Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.4% → 96.7%

Time: 9.9s

Alternatives: 8

Speedup: 1.3×

• 24.8% 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: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -300000000:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -300000000.0) (* (- x z) (* y t)) (* t (- (* y x) (* y z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -300000000.0) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t * ((y * x) - (y * 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 (y <= (-300000000.0d0)) then
        tmp = (x - z) * (y * t)
    else
        tmp = t * ((y * x) - (y * 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 (y <= -300000000.0) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t * ((y * x) - (y * z));
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -300000000.0:
		tmp = (x - z) * (y * t)
	else:
		tmp = t * ((y * x) - (y * z))
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -300000000.0)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	else
		tmp = Float64(t * Float64(Float64(y * x) - Float64(y * z)));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -300000000.0)
		tmp = (x - z) * (y * t);
	else
		tmp = t * ((y * x) - (y * 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[y, -300000000.0], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(y * x), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e8

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. distribute-rgt-out-- 85.2%

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

      3. associate-*r* 98.4%

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

      4. *-commutative 98.4%

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

   3. Simplified 98.4%

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

   if -3e8 < y

   1. Initial program 94.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -300000000:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \]

Alternative 2: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-20} \lor \neg \left(z \leq 9 \cdot 10^{-54}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.3e-20) (not (<= z 9e-54))) (* y (* t (- z))) (* t (* y x))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.3e-20) || !(z <= 9e-54)) {
		tmp = y * (t * -z);
	} 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 ((z <= (-4.3d-20)) .or. (.not. (z <= 9d-54))) then
        tmp = y * (t * -z)
    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 ((z <= -4.3e-20) || !(z <= 9e-54)) {
		tmp = y * (t * -z);
	} else {
		tmp = t * (y * x);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -4.3e-20) or not (z <= 9e-54):
		tmp = y * (t * -z)
	else:
		tmp = t * (y * x)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.3e-20) || !(z <= 9e-54))
		tmp = Float64(y * Float64(t * Float64(-z)));
	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 ((z <= -4.3e-20) || ~((z <= 9e-54)))
		tmp = y * (t * -z);
	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[Or[LessEqual[z, -4.3e-20], N[Not[LessEqual[z, 9e-54]], $MachinePrecision]], N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.30000000000000011e-20 or 8.9999999999999997e-54 < z

   1. Initial program 87.1%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 91.2%

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

      3. *-commutative 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in x around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. distribute-rgt-neg-out 78.6%

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

   6. Simplified 78.6%

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

   if -4.30000000000000011e-20 < z < 8.9999999999999997e-54

   1. Initial program 95.6%

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

      1. distribute-rgt-out-- 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around inf 84.0%

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

      1. *-commutative 84.0%

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

   6. Simplified 84.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-20} \lor \neg \left(z \leq 9 \cdot 10^{-54}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.9 \cdot 10^{-14}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-52}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -6.9e-14)
   (* z (* y (- t)))
   (if (<= z 7.2e-52) (* t (* y x)) (* y (* t (- z))))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.9e-14) {
		tmp = z * (y * -t);
	} else if (z <= 7.2e-52) {
		tmp = t * (y * x);
	} else {
		tmp = y * (t * -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 (z <= (-6.9d-14)) then
        tmp = z * (y * -t)
    else if (z <= 7.2d-52) then
        tmp = t * (y * x)
    else
        tmp = y * (t * -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 (z <= -6.9e-14) {
		tmp = z * (y * -t);
	} else if (z <= 7.2e-52) {
		tmp = t * (y * x);
	} else {
		tmp = y * (t * -z);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if z <= -6.9e-14:
		tmp = z * (y * -t)
	elif z <= 7.2e-52:
		tmp = t * (y * x)
	else:
		tmp = y * (t * -z)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6.9e-14)
		tmp = Float64(z * Float64(y * Float64(-t)));
	elseif (z <= 7.2e-52)
		tmp = Float64(t * Float64(y * x));
	else
		tmp = Float64(y * Float64(t * Float64(-z)));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -6.9e-14)
		tmp = z * (y * -t);
	elseif (z <= 7.2e-52)
		tmp = t * (y * x);
	else
		tmp = y * (t * -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[z, -6.9e-14], N[(z * N[(y * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.2e-52], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.89999999999999963e-14

   1. Initial program 90.3%

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

      1. distribute-rgt-out-- 91.9%

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

   3. Simplified 91.9%

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

      1. flip-- 59.1%

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

      2. associate-*r/ 57.7%

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

      3. pow2 57.7%

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

      4. pow2 57.7%

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

   5. Applied egg-rr 57.7%

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

      1. associate-/l* 59.0%

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

   7. Simplified 59.0%

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

   8. Taylor expanded in x around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. associate-*r* 79.6%

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

      3. *-commutative 79.6%

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

      4. *-commutative 79.6%

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

      5. distribute-rgt-neg-in 79.6%

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

      6. distribute-lft-neg-in 79.6%

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

      7. *-commutative 79.6%

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

   10. Simplified 79.6%

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

   if -6.89999999999999963e-14 < z < 7.19999999999999976e-52

   1. Initial program 95.6%

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

      1. distribute-rgt-out-- 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around inf 84.0%

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

      1. *-commutative 84.0%

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

   6. Simplified 84.0%

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

   if 7.19999999999999976e-52 < z

   1. Initial program 83.8%

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

      1. distribute-rgt-out-- 85.2%

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

      2. associate-*l* 90.6%

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

      3. *-commutative 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around 0 77.5%

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

      1. mul-1-neg 77.5%

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

      2. distribute-rgt-neg-out 77.5%

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

   6. Simplified 77.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.9 \cdot 10^{-14}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-52}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \end{array} \]

Alternative 4: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-106}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 4.2e-106) (* y (* t (- x z))) (* (- x z) (* y t))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.2e-106) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (x - z) * (y * t);
	}
	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 <= 4.2d-106) then
        tmp = y * (t * (x - z))
    else
        tmp = (x - z) * (y * t)
    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 <= 4.2e-106) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (x - z) * (y * t);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 4.2e-106:
		tmp = y * (t * (x - z))
	else:
		tmp = (x - z) * (y * t)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 4.2e-106)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(Float64(x - z) * Float64(y * t));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 4.2e-106)
		tmp = y * (t * (x - z));
	else
		tmp = (x - z) * (y * t);
	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, 4.2e-106], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.20000000000000007e-106

   1. Initial program 87.3%

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

      1. distribute-rgt-out-- 88.0%

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

      2. associate-*l* 90.5%

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

      3. *-commutative 90.5%

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

   3. Simplified 90.5%

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

   if 4.20000000000000007e-106 < t

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. distribute-rgt-out-- 97.9%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-106}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 5: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -300000000:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -300000000.0) (* (- x z) (* y t)) (* t (* y (- x z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -300000000.0) {
		tmp = (x - z) * (y * t);
	} 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 (y <= (-300000000.0d0)) then
        tmp = (x - z) * (y * t)
    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 (y <= -300000000.0) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -300000000.0:
		tmp = (x - z) * (y * t)
	else:
		tmp = t * (y * (x - z))
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -300000000.0)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	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 (y <= -300000000.0)
		tmp = (x - z) * (y * t);
	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[y, -300000000.0], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e8

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. distribute-rgt-out-- 85.2%

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

      3. associate-*r* 98.4%

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

      4. *-commutative 98.4%

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

   3. Simplified 98.4%

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

   if -3e8 < y

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 94.2%

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

   3. Simplified 94.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -300000000:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6: 55.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-120}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 5.4e-120) (* y (* t x)) (* x (* y t))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5.4e-120) {
		tmp = y * (t * x);
	} else {
		tmp = x * (y * t);
	}
	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 <= 5.4d-120) then
        tmp = y * (t * x)
    else
        tmp = x * (y * t)
    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 <= 5.4e-120) {
		tmp = y * (t * x);
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 5.4e-120:
		tmp = y * (t * x)
	else:
		tmp = x * (y * t)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 5.4e-120)
		tmp = Float64(y * Float64(t * x));
	else
		tmp = Float64(x * Float64(y * t));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 5.4e-120)
		tmp = y * (t * x);
	else
		tmp = x * (y * t);
	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, 5.4e-120], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.3999999999999997e-120

   1. Initial program 87.0%

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

      1. distribute-rgt-out-- 87.8%

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

      2. associate-*l* 90.3%

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

      3. *-commutative 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around inf 49.1%

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

   if 5.3999999999999997e-120 < t

   1. Initial program 96.9%

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

      1. distribute-rgt-out-- 97.9%

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

   3. Simplified 97.9%

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

      1. flip-- 75.7%

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

      2. associate-*r/ 71.8%

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

      3. pow2 71.8%

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

      4. pow2 71.8%

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

   5. Applied egg-rr 71.8%

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

      1. associate-/l* 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in x around inf 54.3%

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

      1. *-commutative 54.3%

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

      2. associate-*r* 58.2%

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

      3. *-commutative 58.2%

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

   10. Simplified 58.2%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-120}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 7: 91.9% 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 90.9%

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

   1. distribute-rgt-out-- 91.7%

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

   2. associate-*l* 91.0%

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

   3. *-commutative 91.0%

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

3. Simplified 91.0%

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

4. Final simplification 91.0%

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

Alternative 8: 53.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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 = x * (y * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. distribute-rgt-out-- 91.7%

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

3. Simplified 91.7%

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

   1. flip-- 64.9%

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

   2. associate-*r/ 62.7%

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

   3. pow2 62.7%

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

   4. pow2 62.7%

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

5. Applied egg-rr 62.7%

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

   1. associate-/l* 64.9%

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

7. Simplified 64.9%

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

8. Taylor expanded in x around inf 51.5%

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

   1. *-commutative 51.5%

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

   2. associate-*r* 51.9%

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

   3. *-commutative 51.9%

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

10. Simplified 51.9%

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

11. Final simplification 51.9%

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

Developer target: 95.7% 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 2023318 
(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))