Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.7%

Time: 10.6s

Alternatives: 10

Speedup: 1.3×

• 25.2% 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: 96.2% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := x \cdot y\_m - y\_m \cdot z\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-249}:\\ \;\;\;\;t\_2 \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-251}:\\ \;\;\;\;z \cdot \left(t\_m \cdot \left(x \cdot \frac{y\_m}{z}\right) - y\_m \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot \left(x - z\right)\right)\\ \end{array}\right) \end{array} \end{array} \]

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 (- (* x y_m) (* y_m z))))
   (*
    t_s
    (*
     y_s
     (if (<= t_2 -2e-249)
       (* t_2 t_m)
       (if (<= t_2 2e-251)
         (* z (- (* t_m (* x (/ y_m z))) (* y_m t_m)))
         (* t_m (* y_m (- x z)))))))))

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 = (x * y_m) - (y_m * z);
	double tmp;
	if (t_2 <= -2e-249) {
		tmp = t_2 * t_m;
	} else if (t_2 <= 2e-251) {
		tmp = z * ((t_m * (x * (y_m / z))) - (y_m * t_m));
	} else {
		tmp = t_m * (y_m * (x - z));
	}
	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 = (x * y_m) - (y_m * z)
    if (t_2 <= (-2d-249)) then
        tmp = t_2 * t_m
    else if (t_2 <= 2d-251) then
        tmp = z * ((t_m * (x * (y_m / z))) - (y_m * t_m))
    else
        tmp = t_m * (y_m * (x - z))
    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 = (x * y_m) - (y_m * z);
	double tmp;
	if (t_2 <= -2e-249) {
		tmp = t_2 * t_m;
	} else if (t_2 <= 2e-251) {
		tmp = z * ((t_m * (x * (y_m / z))) - (y_m * t_m));
	} else {
		tmp = t_m * (y_m * (x - z));
	}
	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 = (x * y_m) - (y_m * z)
	tmp = 0
	if t_2 <= -2e-249:
		tmp = t_2 * t_m
	elif t_2 <= 2e-251:
		tmp = z * ((t_m * (x * (y_m / z))) - (y_m * t_m))
	else:
		tmp = t_m * (y_m * (x - z))
	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(Float64(x * y_m) - Float64(y_m * z))
	tmp = 0.0
	if (t_2 <= -2e-249)
		tmp = Float64(t_2 * t_m);
	elseif (t_2 <= 2e-251)
		tmp = Float64(z * Float64(Float64(t_m * Float64(x * Float64(y_m / z))) - Float64(y_m * t_m)));
	else
		tmp = Float64(t_m * Float64(y_m * Float64(x - z)));
	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 = (x * y_m) - (y_m * z);
	tmp = 0.0;
	if (t_2 <= -2e-249)
		tmp = t_2 * t_m;
	elseif (t_2 <= 2e-251)
		tmp = z * ((t_m * (x * (y_m / z))) - (y_m * t_m));
	else
		tmp = t_m * (y_m * (x - z));
	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[(N[(x * y$95$m), $MachinePrecision] - N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[t$95$2, -2e-249], N[(t$95$2 * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e-251], N[(z * N[(N[(t$95$m * N[(x * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y$95$m * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -2.00000000000000011e-249

   1. Initial program 95.1%

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

   if -2.00000000000000011e-249 < (-.f64 (*.f64 x y) (*.f64 z y)) < 2.00000000000000003e-251

   1. Initial program 68.5%

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

      1. distribute-rgt-out-- 68.5%

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

      2. associate-*l* 95.2%

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

      3. *-commutative 95.2%

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

   3. Simplified 95.2%

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

      1. *-commutative 95.2%

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

      2. associate-*l* 68.5%

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

      3. distribute-rgt-out-- 68.5%

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

      4. *-commutative 68.5%

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

      5. sub-neg 68.5%

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

      6. distribute-rgt-in 68.5%

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

      7. *-commutative 68.5%

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

      8. *-commutative 68.5%

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

      9. distribute-rgt-neg-in 68.5%

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

   6. Applied egg-rr 68.5%

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

   7. Taylor expanded in z around inf 76.5%

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

      1. +-commutative 76.5%

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

      2. associate-*r* 76.5%

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

      3. neg-mul-1 76.5%

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

      4. cancel-sign-sub-inv 76.5%

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

      5. associate-/l* 76.5%

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

      6. associate-/l* 99.9%

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

   9. Simplified 99.9%

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

   if 2.00000000000000003e-251 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 94.3%

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

   3. Simplified 94.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -2 \cdot 10^{-249}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 2 \cdot 10^{-251}:\\ \;\;\;\;z \cdot \left(t \cdot \left(x \cdot \frac{y}{z}\right) - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := x \cdot y\_m - y\_m \cdot z\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-249}:\\ \;\;\;\;t\_2 \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-231}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y\_m \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot \left(x - z\right)\right)\\ \end{array}\right) \end{array} \end{array} \]

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 (- (* x y_m) (* y_m z))))
   (*
    t_s
    (*
     y_s
     (if (<= t_2 -2e-249)
       (* t_2 t_m)
       (if (<= t_2 2e-231)
         (* (- x z) (* y_m t_m))
         (* t_m (* y_m (- x z)))))))))

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 = (x * y_m) - (y_m * z);
	double tmp;
	if (t_2 <= -2e-249) {
		tmp = t_2 * t_m;
	} else if (t_2 <= 2e-231) {
		tmp = (x - z) * (y_m * t_m);
	} else {
		tmp = t_m * (y_m * (x - z));
	}
	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 = (x * y_m) - (y_m * z)
    if (t_2 <= (-2d-249)) then
        tmp = t_2 * t_m
    else if (t_2 <= 2d-231) then
        tmp = (x - z) * (y_m * t_m)
    else
        tmp = t_m * (y_m * (x - z))
    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 = (x * y_m) - (y_m * z);
	double tmp;
	if (t_2 <= -2e-249) {
		tmp = t_2 * t_m;
	} else if (t_2 <= 2e-231) {
		tmp = (x - z) * (y_m * t_m);
	} else {
		tmp = t_m * (y_m * (x - z));
	}
	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 = (x * y_m) - (y_m * z)
	tmp = 0
	if t_2 <= -2e-249:
		tmp = t_2 * t_m
	elif t_2 <= 2e-231:
		tmp = (x - z) * (y_m * t_m)
	else:
		tmp = t_m * (y_m * (x - z))
	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(Float64(x * y_m) - Float64(y_m * z))
	tmp = 0.0
	if (t_2 <= -2e-249)
		tmp = Float64(t_2 * t_m);
	elseif (t_2 <= 2e-231)
		tmp = Float64(Float64(x - z) * Float64(y_m * t_m));
	else
		tmp = Float64(t_m * Float64(y_m * Float64(x - z)));
	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 = (x * y_m) - (y_m * z);
	tmp = 0.0;
	if (t_2 <= -2e-249)
		tmp = t_2 * t_m;
	elseif (t_2 <= 2e-231)
		tmp = (x - z) * (y_m * t_m);
	else
		tmp = t_m * (y_m * (x - z));
	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[(N[(x * y$95$m), $MachinePrecision] - N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[t$95$2, -2e-249], N[(t$95$2 * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e-231], N[(N[(x - z), $MachinePrecision] * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y$95$m * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -2.00000000000000011e-249

   1. Initial program 95.1%

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

   if -2.00000000000000011e-249 < (-.f64 (*.f64 x y) (*.f64 z y)) < 2e-231

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. distribute-rgt-out-- 71.3%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 2e-231 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 90.8%

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

      1. distribute-rgt-out-- 94.2%

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

   3. Simplified 94.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -2 \cdot 10^{-249}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 2 \cdot 10^{-231}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(x \cdot y\_m\right) \cdot t\_m\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{+56}:\\ \;\;\;\;\left(y\_m \cdot z\right) \cdot \left(-t\_m\right)\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \mathbf{elif}\;x \leq 86000000000:\\ \;\;\;\;z \cdot \left(y\_m \cdot \left(-t\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

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 (* (* x y_m) t_m)))
   (*
    t_s
    (*
     y_s
     (if (<= x -7.8e+153)
       t_2
       (if (<= x -1.1e+56)
         (* (* y_m z) (- t_m))
         (if (<= x -2.2e-41)
           (* x (* y_m t_m))
           (if (<= x 86000000000.0) (* z (* 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 = (x * y_m) * t_m;
	double tmp;
	if (x <= -7.8e+153) {
		tmp = t_2;
	} else if (x <= -1.1e+56) {
		tmp = (y_m * z) * -t_m;
	} else if (x <= -2.2e-41) {
		tmp = x * (y_m * t_m);
	} else if (x <= 86000000000.0) {
		tmp = z * (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 = (x * y_m) * t_m
    if (x <= (-7.8d+153)) then
        tmp = t_2
    else if (x <= (-1.1d+56)) then
        tmp = (y_m * z) * -t_m
    else if (x <= (-2.2d-41)) then
        tmp = x * (y_m * t_m)
    else if (x <= 86000000000.0d0) then
        tmp = z * (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 = (x * y_m) * t_m;
	double tmp;
	if (x <= -7.8e+153) {
		tmp = t_2;
	} else if (x <= -1.1e+56) {
		tmp = (y_m * z) * -t_m;
	} else if (x <= -2.2e-41) {
		tmp = x * (y_m * t_m);
	} else if (x <= 86000000000.0) {
		tmp = z * (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 = (x * y_m) * t_m
	tmp = 0
	if x <= -7.8e+153:
		tmp = t_2
	elif x <= -1.1e+56:
		tmp = (y_m * z) * -t_m
	elif x <= -2.2e-41:
		tmp = x * (y_m * t_m)
	elif x <= 86000000000.0:
		tmp = z * (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(Float64(x * y_m) * t_m)
	tmp = 0.0
	if (x <= -7.8e+153)
		tmp = t_2;
	elseif (x <= -1.1e+56)
		tmp = Float64(Float64(y_m * z) * Float64(-t_m));
	elseif (x <= -2.2e-41)
		tmp = Float64(x * Float64(y_m * t_m));
	elseif (x <= 86000000000.0)
		tmp = Float64(z * Float64(y_m * Float64(-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 = (x * y_m) * t_m;
	tmp = 0.0;
	if (x <= -7.8e+153)
		tmp = t_2;
	elseif (x <= -1.1e+56)
		tmp = (y_m * z) * -t_m;
	elseif (x <= -2.2e-41)
		tmp = x * (y_m * t_m);
	elseif (x <= 86000000000.0)
		tmp = z * (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[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -7.8e+153], t$95$2, If[LessEqual[x, -1.1e+56], N[(N[(y$95$m * z), $MachinePrecision] * (-t$95$m)), $MachinePrecision], If[LessEqual[x, -2.2e-41], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 86000000000.0], N[(z * N[(y$95$m * (-t$95$m)), $MachinePrecision]), $MachinePrecision], t$95$2]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < -7.79999999999999966e153 or 8.6e10 < x

   1. Initial program 86.8%

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

      1. distribute-rgt-out-- 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in x around inf 79.3%

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

      1. *-commutative 79.3%

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

   7. Simplified 79.3%

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

   if -7.79999999999999966e153 < x < -1.10000000000000008e56

   1. Initial program 91.3%

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

      1. distribute-rgt-out-- 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in x around 0 70.3%

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

      1. mul-1-neg 70.3%

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

      2. distribute-rgt-neg-out 70.3%

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

   7. Simplified 70.3%

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

   if -1.10000000000000008e56 < x < -2.2e-41

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 91.9%

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

      2. associate-*l* 94.9%

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

      3. *-commutative 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around inf 99.5%

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

   6. Taylor expanded in z around 0 75.9%

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

      1. *-commutative 75.9%

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

   8. Simplified 75.9%

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

   if -2.2e-41 < x < 8.6e10

   1. Initial program 94.0%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

      1. distribute-rgt-out-- 94.0%

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

      2. add-cube-cbrt 93.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x \cdot y - z \cdot y} \cdot \sqrt[3]{x \cdot y - z \cdot y}\right) \cdot \sqrt[3]{x \cdot y - z \cdot y}\right)} \cdot t \]

      3. pow3 93.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot y - z \cdot y}\right)}^{3}} \cdot t \]

      4. distribute-rgt-out-- 93.1%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{y \cdot \left(x - z\right)}}\right)}^{3} \cdot t \]

   6. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(x - z\right)}\right)}^{3}} \cdot t \]

   7. Taylor expanded in x around 0 80.0%

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

      1. mul-1-neg 80.0%

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

      2. *-commutative 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-*r* 82.5%

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

      5. distribute-rgt-neg-in 82.5%

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

      6. *-commutative 82.5%

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

      7. distribute-lft-neg-in 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+153}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{+56}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-t\right)\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 86000000000:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := z \cdot \left(y\_m \cdot \left(-t\_m\right)\right)\\ t_3 := \left(x \cdot y\_m\right) \cdot t\_m\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+104}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{+49}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \mathbf{elif}\;x \leq 530000000000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array}\right) \end{array} \end{array} \]

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_3 (* (* x y_m) t_m)))
   (*
    t_s
    (*
     y_s
     (if (<= x -1.22e+104)
       t_3
       (if (<= x -2.7e+49)
         t_2
         (if (<= x -3.4e-40)
           (* x (* y_m t_m))
           (if (<= x 530000000000.0) t_2 t_3))))))))

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 t_3 = (x * y_m) * t_m;
	double tmp;
	if (x <= -1.22e+104) {
		tmp = t_3;
	} else if (x <= -2.7e+49) {
		tmp = t_2;
	} else if (x <= -3.4e-40) {
		tmp = x * (y_m * t_m);
	} else if (x <= 530000000000.0) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_2 = z * (y_m * -t_m)
    t_3 = (x * y_m) * t_m
    if (x <= (-1.22d+104)) then
        tmp = t_3
    else if (x <= (-2.7d+49)) then
        tmp = t_2
    else if (x <= (-3.4d-40)) then
        tmp = x * (y_m * t_m)
    else if (x <= 530000000000.0d0) then
        tmp = t_2
    else
        tmp = t_3
    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 t_3 = (x * y_m) * t_m;
	double tmp;
	if (x <= -1.22e+104) {
		tmp = t_3;
	} else if (x <= -2.7e+49) {
		tmp = t_2;
	} else if (x <= -3.4e-40) {
		tmp = x * (y_m * t_m);
	} else if (x <= 530000000000.0) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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)
	t_3 = (x * y_m) * t_m
	tmp = 0
	if x <= -1.22e+104:
		tmp = t_3
	elif x <= -2.7e+49:
		tmp = t_2
	elif x <= -3.4e-40:
		tmp = x * (y_m * t_m)
	elif x <= 530000000000.0:
		tmp = t_2
	else:
		tmp = t_3
	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)))
	t_3 = Float64(Float64(x * y_m) * t_m)
	tmp = 0.0
	if (x <= -1.22e+104)
		tmp = t_3;
	elseif (x <= -2.7e+49)
		tmp = t_2;
	elseif (x <= -3.4e-40)
		tmp = Float64(x * Float64(y_m * t_m));
	elseif (x <= 530000000000.0)
		tmp = t_2;
	else
		tmp = t_3;
	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);
	t_3 = (x * y_m) * t_m;
	tmp = 0.0;
	if (x <= -1.22e+104)
		tmp = t_3;
	elseif (x <= -2.7e+49)
		tmp = t_2;
	elseif (x <= -3.4e-40)
		tmp = x * (y_m * t_m);
	elseif (x <= 530000000000.0)
		tmp = t_2;
	else
		tmp = t_3;
	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]}, Block[{t$95$3 = N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -1.22e+104], t$95$3, If[LessEqual[x, -2.7e+49], t$95$2, If[LessEqual[x, -3.4e-40], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 530000000000.0], t$95$2, t$95$3]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.22e104 or 5.3e11 < x

   1. Initial program 88.6%

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

      1. distribute-rgt-out-- 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in x around inf 77.1%

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

      1. *-commutative 77.1%

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

   7. Simplified 77.1%

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

   if -1.22e104 < x < -2.7000000000000001e49 or -3.39999999999999984e-40 < x < 5.3e11

   1. Initial program 92.8%

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

      1. distribute-rgt-out-- 93.6%

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

   3. Simplified 93.6%

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

      1. distribute-rgt-out-- 92.8%

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

      2. add-cube-cbrt 91.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x \cdot y - z \cdot y} \cdot \sqrt[3]{x \cdot y - z \cdot y}\right) \cdot \sqrt[3]{x \cdot y - z \cdot y}\right)} \cdot t \]

      3. pow3 91.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot y - z \cdot y}\right)}^{3}} \cdot t \]

      4. distribute-rgt-out-- 92.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{y \cdot \left(x - z\right)}}\right)}^{3} \cdot t \]

   6. Applied egg-rr 92.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(x - z\right)}\right)}^{3}} \cdot t \]

   7. Taylor expanded in x around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. *-commutative 80.5%

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

      3. *-commutative 80.5%

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

      4. associate-*r* 82.8%

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

      5. distribute-rgt-neg-in 82.8%

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

      6. *-commutative 82.8%

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

      7. distribute-lft-neg-in 82.8%

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

   9. Simplified 82.8%

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

   if -2.7000000000000001e49 < x < -3.39999999999999984e-40

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 91.9%

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

      2. associate-*l* 94.9%

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

      3. *-commutative 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around inf 99.5%

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

   6. Taylor expanded in z around 0 75.9%

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

      1. *-commutative 75.9%

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

   8. Simplified 75.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+104}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{+49}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 530000000000:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := y\_m \cdot \left(z \cdot \left(-t\_m\right)\right)\\ t_3 := \left(x \cdot y\_m\right) \cdot t\_m\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+153}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \mathbf{elif}\;x \leq 5200000000000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array}\right) \end{array} \end{array} \]

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 (* y_m (* z (- t_m)))) (t_3 (* (* x y_m) t_m)))
   (*
    t_s
    (*
     y_s
     (if (<= x -7.8e+153)
       t_3
       (if (<= x -4.4e+55)
         t_2
         (if (<= x -5.5e-41)
           (* x (* y_m t_m))
           (if (<= x 5200000000000.0) t_2 t_3))))))))

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 = y_m * (z * -t_m);
	double t_3 = (x * y_m) * t_m;
	double tmp;
	if (x <= -7.8e+153) {
		tmp = t_3;
	} else if (x <= -4.4e+55) {
		tmp = t_2;
	} else if (x <= -5.5e-41) {
		tmp = x * (y_m * t_m);
	} else if (x <= 5200000000000.0) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_2 = y_m * (z * -t_m)
    t_3 = (x * y_m) * t_m
    if (x <= (-7.8d+153)) then
        tmp = t_3
    else if (x <= (-4.4d+55)) then
        tmp = t_2
    else if (x <= (-5.5d-41)) then
        tmp = x * (y_m * t_m)
    else if (x <= 5200000000000.0d0) then
        tmp = t_2
    else
        tmp = t_3
    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 = y_m * (z * -t_m);
	double t_3 = (x * y_m) * t_m;
	double tmp;
	if (x <= -7.8e+153) {
		tmp = t_3;
	} else if (x <= -4.4e+55) {
		tmp = t_2;
	} else if (x <= -5.5e-41) {
		tmp = x * (y_m * t_m);
	} else if (x <= 5200000000000.0) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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 = y_m * (z * -t_m)
	t_3 = (x * y_m) * t_m
	tmp = 0
	if x <= -7.8e+153:
		tmp = t_3
	elif x <= -4.4e+55:
		tmp = t_2
	elif x <= -5.5e-41:
		tmp = x * (y_m * t_m)
	elif x <= 5200000000000.0:
		tmp = t_2
	else:
		tmp = t_3
	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(y_m * Float64(z * Float64(-t_m)))
	t_3 = Float64(Float64(x * y_m) * t_m)
	tmp = 0.0
	if (x <= -7.8e+153)
		tmp = t_3;
	elseif (x <= -4.4e+55)
		tmp = t_2;
	elseif (x <= -5.5e-41)
		tmp = Float64(x * Float64(y_m * t_m));
	elseif (x <= 5200000000000.0)
		tmp = t_2;
	else
		tmp = t_3;
	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 = y_m * (z * -t_m);
	t_3 = (x * y_m) * t_m;
	tmp = 0.0;
	if (x <= -7.8e+153)
		tmp = t_3;
	elseif (x <= -4.4e+55)
		tmp = t_2;
	elseif (x <= -5.5e-41)
		tmp = x * (y_m * t_m);
	elseif (x <= 5200000000000.0)
		tmp = t_2;
	else
		tmp = t_3;
	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[(y$95$m * N[(z * (-t$95$m)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -7.8e+153], t$95$3, If[LessEqual[x, -4.4e+55], t$95$2, If[LessEqual[x, -5.5e-41], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5200000000000.0], t$95$2, t$95$3]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.79999999999999966e153 or 5.2e12 < x

   1. Initial program 86.8%

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

      1. distribute-rgt-out-- 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in x around inf 79.3%

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

      1. *-commutative 79.3%

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

   7. Simplified 79.3%

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

   if -7.79999999999999966e153 < x < -4.40000000000000021e55 or -5.50000000000000022e-41 < x < 5.2e12

   1. Initial program 93.6%

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

      1. distribute-rgt-out-- 94.3%

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

      2. associate-*l* 94.7%

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

      3. *-commutative 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in x around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. distribute-rgt-neg-out 78.6%

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

   7. Simplified 78.6%

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

   if -4.40000000000000021e55 < x < -5.50000000000000022e-41

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 91.9%

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

      2. associate-*l* 94.9%

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

      3. *-commutative 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around inf 99.5%

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

   6. Taylor expanded in z around 0 75.9%

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

      1. *-commutative 75.9%

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

   8. Simplified 75.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+153}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 5200000000000:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+178} \lor \neg \left(x \leq 3.2 \cdot 10^{+200}\right):\\ \;\;\;\;\left(x \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(t\_m \cdot \left(x - z\right)\right)\\ \end{array}\right) \end{array} \]

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 -5.2e+178) (not (<= x 3.2e+200)))
     (* (* x y_m) t_m)
     (* y_m (* t_m (- x z)))))))

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 <= -5.2e+178) || !(x <= 3.2e+200)) {
		tmp = (x * y_m) * t_m;
	} else {
		tmp = y_m * (t_m * (x - z));
	}
	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 <= (-5.2d+178)) .or. (.not. (x <= 3.2d+200))) then
        tmp = (x * y_m) * t_m
    else
        tmp = y_m * (t_m * (x - z))
    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 <= -5.2e+178) || !(x <= 3.2e+200)) {
		tmp = (x * y_m) * t_m;
	} else {
		tmp = y_m * (t_m * (x - z));
	}
	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 <= -5.2e+178) or not (x <= 3.2e+200):
		tmp = (x * y_m) * t_m
	else:
		tmp = y_m * (t_m * (x - z))
	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 <= -5.2e+178) || !(x <= 3.2e+200))
		tmp = Float64(Float64(x * y_m) * t_m);
	else
		tmp = Float64(y_m * Float64(t_m * Float64(x - z)));
	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 <= -5.2e+178) || ~((x <= 3.2e+200)))
		tmp = (x * y_m) * t_m;
	else
		tmp = y_m * (t_m * (x - z));
	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, -5.2e+178], N[Not[LessEqual[x, 3.2e+200]], $MachinePrecision]], N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], N[(y$95$m * N[(t$95$m * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -5.2000000000000001e178 or 3.20000000000000031e200 < x

   1. Initial program 84.6%

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

      1. distribute-rgt-out-- 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in x around inf 82.7%

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

      1. *-commutative 82.7%

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

   7. Simplified 82.7%

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

   if -5.2000000000000001e178 < x < 3.20000000000000031e200

   1. Initial program 92.5%

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

      1. distribute-rgt-out-- 94.4%

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

      2. associate-*l* 94.4%

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

      3. *-commutative 94.4%

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

   3. Simplified 94.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+178} \lor \neg \left(x \leq 3.2 \cdot 10^{+200}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 0.7× speedup

Math

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

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 30.0) (* y_m (* t_m (- x z))) (* (- 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 <= 30.0) {
		tmp = y_m * (t_m * (x - z));
	} 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 <= 30.0d0) then
        tmp = y_m * (t_m * (x - z))
    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 <= 30.0) {
		tmp = y_m * (t_m * (x - z));
	} 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 <= 30.0:
		tmp = y_m * (t_m * (x - z))
	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 <= 30.0)
		tmp = Float64(y_m * Float64(t_m * Float64(x - z)));
	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 <= 30.0)
		tmp = y_m * (t_m * (x - z));
	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, 30.0], N[(y$95$m * N[(t$95$m * N[(x - z), $MachinePrecision]), $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 < 30

   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* 96.5%

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

      3. *-commutative 96.5%

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

   3. Simplified 96.5%

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

   if 30 < t

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

      2. distribute-rgt-out-- 92.7%

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

      3. associate-*r* 97.0%

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

      4. *-commutative 97.0%

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

   3. Simplified 97.0%

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

4. Final simplification 96.6%

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

Alternative 8: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{+14}:\\ \;\;\;\;y\_m \cdot \left(x \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \end{array}\right) \end{array} \]

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 2.75e+14) (* y_m (* x t_m)) (* 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 tmp;
	if (t_m <= 2.75e+14) {
		tmp = y_m * (x * t_m);
	} 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) :: tmp
    if (t_m <= 2.75d+14) then
        tmp = y_m * (x * t_m)
    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 tmp;
	if (t_m <= 2.75e+14) {
		tmp = y_m * (x * t_m);
	} 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):
	tmp = 0
	if t_m <= 2.75e+14:
		tmp = y_m * (x * t_m)
	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)
	tmp = 0.0
	if (t_m <= 2.75e+14)
		tmp = Float64(y_m * Float64(x * t_m));
	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)
	tmp = 0.0;
	if (t_m <= 2.75e+14)
		tmp = y_m * (x * t_m);
	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_] := N[(t$95$s * N[(y$95$s * If[LessEqual[t$95$m, 2.75e+14], N[(y$95$m * N[(x * t$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.75e14

   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* 96.5%

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

      3. *-commutative 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in x around inf 54.1%

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

      1. associate-*r* 55.6%

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

      2. *-commutative 55.6%

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

   7. Simplified 55.6%

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

   if 2.75e14 < t

   1. Initial program 91.1%

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

      1. distribute-rgt-out-- 92.7%

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

      2. associate-*l* 84.8%

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

      3. *-commutative 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around inf 84.2%

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

   6. Taylor expanded in z around 0 61.0%

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

      1. *-commutative 61.0%

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

   8. Simplified 61.0%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.8% accurate, 1.3× speedup

Math

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

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 z))))))

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 - z))));
}

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 - z))))
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 - z))));
}

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 - z))))

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 * Float64(x - z)))))
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 - z))));
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 * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

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

Alternative 10: 54.0% accurate, 1.8× speedup

Math

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

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 91.0%

   \[\left(x \cdot y - z \cdot y\right) \cdot t \]
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* 93.6%

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

   3. *-commutative 93.6%

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

3. Simplified 93.6%

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

5. Taylor expanded in x around inf 83.1%

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

6. Taylor expanded in z around 0 55.4%

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

   1. *-commutative 55.4%

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

8. Simplified 55.4%

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

Developer target: 95.6% 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 2024107 
(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))