Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.2% → 98.1%

Time: 9.7s

Alternatives: 9

Speedup: 1.3×

• 25.5% 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: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-49}:\\ \;\;\;\;y\_m \cdot \left(\left(x - z\right) \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y\_m \cdot t\_m\right)\\ \end{array}\right) \end{array} \]

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 1.5e-49) (* 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 <= 1.5e-49) {
		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 <= 1.5d-49) 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 <= 1.5e-49) {
		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 <= 1.5e-49:
		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 <= 1.5e-49)
		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 <= 1.5e-49)
		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, 1.5e-49], 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 < 1.5e-49

   1. Initial program 89.8%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 93.0%

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

      3. *-commutative 93.0%

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

   3. Simplified 93.0%

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

   if 1.5e-49 < t

   1. Initial program 97.4%

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

      1. *-commutative 97.4%

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

      2. distribute-rgt-out-- 97.3%

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

      3. associate-*r* 97.2%

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

      4. *-commutative 97.2%

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

   3. Simplified 97.2%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-49}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := y\_m \cdot \left(\left(x - z\right) \cdot t\_m\right)\\ t_3 := t\_m \cdot \left(y\_m \cdot \left(-z\right)\right)\\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+126}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -7.9 \cdot 10^{-212}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-304}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+82}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array}\right) \end{array} \end{array} \]

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 (* (- x z) t_m))) (t_3 (* t_m (* y_m (- z)))))
   (*
    t_s
    (*
     y_s
     (if (<= z -7.6e+126)
       t_3
       (if (<= z -7.9e-212)
         t_2
         (if (<= z 4.4e-304) (* t_m (* y_m x)) (if (<= z 8e+82) 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 * ((x - z) * t_m);
	double t_3 = t_m * (y_m * -z);
	double tmp;
	if (z <= -7.6e+126) {
		tmp = t_3;
	} else if (z <= -7.9e-212) {
		tmp = t_2;
	} else if (z <= 4.4e-304) {
		tmp = t_m * (y_m * x);
	} else if (z <= 8e+82) {
		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 * ((x - z) * t_m)
    t_3 = t_m * (y_m * -z)
    if (z <= (-7.6d+126)) then
        tmp = t_3
    else if (z <= (-7.9d-212)) then
        tmp = t_2
    else if (z <= 4.4d-304) then
        tmp = t_m * (y_m * x)
    else if (z <= 8d+82) 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 * ((x - z) * t_m);
	double t_3 = t_m * (y_m * -z);
	double tmp;
	if (z <= -7.6e+126) {
		tmp = t_3;
	} else if (z <= -7.9e-212) {
		tmp = t_2;
	} else if (z <= 4.4e-304) {
		tmp = t_m * (y_m * x);
	} else if (z <= 8e+82) {
		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 * ((x - z) * t_m)
	t_3 = t_m * (y_m * -z)
	tmp = 0
	if z <= -7.6e+126:
		tmp = t_3
	elif z <= -7.9e-212:
		tmp = t_2
	elif z <= 4.4e-304:
		tmp = t_m * (y_m * x)
	elif z <= 8e+82:
		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(Float64(x - z) * t_m))
	t_3 = Float64(t_m * Float64(y_m * Float64(-z)))
	tmp = 0.0
	if (z <= -7.6e+126)
		tmp = t_3;
	elseif (z <= -7.9e-212)
		tmp = t_2;
	elseif (z <= 4.4e-304)
		tmp = Float64(t_m * Float64(y_m * x));
	elseif (z <= 8e+82)
		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 * ((x - z) * t_m);
	t_3 = t_m * (y_m * -z);
	tmp = 0.0;
	if (z <= -7.6e+126)
		tmp = t_3;
	elseif (z <= -7.9e-212)
		tmp = t_2;
	elseif (z <= 4.4e-304)
		tmp = t_m * (y_m * x);
	elseif (z <= 8e+82)
		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[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[z, -7.6e+126], t$95$3, If[LessEqual[z, -7.9e-212], t$95$2, If[LessEqual[z, 4.4e-304], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8e+82], t$95$2, t$95$3]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.60000000000000033e126 or 7.9999999999999997e82 < z

   1. Initial program 94.3%

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

      1. distribute-rgt-out-- 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in x around 0 91.0%

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

      1. associate-*r* 91.0%

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

      2. mul-1-neg 91.0%

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

   7. Simplified 91.0%

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

   if -7.60000000000000033e126 < z < -7.9000000000000004e-212 or 4.4e-304 < z < 7.9999999999999997e82

   1. Initial program 93.3%

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

      1. distribute-rgt-out-- 95.3%

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

      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 -7.9000000000000004e-212 < z < 4.4e-304

   1. Initial program 74.1%

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

      1. distribute-rgt-out-- 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in x around inf 74.1%

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

      1. *-commutative 74.1%

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

   7. Simplified 74.1%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+126}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq -7.9 \cdot 10^{-212}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-304}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+82}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+37} \lor \neg \left(z \leq 5.3 \cdot 10^{+82}\right):\\ \;\;\;\;y\_m \cdot \left(t\_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \end{array}\right) \end{array} \]

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 (or (<= z -1.5e+37) (not (<= z 5.3e+82)))
     (* y_m (* t_m (- z)))
     (* 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 ((z <= -1.5e+37) || !(z <= 5.3e+82)) {
		tmp = y_m * (t_m * -z);
	} 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 ((z <= (-1.5d+37)) .or. (.not. (z <= 5.3d+82))) then
        tmp = y_m * (t_m * -z)
    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 ((z <= -1.5e+37) || !(z <= 5.3e+82)) {
		tmp = y_m * (t_m * -z);
	} 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 (z <= -1.5e+37) or not (z <= 5.3e+82):
		tmp = y_m * (t_m * -z)
	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 ((z <= -1.5e+37) || !(z <= 5.3e+82))
		tmp = Float64(y_m * Float64(t_m * Float64(-z)));
	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 ((z <= -1.5e+37) || ~((z <= 5.3e+82)))
		tmp = y_m * (t_m * -z);
	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[Or[LessEqual[z, -1.5e+37], N[Not[LessEqual[z, 5.3e+82]], $MachinePrecision]], N[(y$95$m * N[(t$95$m * (-z)), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000011e37 or 5.29999999999999977e82 < z

   1. Initial program 95.3%

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

      1. distribute-rgt-out-- 96.2%

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

      2. associate-*l* 88.0%

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

      3. *-commutative 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 81.5%

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

      1. mul-1-neg 81.5%

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

      2. distribute-rgt-neg-out 81.5%

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

   7. Simplified 81.5%

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

   if -1.50000000000000011e37 < z < 5.29999999999999977e82

   1. Initial program 89.7%

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

      1. distribute-rgt-out-- 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in x around inf 74.6%

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

      1. *-commutative 74.6%

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

   7. Simplified 74.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+37} \lor \neg \left(z \leq 5.3 \cdot 10^{+82}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+37} \lor \neg \left(z \leq 7.8 \cdot 10^{+48}\right):\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \end{array}\right) \end{array} \]

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 (or (<= z -1.05e+37) (not (<= z 7.8e+48)))
     (* (* y_m t_m) (- z))
     (* 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 ((z <= -1.05e+37) || !(z <= 7.8e+48)) {
		tmp = (y_m * t_m) * -z;
	} 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 ((z <= (-1.05d+37)) .or. (.not. (z <= 7.8d+48))) then
        tmp = (y_m * t_m) * -z
    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 ((z <= -1.05e+37) || !(z <= 7.8e+48)) {
		tmp = (y_m * t_m) * -z;
	} 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 (z <= -1.05e+37) or not (z <= 7.8e+48):
		tmp = (y_m * t_m) * -z
	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 ((z <= -1.05e+37) || !(z <= 7.8e+48))
		tmp = Float64(Float64(y_m * t_m) * Float64(-z));
	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 ((z <= -1.05e+37) || ~((z <= 7.8e+48)))
		tmp = (y_m * t_m) * -z;
	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[Or[LessEqual[z, -1.05e+37], N[Not[LessEqual[z, 7.8e+48]], $MachinePrecision]], N[(N[(y$95$m * t$95$m), $MachinePrecision] * (-z)), $MachinePrecision], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.0500000000000001e37 or 7.8000000000000002e48 < z

   1. Initial program 93.9%

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

      1. distribute-rgt-out-- 95.7%

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

      2. associate-*l* 88.8%

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

      3. *-commutative 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in x around 0 87.0%

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

      1. mul-1-neg 87.0%

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

      2. distribute-rgt-neg-in 87.0%

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

      3. distribute-rgt-neg-out 87.0%

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

      4. associate-*l* 87.3%

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

   7. Simplified 87.3%

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

   if -1.0500000000000001e37 < z < 7.8000000000000002e48

   1. Initial program 90.5%

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

      1. distribute-rgt-out-- 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+37} \lor \neg \left(z \leq 7.8 \cdot 10^{+48}\right):\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+36} \lor \neg \left(z \leq 8.8 \cdot 10^{+52}\right):\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \end{array}\right) \end{array} \]

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 (or (<= z -9e+36) (not (<= z 8.8e+52)))
     (* t_m (* y_m (- z)))
     (* 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 ((z <= -9e+36) || !(z <= 8.8e+52)) {
		tmp = t_m * (y_m * -z);
	} 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 ((z <= (-9d+36)) .or. (.not. (z <= 8.8d+52))) then
        tmp = t_m * (y_m * -z)
    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 ((z <= -9e+36) || !(z <= 8.8e+52)) {
		tmp = t_m * (y_m * -z);
	} 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 (z <= -9e+36) or not (z <= 8.8e+52):
		tmp = t_m * (y_m * -z)
	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 ((z <= -9e+36) || !(z <= 8.8e+52))
		tmp = Float64(t_m * Float64(y_m * Float64(-z)));
	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 ((z <= -9e+36) || ~((z <= 8.8e+52)))
		tmp = t_m * (y_m * -z);
	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[Or[LessEqual[z, -9e+36], N[Not[LessEqual[z, 8.8e+52]], $MachinePrecision]], N[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999994e36 or 8.7999999999999999e52 < z

   1. Initial program 93.9%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 87.0%

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

      1. associate-*r* 87.0%

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

      2. mul-1-neg 87.0%

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

   7. Simplified 87.0%

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

   if -8.99999999999999994e36 < z < 8.7999999999999999e52

   1. Initial program 90.5%

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

      1. distribute-rgt-out-- 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+36} \lor \neg \left(z \leq 8.8 \cdot 10^{+52}\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;y\_m \cdot \left(x \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \end{array}\right) \end{array} \]

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 4.2e-54) (* 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 <= 4.2e-54) {
		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 <= 4.2d-54) 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 <= 4.2e-54) {
		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 <= 4.2e-54:
		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 <= 4.2e-54)
		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 <= 4.2e-54)
		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, 4.2e-54], 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 < 4.2e-54

   1. Initial program 89.8%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 93.0%

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

      3. *-commutative 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around inf 57.6%

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

      1. associate-*r* 61.0%

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

      2. *-commutative 61.0%

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

   7. Simplified 61.0%

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

   if 4.2e-54 < t

   1. Initial program 97.4%

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

      1. distribute-rgt-out-- 97.4%

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

   3. Simplified 97.4%

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

      1. add-cube-cbrt 96.2%

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

      2. pow3 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in z around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*r* 82.1%

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

      3. rem-cube-cbrt 82.1%

         \[\leadsto z \cdot \left(t \cdot \left(y \cdot \color{blue}{{\left(\sqrt[3]{-1}\right)}^{3}}\right) + \frac{t \cdot \left(x \cdot y\right)}{z}\right) \]

      4. associate-/l* 84.8%

         \[\leadsto z \cdot \left(t \cdot \left(y \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right) + \color{blue}{t \cdot \frac{x \cdot y}{z}}\right) \]

      5. distribute-lft-out 90.2%

         \[\leadsto z \cdot \color{blue}{\left(t \cdot \left(y \cdot {\left(\sqrt[3]{-1}\right)}^{3} + \frac{x \cdot y}{z}\right)\right)} \]

      6. rem-cube-cbrt 90.2%

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

      7. *-commutative 90.2%

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

      8. associate-/l* 89.2%

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

      9. distribute-lft-out 89.2%

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

   9. Simplified 89.2%

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

   10. Taylor expanded in z around 0 42.7%

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

       1. *-commutative 42.7%

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

       2. associate-*r* 41.4%

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

   12. Simplified 41.4%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-54}:\\ \;\;\;\;t\_m \cdot \left(y\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot t\_m\right)\\ \end{array}\right) \end{array} \]

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 2.45e-54) (* t_m (* y_m x)) (* 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 <= 2.45e-54) {
		tmp = t_m * (y_m * x);
	} 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 <= 2.45d-54) then
        tmp = t_m * (y_m * x)
    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 <= 2.45e-54) {
		tmp = t_m * (y_m * x);
	} 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 <= 2.45e-54:
		tmp = t_m * (y_m * x)
	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 <= 2.45e-54)
		tmp = Float64(t_m * Float64(y_m * x));
	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 <= 2.45e-54)
		tmp = t_m * (y_m * x);
	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, 2.45e-54], N[(t$95$m * N[(y$95$m * x), $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 < 2.4500000000000001e-54

   1. Initial program 89.8%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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 57.6%

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

      1. *-commutative 57.6%

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

   7. Simplified 57.6%

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

   if 2.4500000000000001e-54 < t

   1. Initial program 97.4%

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

      1. distribute-rgt-out-- 97.4%

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

   3. Simplified 97.4%

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

      1. add-cube-cbrt 96.2%

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

      2. pow3 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in z around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*r* 82.1%

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

      3. rem-cube-cbrt 82.1%

         \[\leadsto z \cdot \left(t \cdot \left(y \cdot \color{blue}{{\left(\sqrt[3]{-1}\right)}^{3}}\right) + \frac{t \cdot \left(x \cdot y\right)}{z}\right) \]

      4. associate-/l* 84.8%

         \[\leadsto z \cdot \left(t \cdot \left(y \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right) + \color{blue}{t \cdot \frac{x \cdot y}{z}}\right) \]

      5. distribute-lft-out 90.2%

         \[\leadsto z \cdot \color{blue}{\left(t \cdot \left(y \cdot {\left(\sqrt[3]{-1}\right)}^{3} + \frac{x \cdot y}{z}\right)\right)} \]

      6. rem-cube-cbrt 90.2%

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

      7. *-commutative 90.2%

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

      8. associate-/l* 89.2%

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

      9. distribute-lft-out 89.2%

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

   9. Simplified 89.2%

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

   10. Taylor expanded in z around 0 42.7%

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

       1. *-commutative 42.7%

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

       2. associate-*r* 41.4%

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

   12. Simplified 41.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{-54}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% accurate, 1.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #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 92.0%

   \[\left(x \cdot y - z \cdot y\right) \cdot t \]
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. Final simplification 93.6%

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

Alternative 9: 50.4% accurate, 1.8× speedup

Math

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

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 (* 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) {
	return t_s * (y_s * (y_m * (x * 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 * 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 * 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 * 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(y_m * Float64(x * 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 * 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[(y$95$m * N[(x * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.0%

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

   1. distribute-rgt-out-- 93.6%

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

   2. associate-*l* 92.0%

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

   3. *-commutative 92.0%

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

3. Simplified 92.0%

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

5. Taylor expanded in x around inf 53.3%

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

   1. associate-*r* 54.6%

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

   2. *-commutative 54.6%

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

7. Simplified 54.6%

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

8. Final simplification 54.6%

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

Developer target: 96.0% 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 2024075 
(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))