Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.1% → 98.2%

Time: 8.1s

Alternatives: 8

Speedup: 1.3×

• 25.4% 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: 90.1% 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: 98.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+284}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- (* x y) (* y z)) t)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+284)))
     (* y (* t (- x z)))
     (fma (* y (- x z)) t (* t (fma y (- z) (* y z)))))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double t_1 = ((x * y) - (y * z)) * t;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+284)) {
		tmp = y * (t * (x - z));
	} else {
		tmp = fma((y * (x - z)), t, (t * fma(y, -z, (y * z))));
	}
	return tmp;
}

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * y) - Float64(y * z)) * t)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+284))
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = fma(Float64(y * Float64(x - z)), t, Float64(t * fma(y, Float64(-z), Float64(y * z))));
	end
	return tmp
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+284]], $MachinePrecision]], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t + N[(t * N[(y * (-z) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < -inf.0 or 4.9999999999999999e284 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t)

   1. Initial program 71.7%

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

      1. distribute-rgt-out-- 75.3%

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

      2. associate-*l* 98.8%

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

   3. Simplified 98.8%

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

   if -inf.0 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < 4.9999999999999999e284

   1. Initial program 99.1%

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

      1. distribute-rgt-out-- 99.1%

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

      2. associate-*l* 88.3%

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

   3. Simplified 88.3%

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

      1. associate-*r* 99.1%

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

      2. distribute-rgt-out-- 99.1%

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

      3. *-commutative 99.1%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. fma-neg 99.1%

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

      7. distribute-rgt-in 99.2%

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

      8. fma-def 99.3%

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

      9. distribute-rgt-out-- 99.3%

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

      10. *-commutative 99.3%

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

      11. fma-udef 99.1%

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

      12. distribute-lft-neg-in 99.1%

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

      13. distribute-rgt-neg-in 99.1%

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

      14. fma-def 99.3%

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

      15. *-commutative 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq -\infty \lor \neg \left(\left(x \cdot y - y \cdot z\right) \cdot t \leq 5 \cdot 10^{+284}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \end{array} \]

Alternative 2: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq 5 \cdot 10^{+162}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x y) (* y z)) 5e+162) (* t (* y (- x z))) (* y (* t (- x z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * y) - (y * z)) <= 5e+162) {
		tmp = t * (y * (x - z));
	} else {
		tmp = y * (t * (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 (((x * y) - (y * z)) <= 5d+162) then
        tmp = t * (y * (x - z))
    else
        tmp = y * (t * (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 (((x * y) - (y * z)) <= 5e+162) {
		tmp = t * (y * (x - z));
	} else {
		tmp = y * (t * (x - z));
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if ((x * y) - (y * z)) <= 5e+162:
		tmp = t * (y * (x - z))
	else:
		tmp = y * (t * (x - z))
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(y * z)) <= 5e+162)
		tmp = Float64(t * Float64(y * Float64(x - z)));
	else
		tmp = Float64(y * Float64(t * 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 (((x * y) - (y * z)) <= 5e+162)
		tmp = t * (y * (x - z));
	else
		tmp = y * (t * (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[N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], 5e+162], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < 4.9999999999999997e162

   1. Initial program 96.3%

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

      1. distribute-rgt-out-- 96.3%

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

   3. Simplified 96.3%

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

   if 4.9999999999999997e162 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 72.8%

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

      1. distribute-rgt-out-- 77.3%

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

      2. associate-*l* 98.5%

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

   3. Simplified 98.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq 5 \cdot 10^{+162}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 3: 72.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} t_1 := y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-51} \lor \neg \left(z \leq 8 \cdot 10^{+75}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* t (- z)))))
   (if (<= z -4.6e-33)
     t_1
     (if (<= z 2e-99)
       (* x (* y t))
       (if (or (<= z 1.8e-51) (not (<= z 8e+75))) t_1 (* (* x y) t))))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double t_1 = y * (t * -z);
	double tmp;
	if (z <= -4.6e-33) {
		tmp = t_1;
	} else if (z <= 2e-99) {
		tmp = x * (y * t);
	} else if ((z <= 1.8e-51) || !(z <= 8e+75)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = y * (t * -z)
    if (z <= (-4.6d-33)) then
        tmp = t_1
    else if (z <= 2d-99) then
        tmp = x * (y * t)
    else if ((z <= 1.8d-51) .or. (.not. (z <= 8d+75))) then
        tmp = t_1
    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 t_1 = y * (t * -z);
	double tmp;
	if (z <= -4.6e-33) {
		tmp = t_1;
	} else if (z <= 2e-99) {
		tmp = x * (y * t);
	} else if ((z <= 1.8e-51) || !(z <= 8e+75)) {
		tmp = t_1;
	} else {
		tmp = (x * y) * t;
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	t_1 = y * (t * -z)
	tmp = 0
	if z <= -4.6e-33:
		tmp = t_1
	elif z <= 2e-99:
		tmp = x * (y * t)
	elif (z <= 1.8e-51) or not (z <= 8e+75):
		tmp = t_1
	else:
		tmp = (x * y) * t
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	t_1 = Float64(y * Float64(t * Float64(-z)))
	tmp = 0.0
	if (z <= -4.6e-33)
		tmp = t_1;
	elseif (z <= 2e-99)
		tmp = Float64(x * Float64(y * t));
	elseif ((z <= 1.8e-51) || !(z <= 8e+75))
		tmp = t_1;
	else
		tmp = Float64(Float64(x * y) * t);
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = y * (t * -z);
	tmp = 0.0;
	if (z <= -4.6e-33)
		tmp = t_1;
	elseif (z <= 2e-99)
		tmp = x * (y * t);
	elseif ((z <= 1.8e-51) || ~((z <= 8e+75)))
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.6e-33], t$95$1, If[LessEqual[z, 2e-99], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 1.8e-51], N[Not[LessEqual[z, 8e+75]], $MachinePrecision]], t$95$1, N[(N[(x * y), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.59999999999999971e-33 or 2e-99 < z < 1.8e-51 or 7.99999999999999941e75 < z

   1. Initial program 86.4%

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

      1. distribute-rgt-out-- 88.6%

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

      2. associate-*l* 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in x around 0 76.5%

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

      1. associate-*r* 76.5%

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

      2. neg-mul-1 76.5%

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

   6. Simplified 76.5%

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

   if -4.59999999999999971e-33 < z < 2e-99

   1. Initial program 94.7%

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

      1. distribute-rgt-out-- 94.7%

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

      2. flip-- 73.4%

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

      3. associate-*r/ 67.5%

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

   3. Applied egg-rr 67.5%

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

      1. associate-/l* 73.3%

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

      2. difference-of-squares 73.3%

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

      3. associate-/r* 94.6%

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

      4. *-inverses 94.6%

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

   5. Simplified 94.6%

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

      1. associate-*l/ 92.5%

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

      2. associate-/l* 94.4%

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

      3. clear-num 94.4%

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

      4. associate-/r/ 94.5%

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

      5. /-rgt-identity 94.5%

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

      6. *-commutative 94.5%

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

      7. sub-neg 94.5%

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

      8. add-sqr-sqrt 55.5%

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

      9. sqrt-unprod 88.7%

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

      10. sqr-neg 88.7%

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

      11. sqrt-unprod 34.6%

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

      12. add-sqr-sqrt 81.4%

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

   7. Applied egg-rr 81.4%

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

   8. Taylor expanded in x around inf 81.8%

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

      1. *-commutative 81.8%

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

   10. Simplified 81.8%

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

       1. associate-/r/ 81.7%

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

       2. /-rgt-identity 81.7%

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

       3. *-commutative 81.7%

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

       4. associate-*r* 81.9%

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

   12. Applied egg-rr 81.9%

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

   if 1.8e-51 < z < 7.99999999999999941e75

   1. Initial program 94.0%

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

   2. Taylor expanded in x around inf 73.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-33}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-51} \lor \neg \left(z \leq 8 \cdot 10^{+75}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 4: 72.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} t_1 := z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+75}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* y (- t)))))
   (if (<= z -3.8e-33)
     t_1
     (if (<= z 2.5e-99)
       (* x (* y t))
       (if (<= z 8.5e-53)
         t_1
         (if (<= z 2.9e+75) (* (* x y) t) (* y (* t (- z)))))))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double t_1 = z * (y * -t);
	double tmp;
	if (z <= -3.8e-33) {
		tmp = t_1;
	} else if (z <= 2.5e-99) {
		tmp = x * (y * t);
	} else if (z <= 8.5e-53) {
		tmp = t_1;
	} else if (z <= 2.9e+75) {
		tmp = (x * y) * t;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = z * (y * -t)
    if (z <= (-3.8d-33)) then
        tmp = t_1
    else if (z <= 2.5d-99) then
        tmp = x * (y * t)
    else if (z <= 8.5d-53) then
        tmp = t_1
    else if (z <= 2.9d+75) then
        tmp = (x * y) * t
    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 t_1 = z * (y * -t);
	double tmp;
	if (z <= -3.8e-33) {
		tmp = t_1;
	} else if (z <= 2.5e-99) {
		tmp = x * (y * t);
	} else if (z <= 8.5e-53) {
		tmp = t_1;
	} else if (z <= 2.9e+75) {
		tmp = (x * y) * t;
	} else {
		tmp = y * (t * -z);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	t_1 = z * (y * -t)
	tmp = 0
	if z <= -3.8e-33:
		tmp = t_1
	elif z <= 2.5e-99:
		tmp = x * (y * t)
	elif z <= 8.5e-53:
		tmp = t_1
	elif z <= 2.9e+75:
		tmp = (x * y) * t
	else:
		tmp = y * (t * -z)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	t_1 = Float64(z * Float64(y * Float64(-t)))
	tmp = 0.0
	if (z <= -3.8e-33)
		tmp = t_1;
	elseif (z <= 2.5e-99)
		tmp = Float64(x * Float64(y * t));
	elseif (z <= 8.5e-53)
		tmp = t_1;
	elseif (z <= 2.9e+75)
		tmp = Float64(Float64(x * y) * t);
	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)
	t_1 = z * (y * -t);
	tmp = 0.0;
	if (z <= -3.8e-33)
		tmp = t_1;
	elseif (z <= 2.5e-99)
		tmp = x * (y * t);
	elseif (z <= 8.5e-53)
		tmp = t_1;
	elseif (z <= 2.9e+75)
		tmp = (x * y) * t;
	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_] := Block[{t$95$1 = N[(z * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.8e-33], t$95$1, If[LessEqual[z, 2.5e-99], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e-53], t$95$1, If[LessEqual[z, 2.9e+75], N[(N[(x * y), $MachinePrecision] * t), $MachinePrecision], N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.79999999999999994e-33 or 2.49999999999999985e-99 < z < 8.50000000000000044e-53

   1. Initial program 91.1%

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

   2. Taylor expanded in x around 0 76.8%

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

      1. neg-mul-1 76.8%

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

      2. distribute-lft-neg-in 76.8%

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

   4. Simplified 76.8%

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

      1. associate-*l* 75.6%

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

      2. neg-mul-1 75.6%

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

      3. *-commutative 75.6%

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

      4. metadata-eval 75.6%

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

      5. div-inv 75.6%

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

      6. associate-*l* 76.8%

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

      7. associate-/r/ 76.8%

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

      8. associate-*l/ 79.2%

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

      9. associate-/l* 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. associate-/l* 79.2%

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

      2. associate-/r/ 79.2%

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

      3. associate-/l* 79.1%

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

   8. Simplified 79.1%

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

      1. clear-num 79.0%

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

      2. associate-/r/ 79.2%

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

      3. frac-2neg 79.2%

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

      4. metadata-eval 79.2%

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

      5. remove-double-div 79.2%

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

   10. Applied egg-rr 79.2%

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

   if -3.79999999999999994e-33 < z < 2.49999999999999985e-99

   1. Initial program 94.7%

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

      1. distribute-rgt-out-- 94.7%

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

      2. flip-- 73.4%

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

      3. associate-*r/ 67.5%

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

   3. Applied egg-rr 67.5%

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

      1. associate-/l* 73.3%

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

      2. difference-of-squares 73.3%

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

      3. associate-/r* 94.6%

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

      4. *-inverses 94.6%

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

   5. Simplified 94.6%

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

      1. associate-*l/ 92.5%

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

      2. associate-/l* 94.4%

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

      3. clear-num 94.4%

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

      4. associate-/r/ 94.5%

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

      5. /-rgt-identity 94.5%

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

      6. *-commutative 94.5%

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

      7. sub-neg 94.5%

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

      8. add-sqr-sqrt 55.5%

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

      9. sqrt-unprod 88.7%

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

      10. sqr-neg 88.7%

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

      11. sqrt-unprod 34.6%

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

      12. add-sqr-sqrt 81.4%

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

   7. Applied egg-rr 81.4%

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

   8. Taylor expanded in x around inf 81.8%

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

      1. *-commutative 81.8%

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

   10. Simplified 81.8%

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

       1. associate-/r/ 81.7%

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

       2. /-rgt-identity 81.7%

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

       3. *-commutative 81.7%

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

       4. associate-*r* 81.9%

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

   12. Applied egg-rr 81.9%

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

   if 8.50000000000000044e-53 < z < 2.8999999999999998e75

   1. Initial program 94.0%

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

   2. Taylor expanded in x around inf 73.3%

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

   if 2.8999999999999998e75 < z

   1. Initial program 80.2%

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

      1. distribute-rgt-out-- 80.2%

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

      2. associate-*l* 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in x around 0 77.5%

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

      1. associate-*r* 77.5%

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

      2. neg-mul-1 77.5%

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

   6. Simplified 77.5%

      \[\leadsto \color{blue}{\left(-y\right) \cdot \left(t \cdot z\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-33}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-53}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+75}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \end{array} \]

Alternative 5: 56.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.95e-66) (* y (* x t)) (* x (* y t))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.95e-66) {
		tmp = y * (x * t);
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.95d-66) then
        tmp = y * (x * t)
    else
        tmp = x * (y * t)
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.95e-66) {
		tmp = y * (x * t);
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 1.95e-66:
		tmp = y * (x * t)
	else:
		tmp = x * (y * t)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.95e-66)
		tmp = Float64(y * Float64(x * t));
	else
		tmp = Float64(x * Float64(y * t));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.95e-66)
		tmp = y * (x * t);
	else
		tmp = x * (y * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 1.95e-66], N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.94999999999999991e-66

   1. Initial program 88.3%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in x around inf 48.6%

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

   if 1.94999999999999991e-66 < t

   1. Initial program 94.2%

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

      1. distribute-rgt-out-- 96.6%

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

      2. flip-- 66.5%

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

      3. associate-*r/ 66.4%

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

   3. Applied egg-rr 66.4%

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

      1. associate-/l* 66.3%

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

      2. difference-of-squares 71.2%

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

      3. associate-/r* 96.5%

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

      4. *-inverses 96.5%

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

   5. Simplified 96.5%

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

      1. associate-*l/ 96.4%

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

      2. associate-/l* 91.9%

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

      3. clear-num 91.9%

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

      4. associate-/r/ 92.0%

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

      5. /-rgt-identity 92.0%

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

      6. *-commutative 92.0%

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

      7. sub-neg 92.0%

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

      8. add-sqr-sqrt 42.4%

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

      9. sqrt-unprod 57.3%

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

      10. sqr-neg 57.3%

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

      11. sqrt-unprod 24.5%

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

      12. add-sqr-sqrt 41.1%

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

   7. Applied egg-rr 41.1%

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

   8. Taylor expanded in x around inf 44.9%

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

      1. *-commutative 44.9%

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

   10. Simplified 44.9%

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

       1. associate-/r/ 44.9%

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

       2. /-rgt-identity 44.9%

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

       3. *-commutative 44.9%

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

       4. associate-*r* 52.7%

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

   12. Applied egg-rr 52.7%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 6: 56.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -3e-36) (* x (* y t)) (* (* x y) t)))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3e-36) {
		tmp = x * (y * t);
	} 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 (y <= (-3d-36)) then
        tmp = x * (y * t)
    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 (y <= -3e-36) {
		tmp = x * (y * t);
	} else {
		tmp = (x * y) * t;
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -3e-36:
		tmp = x * (y * t)
	else:
		tmp = (x * y) * t
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3e-36)
		tmp = Float64(x * Float64(y * t));
	else
		tmp = Float64(Float64(x * 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 (y <= -3e-36)
		tmp = x * (y * t);
	else
		tmp = (x * y) * t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -3e-36], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.0000000000000002e-36

   1. Initial program 83.4%

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

      1. distribute-rgt-out-- 86.3%

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

      2. flip-- 72.2%

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

      3. associate-*r/ 65.9%

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

   3. Applied egg-rr 65.9%

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

      1. associate-/l* 72.1%

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

      2. difference-of-squares 73.7%

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

      3. associate-/r* 86.2%

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

      4. *-inverses 86.2%

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

   5. Simplified 86.2%

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

      1. associate-*l/ 98.4%

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

      2. associate-/l* 92.9%

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

      3. clear-num 92.8%

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

      4. associate-/r/ 92.9%

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

      5. /-rgt-identity 92.9%

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

      6. *-commutative 92.9%

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

      7. sub-neg 92.9%

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

      8. add-sqr-sqrt 43.6%

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

      9. sqrt-unprod 55.6%

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

      10. sqr-neg 55.6%

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

      11. sqrt-unprod 19.1%

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

      12. add-sqr-sqrt 36.6%

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

   7. Applied egg-rr 36.6%

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

   8. Taylor expanded in x around inf 39.7%

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

      1. *-commutative 39.7%

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

   10. Simplified 39.7%

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

       1. associate-/r/ 39.7%

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

       2. /-rgt-identity 39.7%

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

       3. *-commutative 39.7%

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

       4. associate-*r* 48.0%

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

   12. Applied egg-rr 48.0%

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

   if -3.0000000000000002e-36 < y

   1. Initial program 92.8%

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

   2. Taylor expanded in x around inf 52.2%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 7: 92.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

   1. distribute-rgt-out-- 91.4%

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

   2. associate-*l* 91.7%

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

3. Simplified 91.7%

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

4. Final simplification 91.7%

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

Alternative 8: 53.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

assert(y < t);
double code(double x, double y, double z, double t) {
	return y * (x * 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 = y * (x * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

   1. distribute-rgt-out-- 91.4%

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

   2. associate-*l* 91.7%

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

3. Simplified 91.7%

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

4. Taylor expanded in x around inf 47.4%

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

5. Final simplification 47.4%

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

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 2023240 
(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))