Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.6% → 97.1%

Time: 9.2s

Alternatives: 8

Speedup: 1.3×

• 25.4% 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: 96.6% 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: 97.1% accurate, 1.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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 89.7%

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

   1. distribute-rgt-out-- 92.2%

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

3. Simplified 92.2%

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

Alternative 2: 73.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := z \cdot \left(y\_m \cdot \left(-t\_m\right)\right)\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{-138}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-104}:\\ \;\;\;\;y\_m \cdot \left(z \cdot \left(-t\_m\right)\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+60} \lor \neg \left(z \leq 7.8 \cdot 10^{+89}\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x \cdot t\_m\right)\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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 (* z (* y_m (- t_m)))))
   (*
    t_s
    (*
     y_s
     (if (<= z -1.5e+68)
       t_2
       (if (<= z 1.82e-138)
         (* t_m (* y_m x))
         (if (<= z 2.8e-104)
           (* y_m (* z (- t_m)))
           (if (<= z 1.75e-18)
             (* x (* y_m t_m))
             (if (or (<= z 4e+60) (not (<= z 7.8e+89)))
               t_2
               (* y_m (* x t_m)))))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = z * (y_m * -t_m);
	double tmp;
	if (z <= -1.5e+68) {
		tmp = t_2;
	} else if (z <= 1.82e-138) {
		tmp = t_m * (y_m * x);
	} else if (z <= 2.8e-104) {
		tmp = y_m * (z * -t_m);
	} else if (z <= 1.75e-18) {
		tmp = x * (y_m * t_m);
	} else if ((z <= 4e+60) || !(z <= 7.8e+89)) {
		tmp = t_2;
	} else {
		tmp = y_m * (x * 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 = z * (y_m * -t_m)
    if (z <= (-1.5d+68)) then
        tmp = t_2
    else if (z <= 1.82d-138) then
        tmp = t_m * (y_m * x)
    else if (z <= 2.8d-104) then
        tmp = y_m * (z * -t_m)
    else if (z <= 1.75d-18) then
        tmp = x * (y_m * t_m)
    else if ((z <= 4d+60) .or. (.not. (z <= 7.8d+89))) then
        tmp = t_2
    else
        tmp = y_m * (x * 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 = z * (y_m * -t_m);
	double tmp;
	if (z <= -1.5e+68) {
		tmp = t_2;
	} else if (z <= 1.82e-138) {
		tmp = t_m * (y_m * x);
	} else if (z <= 2.8e-104) {
		tmp = y_m * (z * -t_m);
	} else if (z <= 1.75e-18) {
		tmp = x * (y_m * t_m);
	} else if ((z <= 4e+60) || !(z <= 7.8e+89)) {
		tmp = t_2;
	} else {
		tmp = y_m * (x * 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 = z * (y_m * -t_m)
	tmp = 0
	if z <= -1.5e+68:
		tmp = t_2
	elif z <= 1.82e-138:
		tmp = t_m * (y_m * x)
	elif z <= 2.8e-104:
		tmp = y_m * (z * -t_m)
	elif z <= 1.75e-18:
		tmp = x * (y_m * t_m)
	elif (z <= 4e+60) or not (z <= 7.8e+89):
		tmp = t_2
	else:
		tmp = y_m * (x * 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(z * Float64(y_m * Float64(-t_m)))
	tmp = 0.0
	if (z <= -1.5e+68)
		tmp = t_2;
	elseif (z <= 1.82e-138)
		tmp = Float64(t_m * Float64(y_m * x));
	elseif (z <= 2.8e-104)
		tmp = Float64(y_m * Float64(z * Float64(-t_m)));
	elseif (z <= 1.75e-18)
		tmp = Float64(x * Float64(y_m * t_m));
	elseif ((z <= 4e+60) || !(z <= 7.8e+89))
		tmp = t_2;
	else
		tmp = Float64(y_m * Float64(x * 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 = z * (y_m * -t_m);
	tmp = 0.0;
	if (z <= -1.5e+68)
		tmp = t_2;
	elseif (z <= 1.82e-138)
		tmp = t_m * (y_m * x);
	elseif (z <= 2.8e-104)
		tmp = y_m * (z * -t_m);
	elseif (z <= 1.75e-18)
		tmp = x * (y_m * t_m);
	elseif ((z <= 4e+60) || ~((z <= 7.8e+89)))
		tmp = t_2;
	else
		tmp = y_m * (x * 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[(z * N[(y$95$m * (-t$95$m)), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[z, -1.5e+68], t$95$2, If[LessEqual[z, 1.82e-138], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e-104], N[(y$95$m * N[(z * (-t$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e-18], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 4e+60], N[Not[LessEqual[z, 7.8e+89]], $MachinePrecision]], t$95$2, N[(y$95$m * N[(x * t$95$m), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if z < -1.5000000000000001e68 or 1.7499999999999999e-18 < z < 3.9999999999999998e60 or 7.80000000000000021e89 < z

   1. Initial program 82.7%

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

      1. distribute-rgt-out-- 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in x around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. unsub-neg 75.9%

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

      3. associate-/l* 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in x around 0 74.6%

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

      1. associate-*r* 74.6%

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

      2. associate-*r* 77.6%

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

      3. associate-*r* 77.6%

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

      4. *-commutative 77.6%

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

      5. associate-*r* 77.6%

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

      6. mul-1-neg 77.6%

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

   10. Simplified 77.6%

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

   if -1.5000000000000001e68 < z < 1.8200000000000001e-138

   1. Initial program 93.4%

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

      1. distribute-rgt-out-- 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in x around inf 83.2%

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

      1. *-commutative 83.2%

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

   7. Simplified 83.2%

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

   if 1.8200000000000001e-138 < z < 2.8e-104

   1. Initial program 100.0%

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

      1. distribute-rgt-out-- 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-rgt-neg-out 87.7%

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

   7. Simplified 87.7%

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

   if 2.8e-104 < z < 1.7499999999999999e-18

   1. Initial program 96.5%

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

      1. distribute-rgt-out-- 96.4%

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

      2. associate-*l* 95.7%

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

      3. *-commutative 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in x around inf 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 96.4%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, y, -1 \cdot \frac{t \cdot \left(y \cdot z\right)}{x}\right)} \]

      3. mul-1-neg 96.4%

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

      4. associate-/l* 96.5%

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

      5. distribute-rgt-neg-in 96.5%

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

      6. distribute-frac-neg 96.5%

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

      7. distribute-rgt-neg-out 96.5%

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

      8. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 81.4%

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

      1. *-commutative 81.4%

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

   10. Simplified 81.4%

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

   if 3.9999999999999998e60 < z < 7.80000000000000021e89

   1. Initial program 90.3%

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

      1. distribute-rgt-out-- 99.4%

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

      2. associate-*l* 99.6%

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

      3. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 82.3%

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

      1. associate-*r* 82.3%

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

      2. *-commutative 82.3%

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

   7. Simplified 82.3%

      \[\leadsto \color{blue}{y \cdot \left(t \cdot x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+68}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{-138}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-104}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+60} \lor \neg \left(z \leq 7.8 \cdot 10^{+89}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := t\_m \cdot \left(z \cdot \left(-y\_m\right)\right)\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{-138}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-104}:\\ \;\;\;\;y\_m \cdot \left(z \cdot \left(-t\_m\right)\right)\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+60}:\\ \;\;\;\;z \cdot \left(y\_m \cdot \left(-t\_m\right)\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+91}:\\ \;\;\;\;y\_m \cdot \left(x \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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 (* z (- y_m)))))
   (*
    t_s
    (*
     y_s
     (if (<= z -1.1e+68)
       t_2
       (if (<= z 1.82e-138)
         (* t_m (* y_m x))
         (if (<= z 2.8e-104)
           (* y_m (* z (- t_m)))
           (if (<= z 1.72e-15)
             (* x (* y_m t_m))
             (if (<= z 3.2e+60)
               (* z (* y_m (- t_m)))
               (if (<= z 5e+91) (* y_m (* x 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 * (z * -y_m);
	double tmp;
	if (z <= -1.1e+68) {
		tmp = t_2;
	} else if (z <= 1.82e-138) {
		tmp = t_m * (y_m * x);
	} else if (z <= 2.8e-104) {
		tmp = y_m * (z * -t_m);
	} else if (z <= 1.72e-15) {
		tmp = x * (y_m * t_m);
	} else if (z <= 3.2e+60) {
		tmp = z * (y_m * -t_m);
	} else if (z <= 5e+91) {
		tmp = y_m * (x * 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 * (z * -y_m)
    if (z <= (-1.1d+68)) then
        tmp = t_2
    else if (z <= 1.82d-138) then
        tmp = t_m * (y_m * x)
    else if (z <= 2.8d-104) then
        tmp = y_m * (z * -t_m)
    else if (z <= 1.72d-15) then
        tmp = x * (y_m * t_m)
    else if (z <= 3.2d+60) then
        tmp = z * (y_m * -t_m)
    else if (z <= 5d+91) then
        tmp = y_m * (x * 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 * (z * -y_m);
	double tmp;
	if (z <= -1.1e+68) {
		tmp = t_2;
	} else if (z <= 1.82e-138) {
		tmp = t_m * (y_m * x);
	} else if (z <= 2.8e-104) {
		tmp = y_m * (z * -t_m);
	} else if (z <= 1.72e-15) {
		tmp = x * (y_m * t_m);
	} else if (z <= 3.2e+60) {
		tmp = z * (y_m * -t_m);
	} else if (z <= 5e+91) {
		tmp = y_m * (x * 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 * (z * -y_m)
	tmp = 0
	if z <= -1.1e+68:
		tmp = t_2
	elif z <= 1.82e-138:
		tmp = t_m * (y_m * x)
	elif z <= 2.8e-104:
		tmp = y_m * (z * -t_m)
	elif z <= 1.72e-15:
		tmp = x * (y_m * t_m)
	elif z <= 3.2e+60:
		tmp = z * (y_m * -t_m)
	elif z <= 5e+91:
		tmp = y_m * (x * 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(z * Float64(-y_m)))
	tmp = 0.0
	if (z <= -1.1e+68)
		tmp = t_2;
	elseif (z <= 1.82e-138)
		tmp = Float64(t_m * Float64(y_m * x));
	elseif (z <= 2.8e-104)
		tmp = Float64(y_m * Float64(z * Float64(-t_m)));
	elseif (z <= 1.72e-15)
		tmp = Float64(x * Float64(y_m * t_m));
	elseif (z <= 3.2e+60)
		tmp = Float64(z * Float64(y_m * Float64(-t_m)));
	elseif (z <= 5e+91)
		tmp = Float64(y_m * Float64(x * 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 * (z * -y_m);
	tmp = 0.0;
	if (z <= -1.1e+68)
		tmp = t_2;
	elseif (z <= 1.82e-138)
		tmp = t_m * (y_m * x);
	elseif (z <= 2.8e-104)
		tmp = y_m * (z * -t_m);
	elseif (z <= 1.72e-15)
		tmp = x * (y_m * t_m);
	elseif (z <= 3.2e+60)
		tmp = z * (y_m * -t_m);
	elseif (z <= 5e+91)
		tmp = y_m * (x * 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[(z * (-y$95$m)), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[z, -1.1e+68], t$95$2, If[LessEqual[z, 1.82e-138], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e-104], N[(y$95$m * N[(z * (-t$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.72e-15], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e+60], N[(z * N[(y$95$m * (-t$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e+91], N[(y$95$m * N[(x * t$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 6 regimes

2. if z < -1.09999999999999994e68 or 5.0000000000000002e91 < z

   1. Initial program 81.4%

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

      1. distribute-rgt-out-- 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in x around 0 74.5%

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

      1. mul-1-neg 74.5%

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

      2. distribute-rgt-neg-out 74.5%

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

   7. Simplified 74.5%

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

   if -1.09999999999999994e68 < z < 1.8200000000000001e-138

   1. Initial program 93.4%

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

      1. distribute-rgt-out-- 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in x around inf 83.2%

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

      1. *-commutative 83.2%

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

   7. Simplified 83.2%

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

   if 1.8200000000000001e-138 < z < 2.8e-104

   1. Initial program 100.0%

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

      1. distribute-rgt-out-- 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-rgt-neg-out 87.7%

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

   7. Simplified 87.7%

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

   if 2.8e-104 < z < 1.7199999999999999e-15

   1. Initial program 96.5%

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

      1. distribute-rgt-out-- 96.4%

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

      2. associate-*l* 95.7%

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

      3. *-commutative 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in x around inf 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 96.4%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, y, -1 \cdot \frac{t \cdot \left(y \cdot z\right)}{x}\right)} \]

      3. mul-1-neg 96.4%

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

      4. associate-/l* 96.5%

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

      5. distribute-rgt-neg-in 96.5%

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

      6. distribute-frac-neg 96.5%

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

      7. distribute-rgt-neg-out 96.5%

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

      8. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 81.4%

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

      1. *-commutative 81.4%

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

   10. Simplified 81.4%

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

   if 1.7199999999999999e-15 < z < 3.19999999999999991e60

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 92.0%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in x around inf 84.0%

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

      1. mul-1-neg 84.0%

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

      2. unsub-neg 84.0%

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

      3. associate-/l* 83.8%

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

   7. Simplified 83.8%

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

   8. Taylor expanded in x around 0 75.5%

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

      1. associate-*r* 75.5%

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

      2. associate-*r* 83.4%

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

      3. associate-*r* 83.4%

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

      4. *-commutative 83.4%

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

      5. associate-*r* 83.4%

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

      6. mul-1-neg 83.4%

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

   10. Simplified 83.4%

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

   if 3.19999999999999991e60 < z < 5.0000000000000002e91

   1. Initial program 90.3%

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

      1. distribute-rgt-out-- 99.4%

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

      2. associate-*l* 99.6%

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

      3. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 82.3%

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

      1. associate-*r* 82.3%

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

      2. *-commutative 82.3%

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

   7. Simplified 82.3%

      \[\leadsto \color{blue}{y \cdot \left(t \cdot x\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+68}:\\ \;\;\;\;t \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{-138}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-104}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+60}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+91}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-121} \lor \neg \left(x \leq 11500000000000\right):\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(z \cdot \left(-t\_m\right)\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (*
   y_s
   (if (or (<= x -3.3e-121) (not (<= x 11500000000000.0)))
     (* t_m (* y_m x))
     (* y_m (* z (- t_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((x <= -3.3e-121) || !(x <= 11500000000000.0)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = y_m * (z * -t_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((x <= (-3.3d-121)) .or. (.not. (x <= 11500000000000.0d0))) then
        tmp = t_m * (y_m * x)
    else
        tmp = y_m * (z * -t_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((x <= -3.3e-121) || !(x <= 11500000000000.0)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = y_m * (z * -t_m);
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if (x <= -3.3e-121) or not (x <= 11500000000000.0):
		tmp = t_m * (y_m * x)
	else:
		tmp = y_m * (z * -t_m)
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if ((x <= -3.3e-121) || !(x <= 11500000000000.0))
		tmp = Float64(t_m * Float64(y_m * x));
	else
		tmp = Float64(y_m * Float64(z * Float64(-t_m)));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if ((x <= -3.3e-121) || ~((x <= 11500000000000.0)))
		tmp = t_m * (y_m * x);
	else
		tmp = y_m * (z * -t_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[Or[LessEqual[x, -3.3e-121], N[Not[LessEqual[x, 11500000000000.0]], $MachinePrecision]], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(z * (-t$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -3.3000000000000001e-121 or 1.15e13 < x

   1. Initial program 88.4%

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

      1. distribute-rgt-out-- 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in x around inf 73.2%

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

      1. *-commutative 73.2%

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

   7. Simplified 73.2%

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

   if -3.3000000000000001e-121 < x < 1.15e13

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 92.0%

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

      2. associate-*l* 95.9%

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

      3. *-commutative 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in x around 0 78.3%

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

      1. mul-1-neg 78.3%

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

      2. distribute-rgt-neg-out 78.3%

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

   7. Simplified 78.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-121} \lor \neg \left(x \leq 11500000000000\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.7× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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 150000000.0) (* 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 <= 150000000.0) {
		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 <= 150000000.0d0) 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 <= 150000000.0) {
		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 <= 150000000.0:
		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 <= 150000000.0)
		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 <= 150000000.0)
		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, 150000000.0], N[(y$95$m * N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e8

   1. Initial program 87.9%

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

      1. distribute-rgt-out-- 90.0%

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

      2. associate-*l* 94.1%

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

      3. *-commutative 94.1%

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

   3. Simplified 94.1%

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

   if 1.5e8 < t

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

      2. distribute-rgt-out-- 99.8%

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

      3. associate-*r* 93.2%

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

      4. *-commutative 93.2%

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

   3. Simplified 93.2%

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

4. Final simplification 93.9%

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

Alternative 6: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.35 \cdot 10^{+218}:\\ \;\;\;\;y\_m \cdot \left(\left(x - z\right) \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (* y_s (if (<= x 3.35e+218) (* y_m (* (- x z) t_m)) (* t_m (* y_m x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (x <= 3.35e+218) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = t_m * (y_m * x);
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (x <= 3.35d+218) then
        tmp = y_m * ((x - z) * t_m)
    else
        tmp = t_m * (y_m * x)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (x <= 3.35e+218) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = t_m * (y_m * x);
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if x <= 3.35e+218:
		tmp = y_m * ((x - z) * t_m)
	else:
		tmp = t_m * (y_m * x)
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (x <= 3.35e+218)
		tmp = Float64(y_m * Float64(Float64(x - z) * t_m));
	else
		tmp = Float64(t_m * Float64(y_m * x));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (x <= 3.35e+218)
		tmp = y_m * ((x - z) * t_m);
	else
		tmp = t_m * (y_m * x);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[LessEqual[x, 3.35e+218], N[(y$95$m * N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.35e218

   1. Initial program 90.6%

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

      1. distribute-rgt-out-- 92.0%

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

      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

   if 3.35e218 < x

   1. Initial program 82.4%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.35 \cdot 10^{+218}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.5% 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 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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 89.7%

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

   1. distribute-rgt-out-- 92.2%

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

3. Simplified 92.2%

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

5. Taylor expanded in x around inf 59.8%

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

   1. *-commutative 59.8%

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

7. Simplified 59.8%

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

8. Final simplification 59.8%

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

Alternative 8: 53.8% accurate, 1.8× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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 (* 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) {
	return t_s * (y_s * (x * (y_m * 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 * (x * (y_m * 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 * (x * (y_m * 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 * (x * (y_m * 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(x * Float64(y_m * 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 * (x * (y_m * 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[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

   1. distribute-rgt-out-- 92.2%

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

   2. associate-*l* 92.1%

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

   3. *-commutative 92.1%

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

3. Simplified 92.1%

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

5. Taylor expanded in x around inf 79.4%

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

   1. +-commutative 79.4%

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

   2. fma-define 81.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, y, -1 \cdot \frac{t \cdot \left(y \cdot z\right)}{x}\right)} \]

   3. mul-1-neg 81.4%

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

   4. associate-/l* 79.9%

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

   5. distribute-rgt-neg-in 79.9%

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

   6. distribute-frac-neg 79.9%

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

   7. distribute-rgt-neg-out 79.9%

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

   8. associate-/l* 81.2%

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

7. Simplified 81.2%

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

8. Taylor expanded in z around 0 59.1%

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

   1. *-commutative 59.1%

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

10. Simplified 59.1%

    \[\leadsto x \cdot \color{blue}{\left(y \cdot t\right)} \]
11. 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 2024091 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :alt
  (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))