Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.4% → 98.4%

Time: 9.1s

Alternatives: 8

Speedup: 1.0×

• 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: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 10^{-9}:\\ \;\;\;\;y_m \cdot \left(t_m \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t_m \cdot y_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (* y_s (if (<= t_m 1e-9) (* y_m (* t_m (- x z))) (* (- x z) (* t_m y_m))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1e-9) {
		tmp = y_m * (t_m * (x - z));
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1d-9) then
        tmp = y_m * (t_m * (x - z))
    else
        tmp = (x - z) * (t_m * y_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 1e-9) {
		tmp = y_m * (t_m * (x - z));
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 1e-9:
		tmp = y_m * (t_m * (x - z))
	else:
		tmp = (x - z) * (t_m * y_m)
	return t_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
t_m = abs(t)
t_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 1e-9)
		tmp = Float64(y_m * Float64(t_m * Float64(x - z)));
	else
		tmp = Float64(Float64(x - z) * Float64(t_m * y_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 1e-9)
		tmp = y_m * (t_m * (x - z));
	else
		tmp = (x - z) * (t_m * y_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000006e-9

   1. Initial program 90.8%

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

      1. distribute-rgt-out-- 91.8%

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

      2. associate-*l* 90.6%

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

      3. *-commutative 90.6%

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

   3. Simplified 90.6%

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

   if 1.00000000000000006e-9 < t

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt-out-- 99.9%

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

      3. associate-*r* 96.5%

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

      4. *-commutative 96.5%

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

   3. Simplified 96.5%

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

4. Final simplification 91.9%

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

Alternative 2: 76.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := t_m \cdot \left(y_m \cdot \left(-z\right)\right)\\ t_3 := t_m \cdot \left(y_m \cdot x\right)\\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+59}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(t_m \cdot y_m\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (* y_m (- z)))) (t_3 (* t_m (* y_m x))))
   (*
    t_s
    (*
     y_s
     (if (<= x -1.95e+59)
       t_3
       (if (<= x -7.2e+32)
         t_2
         (if (<= x -4.1e-63)
           (* x (* t_m y_m))
           (if (<= x 4.1e+35) 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 = t_m * (y_m * -z);
	double t_3 = t_m * (y_m * x);
	double tmp;
	if (x <= -1.95e+59) {
		tmp = t_3;
	} else if (x <= -7.2e+32) {
		tmp = t_2;
	} else if (x <= -4.1e-63) {
		tmp = x * (t_m * y_m);
	} else if (x <= 4.1e+35) {
		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 = t_m * (y_m * -z)
    t_3 = t_m * (y_m * x)
    if (x <= (-1.95d+59)) then
        tmp = t_3
    else if (x <= (-7.2d+32)) then
        tmp = t_2
    else if (x <= (-4.1d-63)) then
        tmp = x * (t_m * y_m)
    else if (x <= 4.1d+35) 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 = t_m * (y_m * -z);
	double t_3 = t_m * (y_m * x);
	double tmp;
	if (x <= -1.95e+59) {
		tmp = t_3;
	} else if (x <= -7.2e+32) {
		tmp = t_2;
	} else if (x <= -4.1e-63) {
		tmp = x * (t_m * y_m);
	} else if (x <= 4.1e+35) {
		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 = t_m * (y_m * -z)
	t_3 = t_m * (y_m * x)
	tmp = 0
	if x <= -1.95e+59:
		tmp = t_3
	elif x <= -7.2e+32:
		tmp = t_2
	elif x <= -4.1e-63:
		tmp = x * (t_m * y_m)
	elif x <= 4.1e+35:
		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(t_m * Float64(y_m * Float64(-z)))
	t_3 = Float64(t_m * Float64(y_m * x))
	tmp = 0.0
	if (x <= -1.95e+59)
		tmp = t_3;
	elseif (x <= -7.2e+32)
		tmp = t_2;
	elseif (x <= -4.1e-63)
		tmp = Float64(x * Float64(t_m * y_m));
	elseif (x <= 4.1e+35)
		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 = t_m * (y_m * -z);
	t_3 = t_m * (y_m * x);
	tmp = 0.0;
	if (x <= -1.95e+59)
		tmp = t_3;
	elseif (x <= -7.2e+32)
		tmp = t_2;
	elseif (x <= -4.1e-63)
		tmp = x * (t_m * y_m);
	elseif (x <= 4.1e+35)
		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[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -1.95e+59], t$95$3, If[LessEqual[x, -7.2e+32], t$95$2, If[LessEqual[x, -4.1e-63], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.1e+35], t$95$2, t$95$3]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95000000000000011e59 or 4.0999999999999998e35 < x

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in x around inf 80.2%

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

      1. *-commutative 80.2%

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

   7. Simplified 80.2%

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

   if -1.95000000000000011e59 < x < -7.1999999999999994e32 or -4.0999999999999998e-63 < x < 4.0999999999999998e35

   1. Initial program 93.0%

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

      1. distribute-rgt-out-- 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 79.0%

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

      1. associate-*r* 79.0%

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

      2. *-commutative 79.0%

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

      3. mul-1-neg 79.0%

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

   7. Simplified 79.0%

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

   if -7.1999999999999994e32 < x < -4.0999999999999998e-63

   1. Initial program 99.8%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

      1. flip-- 89.2%

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

      2. associate-*r/ 89.1%

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

      3. pow2 89.1%

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

      4. pow2 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. associate-/l* 89.1%

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

      2. unpow2 89.1%

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

      3. unpow2 89.1%

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

      4. difference-of-squares 89.1%

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

      5. associate-/r* 99.6%

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

      6. *-inverses 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{y}{\frac{1}{x - z}}} \cdot t \]
   9. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. clear-num 98.9%

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

      3. sub-neg 98.9%

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

      4. add-sqr-sqrt 34.9%

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

      5. sqrt-unprod 76.7%

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

      6. sqr-neg 76.7%

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

      7. sqrt-unprod 46.8%

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

      8. add-sqr-sqrt 70.0%

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

   10. Applied egg-rr 70.0%

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

   11. Taylor expanded in x around inf 70.8%

       \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{x}}}{y \cdot t}} \]
   12. Step-by-step derivation

       1. clear-num 71.2%

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

       2. div-inv 71.5%

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

       3. remove-double-div 71.6%

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

       4. *-commutative 71.6%

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

   13. Applied egg-rr 71.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+59}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+32}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+35}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.76 \cdot 10^{-62} \lor \neg \left(x \leq 1.2 \cdot 10^{+16}\right):\\ \;\;\;\;t_m \cdot \left(y_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \left(t_m \cdot \left(-z\right)\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (*
   y_s
   (if (or (<= x -1.76e-62) (not (<= x 1.2e+16)))
     (* t_m (* y_m x))
     (* y_m (* t_m (- 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 <= -1.76e-62) || !(x <= 1.2e+16)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = y_m * (t_m * -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 <= (-1.76d-62)) .or. (.not. (x <= 1.2d+16))) then
        tmp = t_m * (y_m * x)
    else
        tmp = y_m * (t_m * -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 <= -1.76e-62) || !(x <= 1.2e+16)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = y_m * (t_m * -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 <= -1.76e-62) or not (x <= 1.2e+16):
		tmp = t_m * (y_m * x)
	else:
		tmp = y_m * (t_m * -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 <= -1.76e-62) || !(x <= 1.2e+16))
		tmp = Float64(t_m * Float64(y_m * x));
	else
		tmp = Float64(y_m * Float64(t_m * Float64(-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 <= -1.76e-62) || ~((x <= 1.2e+16)))
		tmp = t_m * (y_m * x);
	else
		tmp = y_m * (t_m * -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, -1.76e-62], N[Not[LessEqual[x, 1.2e+16]], $MachinePrecision]], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(t$95$m * (-z)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -1.76e-62 or 1.2e16 < x

   1. Initial program 92.7%

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

      1. distribute-rgt-out-- 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in x around inf 75.3%

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

      1. *-commutative 75.3%

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

   7. Simplified 75.3%

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

   if -1.76e-62 < x < 1.2e16

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 93.0%

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

      2. associate-*l* 89.1%

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

      3. *-commutative 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. distribute-rgt-neg-out 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.76 \cdot 10^{-62} \lor \neg \left(x \leq 1.2 \cdot 10^{+16}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-65} \lor \neg \left(x \leq 2.7 \cdot 10^{+35}\right):\\ \;\;\;\;t_m \cdot \left(y_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t_m \cdot \left(-y_m\right)\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (*
   y_s
   (if (or (<= x -4.9e-65) (not (<= x 2.7e+35)))
     (* t_m (* y_m x))
     (* z (* t_m (- y_m)))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((x <= -4.9e-65) || !(x <= 2.7e+35)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = z * (t_m * -y_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((x <= (-4.9d-65)) .or. (.not. (x <= 2.7d+35))) then
        tmp = t_m * (y_m * x)
    else
        tmp = z * (t_m * -y_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((x <= -4.9e-65) || !(x <= 2.7e+35)) {
		tmp = t_m * (y_m * x);
	} else {
		tmp = z * (t_m * -y_m);
	}
	return t_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if (x <= -4.9e-65) or not (x <= 2.7e+35):
		tmp = t_m * (y_m * x)
	else:
		tmp = z * (t_m * -y_m)
	return t_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
t_m = abs(t)
t_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if ((x <= -4.9e-65) || !(x <= 2.7e+35))
		tmp = Float64(t_m * Float64(y_m * x));
	else
		tmp = Float64(z * Float64(t_m * Float64(-y_m)));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if ((x <= -4.9e-65) || ~((x <= 2.7e+35)))
		tmp = t_m * (y_m * x);
	else
		tmp = z * (t_m * -y_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[Or[LessEqual[x, -4.9e-65], N[Not[LessEqual[x, 2.7e+35]], $MachinePrecision]], N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(t$95$m * (-y$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -4.89999999999999964e-65 or 2.70000000000000003e35 < x

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

   7. Simplified 76.2%

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

   if -4.89999999999999964e-65 < x < 2.70000000000000003e35

   1. Initial program 93.3%

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

      1. distribute-rgt-out-- 93.3%

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

      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 79.2%

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

      1. mul-1-neg 79.2%

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

      2. distribute-rgt-neg-in 79.2%

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

      3. distribute-rgt-neg-out 79.2%

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

      4. associate-*l* 80.0%

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

   7. Simplified 80.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-65} \lor \neg \left(x \leq 2.7 \cdot 10^{+35}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+246}:\\ \;\;\;\;t_m \cdot \left(y_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \left(t_m \cdot \left(x - z\right)\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (* y_s (if (<= z -4.5e+246) (* t_m (* y_m (- z))) (* 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 (z <= -4.5e+246) {
		tmp = t_m * (y_m * -z);
	} 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 (z <= (-4.5d+246)) then
        tmp = t_m * (y_m * -z)
    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 (z <= -4.5e+246) {
		tmp = t_m * (y_m * -z);
	} 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 z <= -4.5e+246:
		tmp = t_m * (y_m * -z)
	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 (z <= -4.5e+246)
		tmp = Float64(t_m * Float64(y_m * Float64(-z)));
	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 (z <= -4.5e+246)
		tmp = t_m * (y_m * -z);
	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[LessEqual[z, -4.5e+246], N[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $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 z < -4.5e246

   1. Initial program 84.7%

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

      1. distribute-rgt-out-- 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 84.7%

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

      1. associate-*r* 84.7%

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

      2. *-commutative 84.7%

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

      3. mul-1-neg 84.7%

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

   7. Simplified 84.7%

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

   if -4.5e246 < z

   1. Initial program 93.4%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 90.9%

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

      3. *-commutative 90.9%

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

   3. Simplified 90.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+246}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5 \cdot 10^{+17}:\\ \;\;\;\;y_m \cdot \left(t_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t_m \cdot y_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (* t_s (* y_s (if (<= t_m 5e+17) (* y_m (* t_m x)) (* x (* t_m y_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 <= 5e+17) {
		tmp = y_m * (t_m * x);
	} else {
		tmp = x * (t_m * y_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 <= 5d+17) then
        tmp = y_m * (t_m * x)
    else
        tmp = x * (t_m * y_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 <= 5e+17) {
		tmp = y_m * (t_m * x);
	} else {
		tmp = x * (t_m * y_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 <= 5e+17:
		tmp = y_m * (t_m * x)
	else:
		tmp = x * (t_m * y_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 <= 5e+17)
		tmp = Float64(y_m * Float64(t_m * x));
	else
		tmp = Float64(x * Float64(t_m * y_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 <= 5e+17)
		tmp = y_m * (t_m * x);
	else
		tmp = x * (t_m * y_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, 5e+17], N[(y$95$m * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5e17

   1. Initial program 90.9%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 90.7%

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

      3. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in x around inf 52.8%

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

      1. associate-*r* 52.6%

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

      2. *-commutative 52.6%

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

   7. Simplified 52.6%

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

   if 5e17 < t

   1. Initial program 99.9%

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

      1. distribute-rgt-out-- 99.9%

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

   3. Simplified 99.9%

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

      1. flip-- 73.0%

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

      2. associate-*r/ 72.9%

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

      3. pow2 72.9%

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

      4. pow2 72.9%

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

   6. Applied egg-rr 72.9%

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

      1. associate-/l* 72.9%

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

      2. unpow2 72.9%

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

      3. unpow2 72.9%

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

      4. difference-of-squares 76.8%

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

      5. associate-/r* 99.7%

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

      6. *-inverses 99.7%

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

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{y}{\frac{1}{x - z}}} \cdot t \]
   9. Step-by-step derivation

      1. associate-*l/ 96.2%

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

      2. clear-num 96.3%

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

      3. sub-neg 96.3%

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

      4. add-sqr-sqrt 50.1%

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

      5. sqrt-unprod 65.5%

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

      6. sqr-neg 65.5%

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

      7. sqrt-unprod 24.2%

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

      8. add-sqr-sqrt 50.4%

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

   10. Applied egg-rr 50.4%

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

   11. Taylor expanded in x around inf 64.1%

       \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{x}}}{y \cdot t}} \]
   12. Step-by-step derivation

       1. clear-num 64.1%

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

       2. div-inv 64.1%

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

       3. remove-double-div 64.2%

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

       4. *-commutative 64.2%

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

   13. Applied egg-rr 64.2%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l* 90.1%

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

   3. *-commutative 90.1%

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

3. Simplified 90.1%

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

5. Taylor expanded in x around inf 54.5%

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

   1. associate-*r* 53.1%

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

   2. *-commutative 53.1%

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

7. Simplified 53.1%

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

8. Final simplification 53.1%

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

Alternative 8: 55.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Taylor expanded in x around inf 54.5%

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

   1. *-commutative 54.5%

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

7. Simplified 54.5%

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

8. Final simplification 54.5%

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

Developer target: 95.6% accurate, 0.8× 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 2024010 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

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

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