Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.6% → 98.5%

Time: 8.7s

Alternatives: 9

Speedup: 1.3×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2 \cdot 10^{-8}:\\ \;\;\;\;y_m \cdot \left(\left(x - z\right) \cdot t_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y_m \cdot t_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (* y_s (if (<= t_m 2e-8) (* 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 <= 2e-8) {
		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 <= 2d-8) 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 <= 2e-8) {
		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 <= 2e-8:
		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 <= 2e-8)
		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 <= 2e-8)
		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, 2e-8], 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 < 2e-8

   1. Initial program 88.4%

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

      1. distribute-rgt-out-- 89.0%

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

      2. associate-*l* 93.9%

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

      3. *-commutative 93.9%

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

   3. Simplified 93.9%

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

   if 2e-8 < t

   1. Initial program 97.6%

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

      1. *-commutative 97.6%

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

      2. distribute-rgt-out-- 97.6%

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

      3. associate-*r* 97.1%

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

      4. *-commutative 97.1%

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

   3. Simplified 97.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := t_m \cdot \left(y_m \cdot x\right)\\ t_3 := \left(y_m \cdot t_m\right) \cdot \left(-z\right)\\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-73}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-175}:\\ \;\;\;\;y_m \cdot \left(x \cdot t_m\right)\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array}\right) \end{array} \end{array} \]

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 x))) (t_3 (* (* y_m t_m) (- z))))
   (*
    t_s
    (*
     y_s
     (if (<= z -3.7e-7)
       t_3
       (if (<= z -9e-46)
         t_2
         (if (<= z -3.3e-73)
           t_3
           (if (<= z 3.9e-175)
             (* y_m (* x t_m))
             (if (<= z 1.18e-29) 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 * x);
	double t_3 = (y_m * t_m) * -z;
	double tmp;
	if (z <= -3.7e-7) {
		tmp = t_3;
	} else if (z <= -9e-46) {
		tmp = t_2;
	} else if (z <= -3.3e-73) {
		tmp = t_3;
	} else if (z <= 3.9e-175) {
		tmp = y_m * (x * t_m);
	} else if (z <= 1.18e-29) {
		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 * x)
    t_3 = (y_m * t_m) * -z
    if (z <= (-3.7d-7)) then
        tmp = t_3
    else if (z <= (-9d-46)) then
        tmp = t_2
    else if (z <= (-3.3d-73)) then
        tmp = t_3
    else if (z <= 3.9d-175) then
        tmp = y_m * (x * t_m)
    else if (z <= 1.18d-29) 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 * x);
	double t_3 = (y_m * t_m) * -z;
	double tmp;
	if (z <= -3.7e-7) {
		tmp = t_3;
	} else if (z <= -9e-46) {
		tmp = t_2;
	} else if (z <= -3.3e-73) {
		tmp = t_3;
	} else if (z <= 3.9e-175) {
		tmp = y_m * (x * t_m);
	} else if (z <= 1.18e-29) {
		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 * x)
	t_3 = (y_m * t_m) * -z
	tmp = 0
	if z <= -3.7e-7:
		tmp = t_3
	elif z <= -9e-46:
		tmp = t_2
	elif z <= -3.3e-73:
		tmp = t_3
	elif z <= 3.9e-175:
		tmp = y_m * (x * t_m)
	elif z <= 1.18e-29:
		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 * x))
	t_3 = Float64(Float64(y_m * t_m) * Float64(-z))
	tmp = 0.0
	if (z <= -3.7e-7)
		tmp = t_3;
	elseif (z <= -9e-46)
		tmp = t_2;
	elseif (z <= -3.3e-73)
		tmp = t_3;
	elseif (z <= 3.9e-175)
		tmp = Float64(y_m * Float64(x * t_m));
	elseif (z <= 1.18e-29)
		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 * x);
	t_3 = (y_m * t_m) * -z;
	tmp = 0.0;
	if (z <= -3.7e-7)
		tmp = t_3;
	elseif (z <= -9e-46)
		tmp = t_2;
	elseif (z <= -3.3e-73)
		tmp = t_3;
	elseif (z <= 3.9e-175)
		tmp = y_m * (x * t_m);
	elseif (z <= 1.18e-29)
		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 * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y$95$m * t$95$m), $MachinePrecision] * (-z)), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[z, -3.7e-7], t$95$3, If[LessEqual[z, -9e-46], t$95$2, If[LessEqual[z, -3.3e-73], t$95$3, If[LessEqual[z, 3.9e-175], N[(y$95$m * N[(x * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.18e-29], t$95$2, t$95$3]]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.70000000000000004e-7 or -9.00000000000000001e-46 < z < -3.30000000000000004e-73 or 1.17999999999999996e-29 < z

   1. Initial program 88.4%

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

      1. distribute-rgt-out-- 89.2%

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

      2. associate-*l* 92.8%

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

      3. *-commutative 92.8%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in x around 0 70.8%

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

      1. mul-1-neg 70.8%

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

      2. distribute-rgt-neg-in 70.8%

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

      3. distribute-rgt-neg-out 70.8%

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

      4. associate-*l* 76.0%

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

   7. Simplified 76.0%

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

   if -3.70000000000000004e-7 < z < -9.00000000000000001e-46 or 3.89999999999999998e-175 < z < 1.17999999999999996e-29

   1. Initial program 90.8%

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

      1. distribute-rgt-out-- 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in x around inf 81.4%

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

      1. *-commutative 81.4%

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

   7. Simplified 81.4%

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

   if -3.30000000000000004e-73 < z < 3.89999999999999998e-175

   1. Initial program 95.8%

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

      1. distribute-rgt-out-- 95.8%

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

      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

   5. Taylor expanded in x around inf 88.8%

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

      1. associate-*r* 86.4%

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

      2. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-46}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-73}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-175}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-29}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := t_m \cdot \left(y_m \cdot x\right)\\ t_3 := t_m \cdot \left(y_m \cdot \left(-z\right)\right)\\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-73}:\\ \;\;\;\;\left(y_m \cdot t_m\right) \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-175}:\\ \;\;\;\;y_m \cdot \left(x \cdot t_m\right)\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array}\right) \end{array} \end{array} \]

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 x))) (t_3 (* t_m (* y_m (- z)))))
   (*
    t_s
    (*
     y_s
     (if (<= z -3.1e-7)
       t_3
       (if (<= z -8e-46)
         t_2
         (if (<= z -5.6e-73)
           (* (* y_m t_m) (- z))
           (if (<= z 8.2e-175)
             (* y_m (* x t_m))
             (if (<= z 1.18e-29) 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 * x);
	double t_3 = t_m * (y_m * -z);
	double tmp;
	if (z <= -3.1e-7) {
		tmp = t_3;
	} else if (z <= -8e-46) {
		tmp = t_2;
	} else if (z <= -5.6e-73) {
		tmp = (y_m * t_m) * -z;
	} else if (z <= 8.2e-175) {
		tmp = y_m * (x * t_m);
	} else if (z <= 1.18e-29) {
		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 * x)
    t_3 = t_m * (y_m * -z)
    if (z <= (-3.1d-7)) then
        tmp = t_3
    else if (z <= (-8d-46)) then
        tmp = t_2
    else if (z <= (-5.6d-73)) then
        tmp = (y_m * t_m) * -z
    else if (z <= 8.2d-175) then
        tmp = y_m * (x * t_m)
    else if (z <= 1.18d-29) 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 * x);
	double t_3 = t_m * (y_m * -z);
	double tmp;
	if (z <= -3.1e-7) {
		tmp = t_3;
	} else if (z <= -8e-46) {
		tmp = t_2;
	} else if (z <= -5.6e-73) {
		tmp = (y_m * t_m) * -z;
	} else if (z <= 8.2e-175) {
		tmp = y_m * (x * t_m);
	} else if (z <= 1.18e-29) {
		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 * x)
	t_3 = t_m * (y_m * -z)
	tmp = 0
	if z <= -3.1e-7:
		tmp = t_3
	elif z <= -8e-46:
		tmp = t_2
	elif z <= -5.6e-73:
		tmp = (y_m * t_m) * -z
	elif z <= 8.2e-175:
		tmp = y_m * (x * t_m)
	elif z <= 1.18e-29:
		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 * x))
	t_3 = Float64(t_m * Float64(y_m * Float64(-z)))
	tmp = 0.0
	if (z <= -3.1e-7)
		tmp = t_3;
	elseif (z <= -8e-46)
		tmp = t_2;
	elseif (z <= -5.6e-73)
		tmp = Float64(Float64(y_m * t_m) * Float64(-z));
	elseif (z <= 8.2e-175)
		tmp = Float64(y_m * Float64(x * t_m));
	elseif (z <= 1.18e-29)
		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 * x);
	t_3 = t_m * (y_m * -z);
	tmp = 0.0;
	if (z <= -3.1e-7)
		tmp = t_3;
	elseif (z <= -8e-46)
		tmp = t_2;
	elseif (z <= -5.6e-73)
		tmp = (y_m * t_m) * -z;
	elseif (z <= 8.2e-175)
		tmp = y_m * (x * t_m);
	elseif (z <= 1.18e-29)
		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 * x), $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, -3.1e-7], t$95$3, If[LessEqual[z, -8e-46], t$95$2, If[LessEqual[z, -5.6e-73], N[(N[(y$95$m * t$95$m), $MachinePrecision] * (-z)), $MachinePrecision], If[LessEqual[z, 8.2e-175], N[(y$95$m * N[(x * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.18e-29], t$95$2, t$95$3]]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.1e-7 or 1.17999999999999996e-29 < z

   1. Initial program 87.8%

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

      1. distribute-rgt-out-- 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in x around 0 70.4%

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

      1. associate-*r* 70.4%

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

      2. *-commutative 70.4%

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

      3. mul-1-neg 70.4%

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

   7. Simplified 70.4%

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

   if -3.1e-7 < z < -8.00000000000000018e-46 or 8.19999999999999997e-175 < z < 1.17999999999999996e-29

   1. Initial program 90.8%

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

      1. distribute-rgt-out-- 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in x around inf 81.4%

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

      1. *-commutative 81.4%

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

   7. Simplified 81.4%

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

   if -8.00000000000000018e-46 < z < -5.60000000000000023e-73

   1. Initial program 99.6%

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

      1. distribute-rgt-out-- 99.3%

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

      2. associate-*l* 86.4%

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

      3. *-commutative 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 80.4%

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

      1. mul-1-neg 80.4%

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

      2. distribute-rgt-neg-in 80.4%

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

      3. distribute-rgt-neg-out 80.4%

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

      4. associate-*l* 80.8%

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

   7. Simplified 80.8%

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

   if -5.60000000000000023e-73 < z < 8.19999999999999997e-175

   1. Initial program 95.8%

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

      1. distribute-rgt-out-- 95.8%

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

      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

   5. Taylor expanded in x around inf 88.8%

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

      1. associate-*r* 86.4%

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

      2. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-46}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-73}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-175}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-29}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := y_m \cdot \left(z \cdot \left(-t_m\right)\right)\\ t_3 := t_m \cdot \left(y_m \cdot x\right)\\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+110}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \left(y_m \cdot t_m\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array}\right) \end{array} \end{array} \]

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 (* y_m (* z (- t_m)))) (t_3 (* t_m (* y_m x))))
   (*
    t_s
    (*
     y_s
     (if (<= x -1.3e+110)
       t_3
       (if (<= x -1.85e+71)
         t_2
         (if (<= x -3.5e-48)
           (* x (* y_m t_m))
           (if (<= x 2.55e-42) t_2 t_3))))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
t_m = fabs(t);
t_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = y_m * (z * -t_m);
	double t_3 = t_m * (y_m * x);
	double tmp;
	if (x <= -1.3e+110) {
		tmp = t_3;
	} else if (x <= -1.85e+71) {
		tmp = t_2;
	} else if (x <= -3.5e-48) {
		tmp = x * (y_m * t_m);
	} else if (x <= 2.55e-42) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return t_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
t_m = abs(t)
t_s = copysign(1.0d0, t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(t_s, y_s, x, y_m, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = y_m * (z * -t_m)
    t_3 = t_m * (y_m * x)
    if (x <= (-1.3d+110)) then
        tmp = t_3
    else if (x <= (-1.85d+71)) then
        tmp = t_2
    else if (x <= (-3.5d-48)) then
        tmp = x * (y_m * t_m)
    else if (x <= 2.55d-42) then
        tmp = t_2
    else
        tmp = t_3
    end if
    code = t_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double t_2 = y_m * (z * -t_m);
	double t_3 = t_m * (y_m * x);
	double tmp;
	if (x <= -1.3e+110) {
		tmp = t_3;
	} else if (x <= -1.85e+71) {
		tmp = t_2;
	} else if (x <= -3.5e-48) {
		tmp = x * (y_m * t_m);
	} else if (x <= 2.55e-42) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return t_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	t_2 = y_m * (z * -t_m)
	t_3 = t_m * (y_m * x)
	tmp = 0
	if x <= -1.3e+110:
		tmp = t_3
	elif x <= -1.85e+71:
		tmp = t_2
	elif x <= -3.5e-48:
		tmp = x * (y_m * t_m)
	elif x <= 2.55e-42:
		tmp = t_2
	else:
		tmp = t_3
	return t_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
t_m = abs(t)
t_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	t_2 = Float64(y_m * Float64(z * Float64(-t_m)))
	t_3 = Float64(t_m * Float64(y_m * x))
	tmp = 0.0
	if (x <= -1.3e+110)
		tmp = t_3;
	elseif (x <= -1.85e+71)
		tmp = t_2;
	elseif (x <= -3.5e-48)
		tmp = Float64(x * Float64(y_m * t_m));
	elseif (x <= 2.55e-42)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	t_2 = y_m * (z * -t_m);
	t_3 = t_m * (y_m * x);
	tmp = 0.0;
	if (x <= -1.3e+110)
		tmp = t_3;
	elseif (x <= -1.85e+71)
		tmp = t_2;
	elseif (x <= -3.5e-48)
		tmp = x * (y_m * t_m);
	elseif (x <= 2.55e-42)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(y$95$m * N[(z * (-t$95$m)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(y$95$s * If[LessEqual[x, -1.3e+110], t$95$3, If[LessEqual[x, -1.85e+71], t$95$2, If[LessEqual[x, -3.5e-48], N[(x * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-42], t$95$2, t$95$3]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e110 or 2.55e-42 < x

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in x around inf 78.0%

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

      1. *-commutative 78.0%

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

   7. Simplified 78.0%

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

   if -1.3e110 < x < -1.85e71 or -3.49999999999999991e-48 < x < 2.55e-42

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 92.4%

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

      2. associate-*l* 95.2%

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

      3. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around 0 79.2%

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

      1. mul-1-neg 79.2%

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

      2. distribute-rgt-neg-out 79.2%

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

   7. Simplified 79.2%

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

   if -1.85e71 < x < -3.49999999999999991e-48

   1. Initial program 90.5%

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

      1. distribute-rgt-out-- 90.5%

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

      2. associate-*l* 85.0%

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

      3. *-commutative 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in x around inf 57.1%

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

      1. associate-*r* 51.5%

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

      2. *-commutative 51.5%

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

   7. Simplified 51.5%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

         \[\leadsto \color{blue}{{\left(\sqrt{y \cdot \left(t \cdot x\right)}\right)}^{2}} \]

      3. *-commutative 30.4%

         \[\leadsto {\left(\sqrt{y \cdot \color{blue}{\left(x \cdot t\right)}}\right)}^{2} \]

      4. associate-*l* 32.8%

         \[\leadsto {\left(\sqrt{\color{blue}{\left(y \cdot x\right) \cdot t}}\right)}^{2} \]

      5. *-commutative 32.8%

         \[\leadsto {\left(\sqrt{\color{blue}{t \cdot \left(y \cdot x\right)}}\right)}^{2} \]

   9. Applied egg-rr 32.8%

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

       1. unpow2 32.8%

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

       2. add-sqr-sqrt 57.1%

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

       3. associate-*r* 57.3%

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

       4. *-commutative 57.3%

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

   11. Applied egg-rr 57.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+110}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+71}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+120}:\\ \;\;\;\;y_m \cdot \left(\left(x - z\right) \cdot t_m\right)\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \left(y_m \cdot \left(-z\right)\right)\\ \end{array}\right) \end{array} \]

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 (<= z 1.3e+120) (* y_m (* (- x z) t_m)) (* t_m (* y_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 (z <= 1.3e+120) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = t_m * (y_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 (z <= 1.3d+120) then
        tmp = y_m * ((x - z) * t_m)
    else
        tmp = t_m * (y_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 (z <= 1.3e+120) {
		tmp = y_m * ((x - z) * t_m);
	} else {
		tmp = t_m * (y_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 z <= 1.3e+120:
		tmp = y_m * ((x - z) * t_m)
	else:
		tmp = t_m * (y_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 (z <= 1.3e+120)
		tmp = Float64(y_m * Float64(Float64(x - z) * t_m));
	else
		tmp = Float64(t_m * Float64(y_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 (z <= 1.3e+120)
		tmp = y_m * ((x - z) * t_m);
	else
		tmp = t_m * (y_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[LessEqual[z, 1.3e+120], N[(y$95$m * N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(y$95$m * (-z)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.2999999999999999e120

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 92.9%

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

      2. associate-*l* 93.2%

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

      3. *-commutative 93.2%

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

   3. Simplified 93.2%

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

   if 1.2999999999999999e120 < z

   1. Initial program 83.1%

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

      1. distribute-rgt-out-- 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in x around 0 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. mul-1-neg 79.3%

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

   7. Simplified 79.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;y_m \cdot \left(x \cdot t_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y_m \cdot t_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 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 3.1e-15) (* 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 <= 3.1e-15) {
		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 <= 3.1d-15) 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 <= 3.1e-15) {
		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 <= 3.1e-15:
		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 <= 3.1e-15)
		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 <= 3.1e-15)
		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, 3.1e-15], 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 < 3.0999999999999999e-15

   1. Initial program 88.3%

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

      1. distribute-rgt-out-- 88.9%

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

      2. associate-*l* 93.9%

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

      3. *-commutative 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around inf 53.8%

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

      1. associate-*r* 54.3%

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

      2. *-commutative 54.3%

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

   7. Simplified 54.3%

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

   if 3.0999999999999999e-15 < t

   1. Initial program 97.6%

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

      1. distribute-rgt-out-- 97.6%

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

      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 inf 59.9%

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

      1. associate-*r* 54.4%

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

      2. *-commutative 54.4%

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

   7. Simplified 54.4%

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

      1. add-sqr-sqrt 21.1%

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

      2. pow2 21.1%

         \[\leadsto \color{blue}{{\left(\sqrt{y \cdot \left(t \cdot x\right)}\right)}^{2}} \]

      3. *-commutative 21.1%

         \[\leadsto {\left(\sqrt{y \cdot \color{blue}{\left(x \cdot t\right)}}\right)}^{2} \]

      4. associate-*l* 23.9%

         \[\leadsto {\left(\sqrt{\color{blue}{\left(y \cdot x\right) \cdot t}}\right)}^{2} \]

      5. *-commutative 23.9%

         \[\leadsto {\left(\sqrt{\color{blue}{t \cdot \left(y \cdot x\right)}}\right)}^{2} \]

   9. Applied egg-rr 23.9%

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

       1. unpow2 23.9%

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

       2. add-sqr-sqrt 59.9%

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

       3. associate-*r* 64.1%

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

       4. *-commutative 64.1%

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

   11. Applied egg-rr 64.1%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t_s \cdot \left(y_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2 \cdot 10^{-17}:\\ \;\;\;\;t_m \cdot \left(y_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y_m \cdot t_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 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 2e-17) (* 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 <= 2e-17) {
		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 <= 2d-17) 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 <= 2e-17) {
		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 <= 2e-17:
		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 <= 2e-17)
		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 <= 2e-17)
		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, 2e-17], 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 < 2.00000000000000014e-17

   1. Initial program 88.3%

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

      1. distribute-rgt-out-- 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in x around inf 54.1%

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

      1. *-commutative 54.1%

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

   7. Simplified 54.1%

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

   if 2.00000000000000014e-17 < t

   1. Initial program 97.7%

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

      1. distribute-rgt-out-- 97.6%

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

      2. associate-*l* 89.0%

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

      3. *-commutative 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in x around inf 59.0%

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

      1. associate-*r* 53.7%

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

      2. *-commutative 53.7%

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

   7. Simplified 53.7%

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

      1. add-sqr-sqrt 20.8%

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

      2. pow2 20.8%

         \[\leadsto \color{blue}{{\left(\sqrt{y \cdot \left(t \cdot x\right)}\right)}^{2}} \]

      3. *-commutative 20.8%

         \[\leadsto {\left(\sqrt{y \cdot \color{blue}{\left(x \cdot t\right)}}\right)}^{2} \]

      4. associate-*l* 23.5%

         \[\leadsto {\left(\sqrt{\color{blue}{\left(y \cdot x\right) \cdot t}}\right)}^{2} \]

      5. *-commutative 23.5%

         \[\leadsto {\left(\sqrt{\color{blue}{t \cdot \left(y \cdot x\right)}}\right)}^{2} \]

   9. Applied egg-rr 23.5%

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

       1. unpow2 23.5%

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

       2. add-sqr-sqrt 59.0%

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

       3. associate-*r* 63.2%

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

       4. *-commutative 63.2%

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

   11. Applied egg-rr 63.2%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-17}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. distribute-rgt-out-- 91.2%

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

3. Simplified 91.2%

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

5. Final simplification 91.2%

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

Alternative 9: 51.4% 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 1 y)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (* t_s (* y_s (* y_m (* x 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 90.8%

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

   1. distribute-rgt-out-- 91.2%

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

   2. associate-*l* 92.5%

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

   3. *-commutative 92.5%

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

3. Simplified 92.5%

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

5. Taylor expanded in x around inf 55.4%

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

   1. associate-*r* 54.4%

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

   2. *-commutative 54.4%

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

7. Simplified 54.4%

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

8. Final simplification 54.4%

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

Developer target: 95.9% 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 2024021 
(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))