Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.4% → 96.2%

Time: 9.0s

Alternatives: 7

Speedup: 1.3×

• 24.6% 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: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+108}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 8.5e+108) (* 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 <= 8.5e+108) {
		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 <= 8.5d+108) 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 <= 8.5e+108) {
		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 <= 8.5e+108:
		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 <= 8.5e+108)
		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 <= 8.5e+108)
		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, 8.5e+108], 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 < 8.50000000000000016e108

   1. Initial program 87.2%

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

      1. distribute-rgt-out-- 88.7%

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

      2. associate-*l* 91.0%

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

   3. Simplified 91.0%

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

   if 8.50000000000000016e108 < t

   1. Initial program 90.4%

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

      1. distribute-rgt-out-- 92.4%

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

      2. associate-*l* 86.8%

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

   3. Simplified 86.8%

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

      1. add-cube-cbrt 86.0%

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

      2. pow3 86.0%

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

   5. Applied egg-rr 86.0%

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

      1. rem-cube-cbrt 86.8%

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

      2. *-commutative 86.8%

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

      3. associate-*r* 94.5%

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

      4. remove-double-div 94.5%

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

      5. div-inv 94.5%

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

      6. *-commutative 94.5%

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

      7. associate-/l* 92.3%

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

      8. clear-num 92.3%

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

      9. associate-/r/ 92.3%

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

      10. /-rgt-identity 92.3%

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

      11. sub-neg 92.3%

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

      12. add-sqr-sqrt 37.3%

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

      13. sqrt-unprod 65.4%

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

      14. sqr-neg 65.4%

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

      15. sqrt-unprod 29.8%

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

      16. add-sqr-sqrt 47.7%

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

   7. Applied egg-rr 47.7%

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

      1. frac-2neg 47.7%

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

      2. distribute-lft-neg-in 47.7%

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

      3. add-sqr-sqrt 19.8%

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

      4. sqrt-unprod 24.4%

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

      5. sqr-neg 24.4%

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

      6. sqrt-unprod 19.7%

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

      7. add-sqr-sqrt 43.7%

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

      8. distribute-neg-frac 43.7%

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

      9. neg-sub0 41.7%

         \[\leadsto \frac{t}{\color{blue}{0 - \frac{1}{y \cdot \left(x + z\right)}}} \]

      10. frac-2neg 41.7%

          \[\leadsto \frac{t}{0 - \color{blue}{\frac{-1}{-y \cdot \left(x + z\right)}}} \]

      11. metadata-eval 41.7%

          \[\leadsto \frac{t}{0 - \frac{\color{blue}{-1}}{-y \cdot \left(x + z\right)}} \]

      12. distribute-lft-neg-in 41.7%

          \[\leadsto \frac{t}{0 - \frac{-1}{\color{blue}{\left(-y\right) \cdot \left(x + z\right)}}} \]

      13. add-sqr-sqrt 20.0%

          \[\leadsto \frac{t}{0 - \frac{-1}{\color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)} \cdot \left(x + z\right)}} \]

      14. sqrt-unprod 38.5%

          \[\leadsto \frac{t}{0 - \frac{-1}{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}} \cdot \left(x + z\right)}} \]

      15. sqr-neg 38.5%

          \[\leadsto \frac{t}{0 - \frac{-1}{\sqrt{\color{blue}{y \cdot y}} \cdot \left(x + z\right)}} \]

      16. sqrt-unprod 23.9%

          \[\leadsto \frac{t}{0 - \frac{-1}{\color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \left(x + z\right)}} \]

      17. add-sqr-sqrt 51.6%

          \[\leadsto \frac{t}{0 - \frac{-1}{\color{blue}{y} \cdot \left(x + z\right)}} \]

      18. distribute-rgt-in 49.6%

          \[\leadsto \frac{t}{0 - \frac{-1}{\color{blue}{x \cdot y + z \cdot y}}} \]

      19. *-commutative 49.6%

          \[\leadsto \frac{t}{0 - \frac{-1}{\color{blue}{y \cdot x} + z \cdot y}} \]

      20. *-commutative 49.6%

          \[\leadsto \frac{t}{0 - \frac{-1}{y \cdot x + \color{blue}{y \cdot z}}} \]

      21. add-sqr-sqrt 28.0%

          \[\leadsto \frac{t}{0 - \frac{-1}{y \cdot x + \color{blue}{\sqrt{y \cdot z} \cdot \sqrt{y \cdot z}}}} \]

      22. sqrt-unprod 61.4%

          \[\leadsto \frac{t}{0 - \frac{-1}{y \cdot x + \color{blue}{\sqrt{\left(y \cdot z\right) \cdot \left(y \cdot z\right)}}}} \]

      23. sqr-neg 61.4%

          \[\leadsto \frac{t}{0 - \frac{-1}{y \cdot x + \sqrt{\color{blue}{\left(-y \cdot z\right) \cdot \left(-y \cdot z\right)}}}} \]

      24. sqrt-unprod 45.4%

          \[\leadsto \frac{t}{0 - \frac{-1}{y \cdot x + \color{blue}{\sqrt{-y \cdot z} \cdot \sqrt{-y \cdot z}}}} \]

      25. add-sqr-sqrt 80.6%

          \[\leadsto \frac{t}{0 - \frac{-1}{y \cdot x + \color{blue}{\left(-y \cdot z\right)}}} \]

      26. distribute-rgt-neg-in 80.6%

          \[\leadsto \frac{t}{0 - \frac{-1}{y \cdot x + \color{blue}{y \cdot \left(-z\right)}}} \]

   9. Applied egg-rr 82.5%

      \[\leadsto \frac{t}{\color{blue}{0 - \frac{-1}{y \cdot \left(x - z\right)}}} \]
   10. Step-by-step derivation

       1. neg-sub0 92.3%

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

       2. distribute-neg-frac 92.3%

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

       3. metadata-eval 92.3%

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

   11. Simplified 92.3%

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

   12. Taylor expanded in t around 0 86.8%

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

       1. associate-*r* 94.5%

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

       2. *-commutative 94.5%

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

   14. Simplified 94.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+108}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 2: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+115} \lor \neg \left(z \leq 1.4 \cdot 10^{+64}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.25e+115) (not (<= z 1.4e+64)))
   (* (- y) (* t z))
   (* x (* t y))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e+115) || !(z <= 1.4e+64)) {
		tmp = -y * (t * z);
	} else {
		tmp = x * (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 ((z <= (-1.25d+115)) .or. (.not. (z <= 1.4d+64))) then
        tmp = -y * (t * z)
    else
        tmp = x * (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 ((z <= -1.25e+115) || !(z <= 1.4e+64)) {
		tmp = -y * (t * z);
	} else {
		tmp = x * (t * y);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -1.25e+115) or not (z <= 1.4e+64):
		tmp = -y * (t * z)
	else:
		tmp = x * (t * y)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.25e+115) || !(z <= 1.4e+64))
		tmp = Float64(Float64(-y) * Float64(t * z));
	else
		tmp = Float64(x * 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 ((z <= -1.25e+115) || ~((z <= 1.4e+64)))
		tmp = -y * (t * z);
	else
		tmp = x * (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[Or[LessEqual[z, -1.25e+115], N[Not[LessEqual[z, 1.4e+64]], $MachinePrecision]], N[((-y) * N[(t * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000002e115 or 1.40000000000000012e64 < z

   1. Initial program 85.9%

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

      1. distribute-rgt-out-- 90.5%

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

      2. associate-*l* 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in x around 0 80.9%

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

      1. associate-*r* 80.9%

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

      2. neg-mul-1 80.9%

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

   6. Simplified 80.9%

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

   if -1.25000000000000002e115 < z < 1.40000000000000012e64

   1. Initial program 88.9%

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

      1. distribute-rgt-out-- 88.9%

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

      2. associate-*l* 91.8%

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

   3. Simplified 91.8%

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

      1. add-cube-cbrt 90.8%

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

      2. pow3 90.7%

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

   5. Applied egg-rr 90.7%

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

      1. rem-cube-cbrt 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*r* 93.4%

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

      4. remove-double-div 93.4%

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

      5. div-inv 93.4%

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

      6. frac-2neg 93.4%

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

      7. metadata-eval 93.4%

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

      8. associate-/r/ 93.4%

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

      9. sub-neg 93.4%

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

      10. distribute-neg-in 93.4%

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

      11. add-sqr-sqrt 52.3%

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

      12. sqrt-unprod 83.0%

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

      13. sqr-neg 83.0%

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

      14. sqrt-unprod 30.9%

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

      15. add-sqr-sqrt 71.1%

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

      16. add-sqr-sqrt 40.1%

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

      17. sqrt-unprod 80.4%

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

      18. sqr-neg 80.4%

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

      19. sqrt-unprod 41.0%

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

      20. add-sqr-sqrt 93.4%

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

   7. Applied egg-rr 93.4%

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

   8. Taylor expanded in x around inf 69.5%

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

      1. associate-*r* 74.0%

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

   10. Simplified 74.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+115} \lor \neg \left(z \leq 1.4 \cdot 10^{+64}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 3: 71.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+115}:\\ \;\;\;\;\left(-y\right) \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.22e+115)
   (* (- y) (* t z))
   (if (<= z 2e+64) (* x (* t y)) (* z (* t (- y))))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.22e+115) {
		tmp = -y * (t * z);
	} else if (z <= 2e+64) {
		tmp = x * (t * y);
	} else {
		tmp = 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 (z <= (-1.22d+115)) then
        tmp = -y * (t * z)
    else if (z <= 2d+64) then
        tmp = x * (t * y)
    else
        tmp = 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 (z <= -1.22e+115) {
		tmp = -y * (t * z);
	} else if (z <= 2e+64) {
		tmp = x * (t * y);
	} else {
		tmp = z * (t * -y);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if z <= -1.22e+115:
		tmp = -y * (t * z)
	elif z <= 2e+64:
		tmp = x * (t * y)
	else:
		tmp = z * (t * -y)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.22e+115)
		tmp = Float64(Float64(-y) * Float64(t * z));
	elseif (z <= 2e+64)
		tmp = Float64(x * Float64(t * y));
	else
		tmp = Float64(z * Float64(t * Float64(-y)));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.22e+115)
		tmp = -y * (t * z);
	elseif (z <= 2e+64)
		tmp = x * (t * y);
	else
		tmp = 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[z, -1.22e+115], N[((-y) * N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+64], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(t * (-y)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.22e115

   1. Initial program 88.2%

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

      1. distribute-rgt-out-- 93.2%

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

      2. associate-*l* 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in x around 0 87.0%

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

      1. associate-*r* 87.0%

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

      2. neg-mul-1 87.0%

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

   6. Simplified 87.0%

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

   if -1.22e115 < z < 2.00000000000000004e64

   1. Initial program 88.9%

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

      1. distribute-rgt-out-- 88.9%

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

      2. associate-*l* 91.8%

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

   3. Simplified 91.8%

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

      1. add-cube-cbrt 90.8%

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

      2. pow3 90.7%

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

   5. Applied egg-rr 90.7%

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

      1. rem-cube-cbrt 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*r* 93.4%

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

      4. remove-double-div 93.4%

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

      5. div-inv 93.4%

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

      6. frac-2neg 93.4%

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

      7. metadata-eval 93.4%

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

      8. associate-/r/ 93.4%

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

      9. sub-neg 93.4%

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

      10. distribute-neg-in 93.4%

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

      11. add-sqr-sqrt 52.3%

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

      12. sqrt-unprod 83.0%

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

      13. sqr-neg 83.0%

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

      14. sqrt-unprod 30.9%

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

      15. add-sqr-sqrt 71.1%

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

      16. add-sqr-sqrt 40.1%

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

      17. sqrt-unprod 80.4%

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

      18. sqr-neg 80.4%

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

      19. sqrt-unprod 41.0%

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

      20. add-sqr-sqrt 93.4%

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

   7. Applied egg-rr 93.4%

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

   8. Taylor expanded in x around inf 69.5%

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

      1. associate-*r* 74.0%

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

   10. Simplified 74.0%

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

   if 2.00000000000000004e64 < 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-- 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in x around 0 79.6%

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

      1. neg-mul-1 79.6%

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

      2. *-commutative 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

   6. Simplified 79.6%

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

      1. associate-*l* 82.3%

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

      2. distribute-lft-neg-in 82.3%

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

      3. /-rgt-identity 82.3%

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

      4. metadata-eval 82.3%

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

      5. frac-2neg 82.3%

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

      6. clear-num 80.8%

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

      7. un-div-inv 80.7%

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

   8. Applied egg-rr 80.7%

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

      1. associate-/r/ 82.3%

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

   10. Simplified 82.3%

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

   11. Taylor expanded in z around 0 75.6%

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

       1. mul-1-neg 75.6%

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

       2. associate-*r* 82.3%

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

       3. distribute-rgt-neg-in 82.3%

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

   13. Simplified 82.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+115}:\\ \;\;\;\;\left(-y\right) \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 4: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.1e-83) (* 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 <= -1.1e-83) {
		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 <= (-1.1d-83)) 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 <= -1.1e-83) {
		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 <= -1.1e-83:
		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 <= -1.1e-83)
		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 <= -1.1e-83)
		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, -1.1e-83], 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 < -1.10000000000000004e-83

   1. Initial program 80.9%

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

      1. distribute-rgt-out-- 85.3%

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

      2. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   if -1.10000000000000004e-83 < y

   1. Initial program 90.5%

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

      1. distribute-rgt-out-- 91.0%

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

   3. Simplified 91.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5: 55.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.1e+36) (* y (* t x)) (* x (* t y))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.1e+36) {
		tmp = y * (t * x);
	} else {
		tmp = x * (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 <= 1.1d+36) then
        tmp = y * (t * x)
    else
        tmp = x * (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 <= 1.1e+36) {
		tmp = y * (t * x);
	} else {
		tmp = x * (t * y);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 1.1e+36:
		tmp = y * (t * x)
	else:
		tmp = x * (t * y)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.1e+36)
		tmp = Float64(y * Float64(t * x));
	else
		tmp = Float64(x * 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 <= 1.1e+36)
		tmp = y * (t * x);
	else
		tmp = x * (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, 1.1e+36], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e36

   1. Initial program 86.7%

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

      1. distribute-rgt-out-- 88.4%

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

      2. associate-*l* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around inf 53.2%

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

   if 1.1e36 < t

   1. Initial program 91.0%

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

      1. distribute-rgt-out-- 92.5%

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

      2. associate-*l* 89.7%

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

   3. Simplified 89.7%

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

      1. add-cube-cbrt 88.8%

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

      2. pow3 88.8%

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

   5. Applied egg-rr 88.8%

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

      1. rem-cube-cbrt 89.7%

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

      2. *-commutative 89.7%

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

      3. associate-*r* 94.2%

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

      4. remove-double-div 94.2%

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

      5. div-inv 94.2%

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

      6. frac-2neg 94.2%

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

      7. metadata-eval 94.2%

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

      8. associate-/r/ 94.2%

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

      9. sub-neg 94.2%

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

      10. distribute-neg-in 94.2%

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

      11. add-sqr-sqrt 40.8%

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

      12. sqrt-unprod 61.6%

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

      13. sqr-neg 61.6%

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

      14. sqrt-unprod 24.9%

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

      15. add-sqr-sqrt 44.8%

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

      16. add-sqr-sqrt 19.8%

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

      17. sqrt-unprod 71.5%

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

      18. sqr-neg 71.5%

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

      19. sqrt-unprod 53.2%

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

      20. add-sqr-sqrt 94.2%

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

   7. Applied egg-rr 94.2%

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

   8. Taylor expanded in x around inf 47.4%

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

      1. associate-*r* 53.2%

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

   10. Simplified 53.2%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 6: 91.7% 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 87.8%

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

   1. distribute-rgt-out-- 89.4%

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

   2. associate-*l* 90.2%

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

3. Simplified 90.2%

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

4. Final simplification 90.2%

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

Alternative 7: 52.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * (t * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. distribute-rgt-out-- 89.4%

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

   2. associate-*l* 90.2%

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

3. Simplified 90.2%

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

4. Taylor expanded in x around inf 51.7%

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

5. Final simplification 51.7%

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

Developer target: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< t -9.231879582886777e-80)
   (* (* y t) (- x z))
   (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t < (-9.231879582886777d-80)) then
        tmp = (y * t) * (x - z)
    else if (t < 2.543067051564877d+83) then
        tmp = y * (t * (x - z))
    else
        tmp = (y * (x - z)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t < -9.231879582886777e-80:
		tmp = (y * t) * (x - z)
	elif t < 2.543067051564877e+83:
		tmp = y * (t * (x - z))
	else:
		tmp = (y * (x - z)) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t < -9.231879582886777e-80)
		tmp = Float64(Float64(y * t) * Float64(x - z));
	elseif (t < 2.543067051564877e+83)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(Float64(y * Float64(x - z)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t < -9.231879582886777e-80)
		tmp = (y * t) * (x - z);
	elseif (t < 2.543067051564877e+83)
		tmp = y * (t * (x - z));
	else
		tmp = (y * (x - z)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[t, -9.231879582886777e-80], N[(N[(y * t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], If[Less[t, 2.543067051564877e+83], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023199 
(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))