Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.4% → 97.5%

Time: 7.0s

Alternatives: 7

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: 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: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.8e-13) (* 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 <= 1.8e-13) {
		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 <= 1.8d-13) 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 <= 1.8e-13) {
		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 <= 1.8e-13:
		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 <= 1.8e-13)
		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 <= 1.8e-13)
		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, 1.8e-13], 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 < 1.7999999999999999e-13

   1. Initial program 85.7%

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

      1. distribute-rgt-out-- 86.7%

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

      2. associate-*l* 94.6%

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

   3. Simplified 94.6%

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

   if 1.7999999999999999e-13 < t

   1. Initial program 96.3%

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

      1. distribute-rgt-out-- 98.1%

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

      2. associate-*l* 86.2%

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

   3. Simplified 86.2%

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

      1. add-cube-cbrt 85.6%

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

      2. pow3 85.5%

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

   5. Applied egg-rr 85.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(\left(x - z\right) \cdot t\right)}\right)}^{3}} \]
   6. Step-by-step derivation

      1. rem-cube-cbrt 86.2%

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

      2. *-commutative 86.2%

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

      3. associate-*r* 94.9%

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

      4. remove-double-div 94.8%

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

      5. div-inv 94.8%

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

      6. frac-2neg 94.8%

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

      7. metadata-eval 94.8%

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

      8. associate-/r/ 94.9%

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

      9. sub-neg 94.9%

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

      10. distribute-neg-in 94.9%

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

      11. add-sqr-sqrt 47.5%

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

      12. sqrt-unprod 59.8%

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

      13. sqr-neg 59.8%

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

      14. sqrt-unprod 22.6%

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

      15. add-sqr-sqrt 42.8%

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

      16. add-sqr-sqrt 20.2%

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

      17. sqrt-unprod 65.8%

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

      18. sqr-neg 65.8%

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

      19. sqrt-unprod 47.2%

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

      20. add-sqr-sqrt 94.9%

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

   7. Applied egg-rr 94.9%

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

   8. Taylor expanded in y around 0 86.2%

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

      1. mul-1-neg 86.2%

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

      2. associate-*r* 94.9%

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

      3. distribute-lft-neg-in 94.9%

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

      4. distribute-rgt-neg-out 94.9%

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

      5. distribute-rgt-out-- 80.1%

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

      6. *-commutative 80.1%

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

      7. distribute-rgt-neg-out 80.1%

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

      8. cancel-sign-sub 80.1%

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

      9. *-commutative 80.1%

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

      10. distribute-rgt-neg-out 80.1%

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

      11. distribute-lft-neg-in 80.1%

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

      12. associate-*r* 75.4%

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

      13. mul-1-neg 75.4%

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

      14. associate-*r* 73.5%

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

      15. +-commutative 73.5%

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

      16. associate-*r* 75.4%

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

      17. mul-1-neg 75.4%

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

      18. associate-*r* 80.1%

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

      19. distribute-rgt-neg-in 80.1%

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

      20. distribute-lft-in 94.9%

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

      21. sub-neg 94.9%

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

      22. *-commutative 94.9%

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

   10. Simplified 94.9%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 2: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-85} \lor \neg \left(x \leq 3.9 \cdot 10^{+25}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.8e-85) (not (<= x 3.9e+25)))
   (* t (* y x))
   (* y (* t (- z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.8e-85) || !(x <= 3.9e+25)) {
		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 ((x <= (-4.8d-85)) .or. (.not. (x <= 3.9d+25))) 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 ((x <= -4.8e-85) || !(x <= 3.9e+25)) {
		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 (x <= -4.8e-85) or not (x <= 3.9e+25):
		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 ((x <= -4.8e-85) || !(x <= 3.9e+25))
		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 ((x <= -4.8e-85) || ~((x <= 3.9e+25)))
		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[Or[LessEqual[x, -4.8e-85], N[Not[LessEqual[x, 3.9e+25]], $MachinePrecision]], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8000000000000001e-85 or 3.9000000000000002e25 < x

   1. Initial program 84.9%

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

      1. distribute-rgt-out-- 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in x around inf 69.2%

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

   if -4.8000000000000001e-85 < x < 3.9000000000000002e25

   1. Initial program 92.1%

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

      1. distribute-rgt-out-- 92.1%

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

      2. associate-*l* 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around 0 86.1%

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

      1. associate-*r* 86.1%

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

      2. neg-mul-1 86.1%

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

   6. Simplified 86.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-85} \lor \neg \left(x \leq 3.9 \cdot 10^{+25}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \end{array} \]

Alternative 3: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+20} \lor \neg \left(z \leq 5800\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

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 -2.8e+20) (not (<= z 5800.0))) (* t (* y (- z))) (* t (* y x))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.8e+20) || !(z <= 5800.0)) {
		tmp = t * (y * -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 <= (-2.8d+20)) .or. (.not. (z <= 5800.0d0))) then
        tmp = t * (y * -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 <= -2.8e+20) || !(z <= 5800.0)) {
		tmp = t * (y * -z);
	} else {
		tmp = t * (y * x);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -2.8e+20) or not (z <= 5800.0):
		tmp = t * (y * -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 <= -2.8e+20) || !(z <= 5800.0))
		tmp = Float64(t * Float64(y * 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 <= -2.8e+20) || ~((z <= 5800.0)))
		tmp = t * (y * -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, -2.8e+20], N[Not[LessEqual[z, 5800.0]], $MachinePrecision]], N[(t * N[(y * (-z)), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8e20 or 5800 < z

   1. Initial program 84.3%

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

      1. distribute-rgt-out-- 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around 0 73.5%

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

      1. neg-mul-1 73.5%

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

      2. distribute-lft-neg-in 73.5%

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

   6. Simplified 73.5%

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

   if -2.8e20 < z < 5800

   1. Initial program 91.7%

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

      1. distribute-rgt-out-- 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around inf 77.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+20} \lor \neg \left(z \leq 5800\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+106}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -9e+106) (* 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 (y <= -9e+106) {
		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 (y <= (-9d+106)) 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 (y <= -9e+106) {
		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 y <= -9e+106:
		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 (y <= -9e+106)
		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 (y <= -9e+106)
		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[y, -9e+106], 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 y < -8.9999999999999994e106

   1. Initial program 68.1%

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

      1. distribute-rgt-out-- 73.4%

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

      2. associate-*l* 97.3%

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

   3. Simplified 97.3%

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

   if -8.9999999999999994e106 < y

   1. Initial program 91.5%

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

      1. distribute-rgt-out-- 92.0%

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

   3. Simplified 92.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+106}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5: 56.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 9.2e-38) (* y (* t x)) (* t (* y x))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.2e-38) {
		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 (t <= 9.2d-38) 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 (t <= 9.2e-38) {
		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 t <= 9.2e-38:
		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 (t <= 9.2e-38)
		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 (t <= 9.2e-38)
		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[t, 9.2e-38], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.20000000000000007e-38

   1. Initial program 85.4%

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

      1. distribute-rgt-out-- 86.5%

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

      2. associate-*l* 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in x around inf 56.7%

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

   if 9.20000000000000007e-38 < t

   1. Initial program 96.5%

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

      1. distribute-rgt-out-- 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in x around inf 47.9%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 6: 92.5% 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 87.9%

   \[\left(x \cdot y - z \cdot y\right) \cdot t \]
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. Simplified 92.8%

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

4. Final simplification 92.8%

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

Alternative 7: 54.1% 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 87.9%

   \[\left(x \cdot y - z \cdot y\right) \cdot t \]
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. Simplified 92.8%

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

4. Taylor expanded in x around inf 53.5%

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

5. Final simplification 53.5%

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

Developer target: 96.0% 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 2023185 
(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))