Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.6% → 96.7%

Time: 11.0s

Alternatives: 7

Speedup: 1.3×

• 24.7% 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: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-75}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 4.8e-75) (* y (* (- x z) t)) (* (- x z) (* y t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.8e-75) {
		tmp = y * ((x - z) * t);
	} else {
		tmp = (x - z) * (y * t);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 4.8d-75) then
        tmp = y * ((x - z) * t)
    else
        tmp = (x - z) * (y * t)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 4.8e-75) {
		tmp = y * ((x - z) * t);
	} else {
		tmp = (x - z) * (y * t);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 4.8e-75:
		tmp = y * ((x - z) * t)
	else:
		tmp = (x - z) * (y * t)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 4.8e-75)
		tmp = Float64(y * Float64(Float64(x - z) * t));
	else
		tmp = Float64(Float64(x - z) * Float64(y * t));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 4.8e-75)
		tmp = y * ((x - z) * t);
	else
		tmp = (x - z) * (y * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 4.8e-75], N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.80000000000000039e-75

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 88.0%

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

      2. associate-*l* 97.3%

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

      3. *-commutative 97.3%

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

   3. Simplified 97.3%

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

   if 4.80000000000000039e-75 < t

   1. Initial program 90.3%

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

      1. *-commutative 90.3%

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

      2. distribute-rgt-out-- 94.9%

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

      3. associate-*r* 97.7%

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

      4. *-commutative 97.7%

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

   3. Simplified 97.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-75}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+154}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-186} \lor \neg \left(x \leq 6.6 \cdot 10^{-177}\right):\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.8e+154)
   (* t (* y x))
   (if (or (<= x -2e-186) (not (<= x 6.6e-177)))
     (* y (* (- x z) t))
     (* (* y t) (- z)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.8e+154) {
		tmp = t * (y * x);
	} else if ((x <= -2e-186) || !(x <= 6.6e-177)) {
		tmp = y * ((x - z) * t);
	} else {
		tmp = (y * t) * -z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, 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 <= (-3.8d+154)) then
        tmp = t * (y * x)
    else if ((x <= (-2d-186)) .or. (.not. (x <= 6.6d-177))) then
        tmp = y * ((x - z) * t)
    else
        tmp = (y * t) * -z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.8e+154) {
		tmp = t * (y * x);
	} else if ((x <= -2e-186) || !(x <= 6.6e-177)) {
		tmp = y * ((x - z) * t);
	} else {
		tmp = (y * t) * -z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= -3.8e+154:
		tmp = t * (y * x)
	elif (x <= -2e-186) or not (x <= 6.6e-177):
		tmp = y * ((x - z) * t)
	else:
		tmp = (y * t) * -z
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.8e+154)
		tmp = Float64(t * Float64(y * x));
	elseif ((x <= -2e-186) || !(x <= 6.6e-177))
		tmp = Float64(y * Float64(Float64(x - z) * t));
	else
		tmp = Float64(Float64(y * t) * Float64(-z));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.8e+154)
		tmp = t * (y * x);
	elseif ((x <= -2e-186) || ~((x <= 6.6e-177)))
		tmp = y * ((x - z) * t);
	else
		tmp = (y * t) * -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, -3.8e+154], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -2e-186], N[Not[LessEqual[x, 6.6e-177]], $MachinePrecision]], N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7999999999999998e154

   1. Initial program 79.0%

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

      1. distribute-rgt-out-- 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in x around inf 82.9%

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

      1. *-commutative 82.9%

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

   7. Simplified 82.9%

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

   if -3.7999999999999998e154 < x < -1.9999999999999998e-186 or 6.5999999999999999e-177 < x

   1. Initial program 86.9%

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

      1. distribute-rgt-out-- 91.5%

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

      2. associate-*l* 96.2%

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

      3. *-commutative 96.2%

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

   3. Simplified 96.2%

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

   if -1.9999999999999998e-186 < x < 6.5999999999999999e-177

   1. Initial program 90.6%

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

      1. distribute-rgt-out-- 90.6%

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

      2. associate-*l* 89.7%

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

      3. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in x around 0 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-*r* 88.4%

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

      3. associate-*r* 88.4%

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

      4. neg-mul-1 88.4%

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

      5. associate-*l* 87.5%

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

      6. *-commutative 87.5%

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

   7. Simplified 87.5%

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

      1. distribute-lft-neg-out 87.5%

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

      2. add-sqr-sqrt 43.2%

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

      3. sqrt-unprod 41.2%

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

      4. sqr-neg 41.2%

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

      5. sqrt-unprod 6.5%

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

      6. add-sqr-sqrt 15.7%

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

      7. associate-*r* 15.7%

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

      8. *-commutative 15.7%

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

      9. add-sqr-sqrt 6.5%

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

      10. sqrt-unprod 42.9%

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

      11. sqr-neg 42.9%

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

      12. sqrt-unprod 49.8%

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

      13. add-sqr-sqrt 97.3%

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

   9. Applied egg-rr 97.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+154}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-186} \lor \neg \left(x \leq 6.6 \cdot 10^{-177}\right):\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-74} \lor \neg \left(x \leq 1.7 \cdot 10^{+15}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.3e-74) (not (<= x 1.7e+15)))
   (* t (* y x))
   (* (* y t) (- z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.3e-74) || !(x <= 1.7e+15)) {
		tmp = t * (y * x);
	} else {
		tmp = (y * t) * -z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, 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 <= (-1.3d-74)) .or. (.not. (x <= 1.7d+15))) then
        tmp = t * (y * x)
    else
        tmp = (y * t) * -z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.3e-74) || !(x <= 1.7e+15)) {
		tmp = t * (y * x);
	} else {
		tmp = (y * t) * -z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (x <= -1.3e-74) or not (x <= 1.7e+15):
		tmp = t * (y * x)
	else:
		tmp = (y * t) * -z
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.3e-74) || !(x <= 1.7e+15))
		tmp = Float64(t * Float64(y * x));
	else
		tmp = Float64(Float64(y * t) * Float64(-z));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.3e-74) || ~((x <= 1.7e+15)))
		tmp = t * (y * x);
	else
		tmp = (y * t) * -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.3e-74], N[Not[LessEqual[x, 1.7e+15]], $MachinePrecision]], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * t), $MachinePrecision] * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e-74 or 1.7e15 < x

   1. Initial program 84.2%

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

      1. distribute-rgt-out-- 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around inf 72.2%

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

      1. *-commutative 72.2%

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

   7. Simplified 72.2%

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

   if -1.3e-74 < x < 1.7e15

   1. Initial program 90.5%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 94.1%

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

      3. *-commutative 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in x around 0 77.9%

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

      1. *-commutative 77.9%

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

      2. associate-*r* 77.9%

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

      3. associate-*r* 77.9%

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

      4. neg-mul-1 77.9%

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

      5. associate-*l* 81.9%

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

      6. *-commutative 81.9%

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

   7. Simplified 81.9%

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

      1. distribute-lft-neg-out 81.9%

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

      2. add-sqr-sqrt 36.3%

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

      3. sqrt-unprod 36.8%

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

      4. sqr-neg 36.8%

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

      5. sqrt-unprod 7.6%

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

      6. add-sqr-sqrt 15.8%

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

      7. associate-*r* 15.8%

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

      8. *-commutative 15.8%

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

      9. add-sqr-sqrt 7.7%

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

      10. sqrt-unprod 36.9%

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

      11. sqr-neg 36.9%

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

      12. sqrt-unprod 38.9%

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

      13. add-sqr-sqrt 85.3%

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

   9. Applied egg-rr 85.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-74} \lor \neg \left(x \leq 1.7 \cdot 10^{+15}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 3.45 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.45e-93) (* y (* x t)) (* x (* y t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.45e-93) {
		tmp = y * (x * t);
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, 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 <= 3.45d-93) then
        tmp = y * (x * t)
    else
        tmp = x * (y * t)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.45e-93) {
		tmp = y * (x * t);
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 3.45e-93:
		tmp = y * (x * t)
	else:
		tmp = x * (y * t)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.45e-93)
		tmp = Float64(y * Float64(x * t));
	else
		tmp = Float64(x * Float64(y * t));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 3.45e-93)
		tmp = y * (x * t);
	else
		tmp = x * (y * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 3.45e-93], N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.45000000000000015e-93

   1. Initial program 84.4%

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

      1. distribute-rgt-out-- 87.5%

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

      2. associate-*l* 97.2%

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

      3. *-commutative 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in x around inf 57.8%

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

   if 3.45000000000000015e-93 < 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-- 95.3%

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

      2. associate-*l* 88.8%

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

      3. *-commutative 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in x around inf 52.2%

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

      1. add-log-exp 33.0%

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

      2. associate-*r* 33.0%

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

      3. exp-prod 32.5%

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

      4. exp-prod 24.1%

         \[\leadsto \log \left({\color{blue}{\left({\left(e^{y}\right)}^{t}\right)}}^{x}\right) \]

   7. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{\log \left({\left({\left(e^{y}\right)}^{t}\right)}^{x}\right)} \]
   8. Step-by-step derivation

      1. log-pow 24.1%

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

      2. log-pow 25.4%

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

      3. rem-log-exp 57.0%

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

      4. *-commutative 57.0%

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

   9. Simplified 57.0%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.45 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, 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 - z)) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. distribute-rgt-out-- 90.4%

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

3. Simplified 90.4%

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

5. Final simplification 90.4%

   \[\leadsto \left(y \cdot \left(x - z\right)\right) \cdot t \]
6. Add Preprocessing

Alternative 6: 54.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. distribute-rgt-out-- 90.4%

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

   2. associate-*l* 94.1%

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

   3. *-commutative 94.1%

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

3. Simplified 94.1%

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

5. Taylor expanded in x around inf 55.7%

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

   1. add-log-exp 30.5%

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

   2. associate-*r* 30.5%

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

   3. exp-prod 30.5%

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

   4. exp-prod 24.7%

      \[\leadsto \log \left({\color{blue}{\left({\left(e^{y}\right)}^{t}\right)}}^{x}\right) \]

7. Applied egg-rr 24.7%

   \[\leadsto \color{blue}{\log \left({\left({\left(e^{y}\right)}^{t}\right)}^{x}\right)} \]
8. Step-by-step derivation

   1. log-pow 25.1%

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

   2. log-pow 26.1%

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

   3. rem-log-exp 56.5%

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

   4. *-commutative 56.5%

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

9. Simplified 56.5%

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

10. Final simplification 56.5%

    \[\leadsto x \cdot \left(y \cdot t\right) \]
11. Add Preprocessing

Alternative 7: 53.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, 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 = t * (y * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. distribute-rgt-out-- 90.4%

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

3. Simplified 90.4%

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

5. Taylor expanded in x around inf 54.4%

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

   1. *-commutative 54.4%

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

7. Simplified 54.4%

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

8. Final simplification 54.4%

   \[\leadsto t \cdot \left(y \cdot x\right) \]
9. Add Preprocessing

Developer target: 96.2% accurate, 0.5× 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 2024039 
(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))