Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.8% → 96.3%

Time: 10.3s

Alternatives: 6

Speedup: 1.3×

• 25.0% 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: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+79}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -4e+79) (* y (* t (- x z))) (* t (- (* y x) (* y z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4e+79) {
		tmp = y * (t * (x - z));
	} 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 <= (-4d+79)) then
        tmp = y * (t * (x - z))
    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 <= -4e+79) {
		tmp = y * (t * (x - z));
	} 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 <= -4e+79:
		tmp = y * (t * (x - z))
	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 <= -4e+79)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	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 <= -4e+79)
		tmp = y * (t * (x - z));
	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, -4e+79], N[(y * N[(t * N[(x - z), $MachinePrecision]), $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 < -3.99999999999999987e79

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. distribute-rgt-out-- 81.7%

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

      3. associate-*r* 93.7%

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

      4. *-commutative 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in y around 0 97.7%

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

   if -3.99999999999999987e79 < y

   1. Initial program 96.1%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+79}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \]

Alternative 2: 72.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-28} \lor \neg \left(z \leq 1.55 \cdot 10^{+99}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.2e-28) (not (<= z 1.55e+99)))
   (* y (* t (- z)))
   (* x (* y t))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.2e-28) || !(z <= 1.55e+99)) {
		tmp = y * (t * -z);
	} 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 ((z <= (-7.2d-28)) .or. (.not. (z <= 1.55d+99))) then
        tmp = y * (t * -z)
    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 ((z <= -7.2e-28) || !(z <= 1.55e+99)) {
		tmp = y * (t * -z);
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -7.2e-28) or not (z <= 1.55e+99):
		tmp = y * (t * -z)
	else:
		tmp = x * (y * t)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7.2e-28) || !(z <= 1.55e+99))
		tmp = Float64(y * Float64(t * Float64(-z)));
	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 ((z <= -7.2e-28) || ~((z <= 1.55e+99)))
		tmp = y * (t * -z);
	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[Or[LessEqual[z, -7.2e-28], N[Not[LessEqual[z, 1.55e+99]], $MachinePrecision]], N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.1999999999999997e-28 or 1.55e99 < z

   1. Initial program 93.0%

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

      1. *-commutative 93.0%

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

      2. distribute-rgt-out-- 93.9%

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

      3. associate-*r* 94.5%

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

      4. *-commutative 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in x around 0 74.8%

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

      1. mul-1-neg 74.8%

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

      2. distribute-rgt-neg-in 74.8%

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

      3. distribute-rgt-neg-out 74.8%

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

   6. Simplified 74.8%

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

   if -7.1999999999999997e-28 < z < 1.55e99

   1. Initial program 93.2%

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

      1. *-commutative 93.2%

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

      2. distribute-rgt-out-- 93.2%

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

      3. associate-*r* 94.0%

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

      4. *-commutative 94.0%

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

   3. Simplified 94.0%

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

      1. flip-- 73.1%

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

      2. associate-*r/ 66.4%

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

   5. Applied egg-rr 66.4%

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

      1. associate-/l* 73.0%

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

      2. associate-/l* 74.6%

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

      3. difference-of-squares 74.6%

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

      4. associate-/r* 92.3%

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

      5. *-inverses 92.3%

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

   7. Simplified 92.3%

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

   8. Taylor expanded in x around inf 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. div-inv 79.3%

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

      3. times-frac 81.6%

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

      4. clear-num 81.7%

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

      5. /-rgt-identity 81.7%

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

      6. add-sqr-sqrt 36.3%

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

      7. sqrt-unprod 38.2%

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

      8. frac-times 38.2%

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

      9. metadata-eval 38.2%

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

      10. metadata-eval 38.2%

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

      11. frac-times 38.2%

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

      12. sqrt-unprod 10.8%

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

      13. add-sqr-sqrt 14.8%

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

      14. div-inv 14.8%

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

      15. add-sqr-sqrt 10.8%

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

      16. sqrt-unprod 38.2%

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

      17. frac-times 38.2%

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

      18. metadata-eval 38.2%

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

      19. metadata-eval 38.2%

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

      20. frac-times 38.2%

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

      21. sqrt-unprod 36.2%

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

      22. add-sqr-sqrt 81.8%

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

      23. remove-double-div 81.8%

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

   10. Applied egg-rr 81.8%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-28} \lor \neg \left(z \leq 1.55 \cdot 10^{+99}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 3: 97.2% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 1e-5) (* 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 <= 1e-5) {
		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 <= 1d-5) 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 <= 1e-5) {
		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 <= 1e-5:
		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 <= 1e-5)
		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 <= 1e-5)
		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, 1e-5], 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 < 1.00000000000000008e-5

   1. Initial program 92.1%

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

      1. *-commutative 92.1%

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

      2. distribute-rgt-out-- 92.1%

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

      3. associate-*r* 93.9%

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

      4. *-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in y around 0 90.7%

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

   if 1.00000000000000008e-5 < t

   1. Initial program 96.4%

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

      1. *-commutative 96.4%

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

      2. distribute-rgt-out-- 97.9%

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

      3. associate-*r* 95.4%

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

      4. *-commutative 95.4%

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

   3. Simplified 95.4%

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

4. Final simplification 91.9%

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

Alternative 4: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+78}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.5e+78) (* 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 <= -1.5e+78) {
		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 <= (-1.5d+78)) 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 <= -1.5e+78) {
		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 <= -1.5e+78:
		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 <= -1.5e+78)
		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 <= -1.5e+78)
		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, -1.5e+78], 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 < -1.49999999999999991e78

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. distribute-rgt-out-- 82.1%

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

      3. associate-*r* 93.8%

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

      4. *-commutative 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in y around 0 97.8%

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

   if -1.49999999999999991e78 < y

   1. Initial program 96.1%

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

      1. distribute-rgt-out-- 96.0%

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

   3. Simplified 96.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+78}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5: 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 93.1%

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

   1. *-commutative 93.1%

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

   2. distribute-rgt-out-- 93.5%

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

   3. associate-*r* 94.2%

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

   4. *-commutative 94.2%

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

3. Simplified 94.2%

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

4. Taylor expanded in y around 0 91.1%

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

5. Final simplification 91.1%

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

Alternative 6: 54.0% 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 93.1%

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

   1. *-commutative 93.1%

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

   2. distribute-rgt-out-- 93.5%

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

   3. associate-*r* 94.2%

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

   4. *-commutative 94.2%

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

3. Simplified 94.2%

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

   1. flip-- 67.7%

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

   2. associate-*r/ 62.0%

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

5. Applied egg-rr 62.0%

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

   1. associate-/l* 67.7%

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

   2. associate-/l* 67.5%

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

   3. difference-of-squares 70.4%

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

   4. associate-/r* 91.1%

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

   5. *-inverses 91.1%

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

7. Simplified 91.1%

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

8. Taylor expanded in x around inf 56.2%

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

   1. *-un-lft-identity 56.2%

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

   2. div-inv 56.1%

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

   3. times-frac 59.3%

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

   4. clear-num 59.4%

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

   5. /-rgt-identity 59.4%

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

   6. add-sqr-sqrt 27.1%

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

   7. sqrt-unprod 31.6%

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

   8. frac-times 31.5%

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

   9. metadata-eval 31.5%

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

   10. metadata-eval 31.5%

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

   11. frac-times 31.6%

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

   12. sqrt-unprod 9.8%

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

   13. add-sqr-sqrt 13.5%

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

   14. div-inv 13.5%

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

   15. add-sqr-sqrt 9.8%

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

   16. sqrt-unprod 31.5%

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

   17. frac-times 31.5%

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

   18. metadata-eval 31.5%

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

   19. metadata-eval 31.5%

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

   20. frac-times 31.5%

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

   21. sqrt-unprod 27.1%

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

   22. add-sqr-sqrt 59.4%

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

   23. remove-double-div 59.4%

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

10. Applied egg-rr 59.4%

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

11. Final simplification 59.4%

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

Developer target: 96.1% 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 2023274 
(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))