Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.2% → 97.2%

Time: 7.1s

Alternatives: 7

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: 90.2% 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: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 9.5e-42) (* y (* t (- x z))) (* (- x z) (* t y))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.5e-42) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (x - z) * (t * y);
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 9.5d-42) then
        tmp = y * (t * (x - z))
    else
        tmp = (x - z) * (t * y)
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.5e-42) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (x - z) * (t * y);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 9.5e-42:
		tmp = y * (t * (x - z))
	else:
		tmp = (x - z) * (t * y)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 9.5e-42)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(Float64(x - z) * Float64(t * y));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 9.5e-42)
		tmp = y * (t * (x - z));
	else
		tmp = (x - z) * (t * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 9.5e-42], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(t * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.49999999999999948e-42

   1. Initial program 91.6%

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

      1. distribute-rgt-out-- 92.2%

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

      2. associate-*l* 92.8%

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

   3. Simplified 92.8%

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

   if 9.49999999999999948e-42 < t

   1. Initial program 94.9%

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

      1. distribute-rgt-out-- 96.1%

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

      2. associate-*l* 86.9%

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

   3. Simplified 86.9%

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

      1. add-cube-cbrt 86.4%

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

      2. pow3 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. rem-cube-cbrt 86.9%

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

      2. *-commutative 86.9%

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

      3. associate-*l* 98.6%

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

      4. *-commutative 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 2: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-24}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 0.00106:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.8e-24)
   (* t (* y x))
   (if (<= x 0.00106) (* y (* t (- z))) (* x (* t y)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.8e-24) {
		tmp = t * (y * x);
	} else if (x <= 0.00106) {
		tmp = y * (t * -z);
	} else {
		tmp = x * (t * y);
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.8d-24)) then
        tmp = t * (y * x)
    else if (x <= 0.00106d0) then
        tmp = y * (t * -z)
    else
        tmp = x * (t * y)
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.8e-24) {
		tmp = t * (y * x);
	} else if (x <= 0.00106) {
		tmp = y * (t * -z);
	} else {
		tmp = x * (t * y);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if x <= -2.8e-24:
		tmp = t * (y * x)
	elif x <= 0.00106:
		tmp = y * (t * -z)
	else:
		tmp = x * (t * y)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.8e-24)
		tmp = Float64(t * Float64(y * x));
	elseif (x <= 0.00106)
		tmp = Float64(y * Float64(t * Float64(-z)));
	else
		tmp = Float64(x * Float64(t * y));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.8e-24)
		tmp = t * (y * x);
	elseif (x <= 0.00106)
		tmp = y * (t * -z);
	else
		tmp = x * (t * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, -2.8e-24], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00106], N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8000000000000002e-24

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in x around inf 83.9%

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

   if -2.8000000000000002e-24 < x < 0.00105999999999999996

   1. Initial program 91.5%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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* 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in x around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. neg-mul-1 74.8%

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

   6. Simplified 74.8%

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

   if 0.00105999999999999996 < x

   1. Initial program 94.0%

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

      1. distribute-rgt-out-- 95.5%

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

      2. associate-*l* 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in x around inf 68.4%

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

      1. add-log-exp 29.5%

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

      2. associate-*r* 29.5%

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

      3. exp-prod 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. log-pow 30.9%

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

      2. rem-log-exp 76.5%

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

   8. Simplified 76.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-24}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 0.00106:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 3: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-24}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.2e-24)
   (* t (* y x))
   (if (<= x 1.25e-23) (* (* t y) (- z)) (* x (* t y)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.2e-24) {
		tmp = t * (y * x);
	} else if (x <= 1.25e-23) {
		tmp = (t * y) * -z;
	} else {
		tmp = x * (t * y);
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-6.2d-24)) then
        tmp = t * (y * x)
    else if (x <= 1.25d-23) then
        tmp = (t * y) * -z
    else
        tmp = x * (t * y)
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.2e-24) {
		tmp = t * (y * x);
	} else if (x <= 1.25e-23) {
		tmp = (t * y) * -z;
	} else {
		tmp = x * (t * y);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if x <= -6.2e-24:
		tmp = t * (y * x)
	elif x <= 1.25e-23:
		tmp = (t * y) * -z
	else:
		tmp = x * (t * y)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.2e-24)
		tmp = Float64(t * Float64(y * x));
	elseif (x <= 1.25e-23)
		tmp = Float64(Float64(t * y) * Float64(-z));
	else
		tmp = Float64(x * Float64(t * y));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -6.2e-24)
		tmp = t * (y * x);
	elseif (x <= 1.25e-23)
		tmp = (t * y) * -z;
	else
		tmp = x * (t * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, -6.2e-24], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-23], N[(N[(t * y), $MachinePrecision] * (-z)), $MachinePrecision], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2000000000000001e-24

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in x around inf 83.9%

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

   if -6.2000000000000001e-24 < x < 1.2500000000000001e-23

   1. Initial program 90.8%

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

      1. distribute-rgt-out-- 90.8%

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

      2. associate-*l* 91.8%

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

   3. Simplified 91.8%

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

      1. add-cube-cbrt 90.9%

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

      2. pow3 91.0%

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

   5. Applied egg-rr 91.0%

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

      1. rem-cube-cbrt 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*l* 93.6%

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

      4. *-commutative 93.6%

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

   7. Applied egg-rr 93.6%

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

   8. Taylor expanded in x around 0 75.2%

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

      1. mul-1-neg 75.2%

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

      2. associate-*r* 78.8%

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

      3. distribute-rgt-neg-out 78.8%

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

   10. Simplified 78.8%

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

   if 1.2500000000000001e-23 < x

   1. Initial program 94.8%

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

      1. distribute-rgt-out-- 96.1%

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

      2. associate-*l* 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in x around inf 67.6%

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

      1. add-log-exp 31.8%

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

      2. associate-*r* 31.8%

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

      3. exp-prod 31.0%

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

   6. Applied egg-rr 31.0%

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

      1. log-pow 33.7%

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

      2. rem-log-exp 74.7%

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

   8. Simplified 74.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-24}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 4: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-251}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.35e-251) (* y (* t (- x z))) (* t (* y (- x z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.35e-251) {
		tmp = y * (t * (x - z));
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.35d-251)) then
        tmp = y * (t * (x - z))
    else
        tmp = t * (y * (x - z))
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.35e-251) {
		tmp = y * (t * (x - z));
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -1.35e-251:
		tmp = y * (t * (x - z))
	else:
		tmp = t * (y * (x - z))
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.35e-251)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(t * Float64(y * Float64(x - z)));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.35e-251)
		tmp = y * (t * (x - z));
	else
		tmp = t * (y * (x - z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -1.35e-251], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35000000000000005e-251

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 94.1%

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

      2. associate-*l* 95.1%

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

   3. Simplified 95.1%

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

   if -1.35000000000000005e-251 < y

   1. Initial program 91.3%

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

      1. distribute-rgt-out-- 92.8%

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

   3. Simplified 92.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-251}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5: 56.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 3e-42) (* y (* t x)) (* x (* t y))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3e-42) {
		tmp = y * (t * x);
	} else {
		tmp = x * (t * y);
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 3d-42) then
        tmp = y * (t * x)
    else
        tmp = x * (t * y)
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3e-42) {
		tmp = y * (t * x);
	} else {
		tmp = x * (t * y);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 3e-42:
		tmp = y * (t * x)
	else:
		tmp = x * (t * y)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3e-42)
		tmp = Float64(y * Float64(t * x));
	else
		tmp = Float64(x * Float64(t * y));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 3e-42)
		tmp = y * (t * x);
	else
		tmp = x * (t * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 3e-42], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.00000000000000027e-42

   1. Initial program 91.6%

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

      1. distribute-rgt-out-- 92.2%

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

      2. associate-*l* 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around inf 61.3%

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

   if 3.00000000000000027e-42 < t

   1. Initial program 94.9%

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

      1. distribute-rgt-out-- 96.1%

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

      2. associate-*l* 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in x around inf 47.7%

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

      1. add-log-exp 33.6%

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

      2. associate-*r* 33.6%

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

      3. exp-prod 34.5%

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

   6. Applied egg-rr 34.5%

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

      1. log-pow 34.5%

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

      2. rem-log-exp 54.7%

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

   8. Simplified 54.7%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 6: 92.1% 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 92.6%

   \[\left(x \cdot y - z \cdot y\right) \cdot t \]
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* 91.0%

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

3. Simplified 91.0%

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

4. Final simplification 91.0%

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

Alternative 7: 54.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

   \[\left(x \cdot y - z \cdot y\right) \cdot t \]
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* 91.0%

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

3. Simplified 91.0%

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

4. Taylor expanded in x around inf 57.0%

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

   1. add-log-exp 31.5%

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

   2. associate-*r* 31.5%

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

   3. exp-prod 31.4%

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

6. Applied egg-rr 31.4%

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

   1. log-pow 32.2%

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

   2. rem-log-exp 58.3%

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

8. Simplified 58.3%

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

9. Final simplification 58.3%

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

Developer target: 96.4% 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 2023257 
(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))