       2. associate-*r* 91.8%

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

   11. Simplified 91.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+48}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-74}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+158} \lor \neg \left(z \leq 5.2 \cdot 10^{+114}\right):\\ \;\;\;\;t_m \cdot \left(y_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \left(\left(x - z\right) \cdot t_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (*
   y_s
   (if (or (<= z -2.45e+158) (not (<= z 5.2e+114)))
     (* t_m (* y_m (- z)))
     (* y_m (* (- x z) t_m))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((z <= -2.45e+158) || !(z <= 5.2e+114)) {
		tmp = t_m * (y_m * -z);
	} else {
		tmp = y_m * ((x - z) * t_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((z <= (-2.45d+158)) .or. (.not. (z <= 5.2d+114))) then
        tmp = t_m * (y_m * -z)
    else
        tmp = y_m * ((x - z) * t_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((z <= -2.45e+158) || !(z <= 5.2e+114)) {
		tmp = t_m * (y_m * -z);
	} else {
		tmp = y_m * ((x - z) * t_m);
	}
	return t_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if (z <= -2.45e+158) or not (z <= 5.2e+114):
		tmp = t_m * (y_m * -z)
	else:
		tmp = y_m * ((x - z) * t_m)
	return t_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
t_m = abs(t)
t_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if ((z <= -2.45e+158) || !(z <= 5.2e+114))
		tmp = Float64(t_m * Float64(y_m * Float64(-z)));
	else
		tmp = Float64(y_m * Float64(Float64(x - z) * t_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if ((z <= -2.45e+158) || ~((z <= 5.2e+114)))
		tmp = t_m * (y_m * -z);
	else
		tmp = y_m * ((x - z) * t_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[Or[LessEqual[z, -2.45e+158], N[Not[LessEqual[z, 5.2e+114]], $MachinePrecision]], N[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.4500000000000002e158 or 5.2000000000000001e114 < z

   1. Initial program 81.0%

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

      1. distribute-rgt-out-- 86.7%

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

      2. associate-*l* 79.9%

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

      3. *-commutative 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in x around 0 80.0%

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

      1. mul-1-neg 80.0%

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

      2. distribute-rgt-neg-in 80.0%

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

      3. distribute-rgt-neg-in 80.0%

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

   7. Simplified 80.0%

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

   if -2.4500000000000002e158 < z < 5.2000000000000001e114

   1. Initial program 94.7%

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

      1. distribute-rgt-out-- 94.7%

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

      2. associate-*l* 93.2%

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

      3. *-commutative 93.2%

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

   3. Simplified 93.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+158} \lor \neg \left(z \leq 5.2 \cdot 10^{+114}\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. distribute-rgt-out-- 92.5%

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

3. Simplified 92.5%

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

5. Final simplification 92.5%

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

Alternative 6: 55.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. distribute-rgt-out-- 92.5%

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

   2. associate-*l* 89.5%

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

   3. *-commutative 89.5%

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

3. Simplified 89.5%

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

5. Taylor expanded in x around inf 56.5%

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

   1. *-commutative 56.5%

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

7. Simplified 56.5%

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

8. Final simplification 56.5%

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

Developer target: 95.8% 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 2024018 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

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