Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.8% → 96.9%

Time: 7.7s

Alternatives: 7

Speedup: 1.3×

• 24.3% 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: 90.8% 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: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.56 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.56e+54) (* y (* t (- x z))) (* (- x z) (* t y))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.56e+54) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (x - z) * (t * y);
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
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 <= 2.56d+54) then
        tmp = y * (t * (x - z))
    else
        tmp = (x - z) * (t * y)
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.56e+54) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (x - z) * (t * y);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 2.56e+54:
		tmp = y * (t * (x - z))
	else:
		tmp = (x - z) * (t * y)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.56e+54)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(Float64(x - z) * Float64(t * y));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.56e+54)
		tmp = y * (t * (x - z));
	else
		tmp = (x - z) * (t * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 2.56e+54], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(t * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.56e54

   1. Initial program 87.0%

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

      1. distribute-rgt-out-- 88.6%

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

      2. associate-*l* 94.5%

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

   3. Simplified 94.5%

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

   if 2.56e54 < t

   1. Initial program 97.0%

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

      1. distribute-rgt-out-- 98.7%

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

      2. associate-*l* 81.9%

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

   3. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. flip-- 70.1%

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

      3. associate-*r/ 63.5%

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

   5. Applied egg-rr 63.5%

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

      1. associate-/l* 70.1%

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

      2. difference-of-squares 75.2%

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

      3. associate-/r* 81.9%

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

      4. *-inverses 81.9%

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

   7. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-/r/ 81.9%

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

      3. /-rgt-identity 81.9%

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

      4. associate-*r* 98.7%

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

      5. flip-- 75.0%

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

      6. associate-/r/ 54.1%

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

      7. clear-num 54.1%

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

      8. un-div-inv 54.1%

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

      9. clear-num 54.1%

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

      10. associate-/r/ 75.0%

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

      11. flip-- 97.9%

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

      12. *-commutative 97.9%

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

   9. Applied egg-rr 97.9%

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

       1. associate-/r/ 98.7%

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

       2. /-rgt-identity 98.7%

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

       3. associate-*r* 98.2%

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

       4. *-commutative 98.2%

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

       5. *-commutative 98.2%

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

   11. Applied egg-rr 98.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.56 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 2: 74.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-11}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+18}:\\ \;\;\;\;\left(-y\right) \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.7e-11)
   (* t (* y x))
   (if (<= x 1.06e+18) (* (- y) (* t z)) (* y (* t x)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e-11) {
		tmp = t * (y * x);
	} else if (x <= 1.06e+18) {
		tmp = -y * (t * z);
	} else {
		tmp = y * (t * x);
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
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 (x <= (-1.7d-11)) then
        tmp = t * (y * x)
    else if (x <= 1.06d+18) then
        tmp = -y * (t * z)
    else
        tmp = y * (t * x)
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e-11) {
		tmp = t * (y * x);
	} else if (x <= 1.06e+18) {
		tmp = -y * (t * z);
	} else {
		tmp = y * (t * x);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if x <= -1.7e-11:
		tmp = t * (y * x)
	elif x <= 1.06e+18:
		tmp = -y * (t * z)
	else:
		tmp = y * (t * x)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.7e-11)
		tmp = Float64(t * Float64(y * x));
	elseif (x <= 1.06e+18)
		tmp = Float64(Float64(-y) * Float64(t * z));
	else
		tmp = Float64(y * Float64(t * x));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.7e-11)
		tmp = t * (y * x);
	elseif (x <= 1.06e+18)
		tmp = -y * (t * z);
	else
		tmp = y * (t * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, -1.7e-11], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.06e+18], N[((-y) * N[(t * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6999999999999999e-11

   1. Initial program 88.2%

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

      1. distribute-rgt-out-- 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x around inf 73.2%

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

   if -1.6999999999999999e-11 < x < 1.06e18

   1. Initial program 91.3%

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

      1. distribute-rgt-out-- 91.3%

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

      2. associate-*l* 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in x around 0 76.4%

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

      1. associate-*r* 76.4%

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

      2. neg-mul-1 76.4%

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

   6. Simplified 76.4%

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

   if 1.06e18 < x

   1. Initial program 86.9%

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

      1. distribute-rgt-out-- 91.6%

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

      2. associate-*l* 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in x around inf 74.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-11}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+18}:\\ \;\;\;\;\left(-y\right) \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \end{array} \]

Alternative 3: 74.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.6e-8)
   (* t (* y x))
   (if (<= x 2.9e+18) (* z (* t (- y))) (* y (* t x)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.6e-8) {
		tmp = t * (y * x);
	} else if (x <= 2.9e+18) {
		tmp = z * (t * -y);
	} else {
		tmp = y * (t * x);
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
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 (x <= (-1.6d-8)) then
        tmp = t * (y * x)
    else if (x <= 2.9d+18) then
        tmp = z * (t * -y)
    else
        tmp = y * (t * x)
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.6e-8) {
		tmp = t * (y * x);
	} else if (x <= 2.9e+18) {
		tmp = z * (t * -y);
	} else {
		tmp = y * (t * x);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if x <= -1.6e-8:
		tmp = t * (y * x)
	elif x <= 2.9e+18:
		tmp = z * (t * -y)
	else:
		tmp = y * (t * x)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.6e-8)
		tmp = Float64(t * Float64(y * x));
	elseif (x <= 2.9e+18)
		tmp = Float64(z * Float64(t * Float64(-y)));
	else
		tmp = Float64(y * Float64(t * x));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.6e-8)
		tmp = t * (y * x);
	elseif (x <= 2.9e+18)
		tmp = z * (t * -y);
	else
		tmp = y * (t * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, -1.6e-8], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e+18], N[(z * N[(t * (-y)), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001e-8

   1. Initial program 88.2%

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

      1. distribute-rgt-out-- 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x around inf 73.2%

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

   if -1.6000000000000001e-8 < x < 2.9e18

   1. Initial program 91.3%

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

      1. distribute-rgt-out-- 91.3%

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

      2. associate-*l* 89.0%

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

   3. Simplified 89.0%

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

      1. *-commutative 89.0%

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

      2. flip-- 72.3%

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

      3. associate-*r/ 69.8%

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

   5. Applied egg-rr 69.8%

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

      1. associate-/l* 72.3%

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

      2. difference-of-squares 72.3%

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

      3. associate-/r* 89.0%

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

      4. *-inverses 89.0%

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

   7. Simplified 89.0%

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

      1. associate-*r/ 94.0%

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

      2. frac-2neg 94.0%

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

      3. distribute-neg-frac 94.0%

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

      4. metadata-eval 94.0%

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

   9. Applied egg-rr 94.0%

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

       1. associate-/r/ 94.1%

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

       2. distribute-lft-neg-in 94.1%

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

   11. Simplified 94.1%

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

   12. Taylor expanded in x around 0 76.4%

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

       1. mul-1-neg 76.4%

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

       2. associate-*r* 83.1%

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

       3. distribute-rgt-neg-in 83.1%

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

   14. Simplified 83.1%

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

   if 2.9e18 < x

   1. Initial program 86.9%

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

      1. distribute-rgt-out-- 91.6%

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

      2. associate-*l* 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in x around inf 74.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \end{array} \]

Alternative 4: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-26}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 4.6e-26) (* y (* t (- x z))) (* t (* y (- x z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.6e-26) {
		tmp = y * (t * (x - z));
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
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 <= 4.6d-26) then
        tmp = y * (t * (x - z))
    else
        tmp = t * (y * (x - z))
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.6e-26) {
		tmp = y * (t * (x - z));
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 4.6e-26:
		tmp = y * (t * (x - z))
	else:
		tmp = t * (y * (x - z))
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 4.6e-26)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(t * Float64(y * Float64(x - z)));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 4.6e-26)
		tmp = y * (t * (x - z));
	else
		tmp = t * (y * (x - z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 4.6e-26], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.60000000000000018e-26

   1. Initial program 86.7%

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

      1. distribute-rgt-out-- 87.4%

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

      2. associate-*l* 93.9%

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

   3. Simplified 93.9%

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

   if 4.60000000000000018e-26 < t

   1. Initial program 95.1%

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

      1. distribute-rgt-out-- 99.0%

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

   3. Simplified 99.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-26}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5: 55.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+115}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -5e+115) (* y (* t x)) (* t (* y x))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5e+115) {
		tmp = y * (t * x);
	} else {
		tmp = t * (y * x);
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
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 (y <= (-5d+115)) then
        tmp = y * (t * x)
    else
        tmp = t * (y * x)
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5e+115) {
		tmp = y * (t * x);
	} else {
		tmp = t * (y * x);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -5e+115:
		tmp = y * (t * x)
	else:
		tmp = t * (y * x)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5e+115)
		tmp = Float64(y * Float64(t * x));
	else
		tmp = Float64(t * Float64(y * x));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -5e+115)
		tmp = y * (t * x);
	else
		tmp = t * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -5e+115], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000008e115

   1. Initial program 72.3%

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

      1. distribute-rgt-out-- 75.4%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 53.7%

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

   if -5.00000000000000008e115 < y

   1. Initial program 92.2%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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. Taylor expanded in x around inf 54.9%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+115}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 6: 92.2% accurate, 1.3× speedup

Math

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

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* y (* t (- x z))))

C

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

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
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 = y * (t * (x - z))
end function

Java

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

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	return y * (t * (x - z))

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	return Float64(y * Float64(t * Float64(x - z)))
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp = code(x, y, z, t)
	tmp = y * (t * (x - z));
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

   1. distribute-rgt-out-- 90.9%

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

   2. associate-*l* 91.6%

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

3. Simplified 91.6%

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

4. Final simplification 91.6%

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

Alternative 7: 54.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ y \cdot \left(t \cdot x\right) \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* y (* t x)))

C

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

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
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 = y * (t * x)
end function

Java

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

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	return y * (t * x)

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	return Float64(y * Float64(t * x))
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp = code(x, y, z, t)
	tmp = y * (t * x);
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

   1. distribute-rgt-out-- 90.9%

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

   2. associate-*l* 91.6%

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

3. Simplified 91.6%

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

4. Taylor expanded in x around inf 54.3%

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

5. Final simplification 54.3%

   \[\leadsto y \cdot \left(t \cdot x\right) \]

Developer target: 96.2% 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 2023196 
(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))