Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.9% → 95.8%

Time: 6.6s

Alternatives: 5

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: 90.9% 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: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+128}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -8e+128) (* y (* (- x z) t)) (* t (* y (- x z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8e+128) {
		tmp = y * ((x - z) * t);
	} 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 <= (-8d+128)) then
        tmp = y * ((x - z) * t)
    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 <= -8e+128) {
		tmp = y * ((x - z) * t);
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -8e+128:
		tmp = y * ((x - z) * t)
	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 <= -8e+128)
		tmp = Float64(y * Float64(Float64(x - z) * t));
	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 <= -8e+128)
		tmp = y * ((x - z) * t);
	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, -8e+128], N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.0000000000000006e128

   1. Initial program 73.7%

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

      1. distribute-rgt-out-- 81.3%

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

      2. associate-*l* 97.4%

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

   3. Simplified 97.4%

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

   if -8.0000000000000006e128 < y

   1. Initial program 95.2%

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

      1. distribute-rgt-out-- 95.7%

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

   3. Simplified 95.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+128}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 2: 74.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+45} \lor \neg \left(x \leq 5.2 \cdot 10^{-52}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8.2e+45) (not (<= x 5.2e-52)))
   (* t (* y x))
   (* z (* t (- y)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.2e+45) || !(x <= 5.2e-52)) {
		tmp = t * (y * x);
	} 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 ((x <= (-8.2d+45)) .or. (.not. (x <= 5.2d-52))) then
        tmp = t * (y * x)
    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 ((x <= -8.2e+45) || !(x <= 5.2e-52)) {
		tmp = t * (y * x);
	} else {
		tmp = z * (t * -y);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if (x <= -8.2e+45) or not (x <= 5.2e-52):
		tmp = t * (y * x)
	else:
		tmp = z * (t * -y)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -8.2e+45) || !(x <= 5.2e-52))
		tmp = Float64(t * Float64(y * x));
	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 ((x <= -8.2e+45) || ~((x <= 5.2e-52)))
		tmp = t * (y * x);
	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[Or[LessEqual[x, -8.2e+45], N[Not[LessEqual[x, 5.2e-52]], $MachinePrecision]], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(t * (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.20000000000000025e45 or 5.1999999999999997e-52 < x

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in x around inf 75.6%

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

   if -8.20000000000000025e45 < x < 5.1999999999999997e-52

   1. Initial program 94.0%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in x around 0 80.7%

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

      1. neg-mul-1 80.7%

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

      2. distribute-lft-neg-in 80.7%

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

   6. Simplified 80.7%

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

      1. distribute-lft-neg-in 80.7%

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

      2. distribute-lft-neg-out 80.7%

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

      3. *-commutative 80.7%

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

      4. associate-*l* 82.6%

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

      5. *-commutative 82.6%

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

   8. Applied egg-rr 82.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+45} \lor \neg \left(x \leq 5.2 \cdot 10^{-52}\right):\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 3: 91.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 91.8%

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

   1. distribute-rgt-out-- 93.4%

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

   2. associate-*l* 92.9%

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

3. Simplified 92.9%

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

4. Final simplification 92.9%

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

Alternative 4: 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 91.8%

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

   1. distribute-rgt-out-- 93.4%

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

   2. associate-*l* 92.9%

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

3. Simplified 92.9%

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

4. Taylor expanded in x around inf 52.4%

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

5. Final simplification 52.4%

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

Alternative 5: 53.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y 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 91.8%

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

   1. distribute-rgt-out-- 93.4%

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

3. Simplified 93.4%

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

4. Taylor expanded in x around inf 53.1%

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

5. Final simplification 53.1%

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

Developer target: 96.3% 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 2023238 
(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))