Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.2% → 97.4%

Time: 8.4s

Alternatives: 6

Speedup: 1.0×

• 25.0% 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: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-34}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 5e-34) (* 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 <= 5e-34) {
		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 <= 5d-34) 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 <= 5e-34) {
		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 <= 5e-34:
		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 <= 5e-34)
		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 <= 5e-34)
		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, 5e-34], 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 < 5.0000000000000003e-34

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 92.9%

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

      2. associate-*l* 93.1%

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

      3. *-commutative 93.1%

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

   3. Simplified 93.1%

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

   if 5.0000000000000003e-34 < t

   1. Initial program 95.0%

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

      1. *-commutative 95.0%

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

      2. distribute-rgt-out-- 97.7%

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

      3. associate-*r* 94.8%

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

      4. *-commutative 94.8%

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

   3. Simplified 94.8%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-34}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 2: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;x \leq 340:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.8e+49)
   (* x (* t y))
   (if (<= x 340.0) (* t (* y (- z))) (* t (* y x)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.8e+49) {
		tmp = x * (t * y);
	} else if (x <= 340.0) {
		tmp = t * (y * -z);
	} else {
		tmp = t * (y * x);
	}
	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 <= (-3.8d+49)) then
        tmp = x * (t * y)
    else if (x <= 340.0d0) then
        tmp = t * (y * -z)
    else
        tmp = t * (y * x)
    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 <= -3.8e+49) {
		tmp = x * (t * y);
	} else if (x <= 340.0) {
		tmp = t * (y * -z);
	} else {
		tmp = t * (y * x);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if x <= -3.8e+49:
		tmp = x * (t * y)
	elif x <= 340.0:
		tmp = t * (y * -z)
	else:
		tmp = t * (y * x)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.8e+49)
		tmp = Float64(x * Float64(t * y));
	elseif (x <= 340.0)
		tmp = Float64(t * Float64(y * Float64(-z)));
	else
		tmp = Float64(t * Float64(y * x));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.8e+49)
		tmp = x * (t * y);
	elseif (x <= 340.0)
		tmp = t * (y * -z);
	else
		tmp = t * (y * x);
	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, -3.8e+49], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 340.0], N[(t * N[(y * (-z)), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7999999999999999e49

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 92.0%

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

      2. associate-*l* 84.1%

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

      3. *-commutative 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in x around inf 89.9%

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

      1. associate-*r* 80.1%

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

      2. *-commutative 80.1%

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

   6. Simplified 80.1%

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

      1. associate-*r* 91.8%

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

      2. *-commutative 91.8%

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

      3. remove-double-div 91.8%

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

      4. div-inv 91.8%

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

      5. add-cube-cbrt 90.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{t \cdot y} \cdot \sqrt[3]{t \cdot y}\right) \cdot \sqrt[3]{t \cdot y}}}{\frac{1}{x}} \]

      6. associate-/l* 90.6%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{t \cdot y} \cdot \sqrt[3]{t \cdot y}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}} \]

      7. cbrt-unprod 79.6%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(t \cdot y\right) \cdot \left(t \cdot y\right)}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}} \]

      8. pow2 79.6%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(t \cdot y\right)}^{2}}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}} \]

   8. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 37.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}\right)\right)} \]

      2. expm1-udef 33.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}\right)} - 1} \]

      3. associate-/r/ 33.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{1}{x}} \cdot \sqrt[3]{t \cdot y}}\right)} - 1 \]

      4. associate-*l/ 33.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}} \cdot \sqrt[3]{t \cdot y}}{\frac{1}{x}}}\right)} - 1 \]

      5. cbrt-prod 29.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt[3]{{\left(t \cdot y\right)}^{2} \cdot \left(t \cdot y\right)}}}{\frac{1}{x}}\right)} - 1 \]

      6. unpow2 29.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt[3]{\color{blue}{\left(\left(t \cdot y\right) \cdot \left(t \cdot y\right)\right)} \cdot \left(t \cdot y\right)}}{\frac{1}{x}}\right)} - 1 \]

      7. add-cbrt-cube 38.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{t \cdot y}}{\frac{1}{x}}\right)} - 1 \]

   10. Applied egg-rr 38.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot y}{\frac{1}{x}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 48.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot y}{\frac{1}{x}}\right)\right)} \]

       2. expm1-log1p 91.8%

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

       3. associate-/r/ 91.8%

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

       4. /-rgt-identity 91.8%

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

       5. *-commutative 91.8%

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

   12. Simplified 91.8%

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

   if -3.7999999999999999e49 < x < 340

   1. Initial program 94.5%

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

      1. distribute-rgt-out-- 94.5%

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

      2. associate-*l* 93.3%

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

      3. *-commutative 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in x around 0 79.8%

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

      1. mul-1-neg 79.8%

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

      2. distribute-rgt-neg-in 79.8%

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

      3. *-commutative 79.8%

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

      4. distribute-lft-neg-out 79.8%

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

      5. *-commutative 79.8%

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

   6. Simplified 79.8%

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

   if 340 < x

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 96.0%

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

      2. associate-*l* 90.6%

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

      3. *-commutative 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around inf 86.3%

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

      1. *-commutative 86.3%

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

   6. Simplified 86.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;x \leq 340:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;x \leq 130:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.3e+50)
   (* x (* t y))
   (if (<= x 130.0) (* y (* z (- t))) (* t (* y x)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.3e+50) {
		tmp = x * (t * y);
	} else if (x <= 130.0) {
		tmp = y * (z * -t);
	} else {
		tmp = t * (y * x);
	}
	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 <= (-3.3d+50)) then
        tmp = x * (t * y)
    else if (x <= 130.0d0) then
        tmp = y * (z * -t)
    else
        tmp = t * (y * x)
    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 <= -3.3e+50) {
		tmp = x * (t * y);
	} else if (x <= 130.0) {
		tmp = y * (z * -t);
	} else {
		tmp = t * (y * x);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if x <= -3.3e+50:
		tmp = x * (t * y)
	elif x <= 130.0:
		tmp = y * (z * -t)
	else:
		tmp = t * (y * x)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.3e+50)
		tmp = Float64(x * Float64(t * y));
	elseif (x <= 130.0)
		tmp = Float64(y * Float64(z * Float64(-t)));
	else
		tmp = Float64(t * Float64(y * x));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.3e+50)
		tmp = x * (t * y);
	elseif (x <= 130.0)
		tmp = y * (z * -t);
	else
		tmp = t * (y * x);
	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, -3.3e+50], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 130.0], N[(y * N[(z * (-t)), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3e50

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 92.0%

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

      2. associate-*l* 84.1%

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

      3. *-commutative 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in x around inf 89.9%

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

      1. associate-*r* 80.1%

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

      2. *-commutative 80.1%

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

   6. Simplified 80.1%

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

      1. associate-*r* 91.8%

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

      2. *-commutative 91.8%

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

      3. remove-double-div 91.8%

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

      4. div-inv 91.8%

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

      5. add-cube-cbrt 90.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{t \cdot y} \cdot \sqrt[3]{t \cdot y}\right) \cdot \sqrt[3]{t \cdot y}}}{\frac{1}{x}} \]

      6. associate-/l* 90.6%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{t \cdot y} \cdot \sqrt[3]{t \cdot y}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}} \]

      7. cbrt-unprod 79.6%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(t \cdot y\right) \cdot \left(t \cdot y\right)}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}} \]

      8. pow2 79.6%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(t \cdot y\right)}^{2}}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}} \]

   8. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 37.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}\right)\right)} \]

      2. expm1-udef 33.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}\right)} - 1} \]

      3. associate-/r/ 33.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{1}{x}} \cdot \sqrt[3]{t \cdot y}}\right)} - 1 \]

      4. associate-*l/ 33.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}} \cdot \sqrt[3]{t \cdot y}}{\frac{1}{x}}}\right)} - 1 \]

      5. cbrt-prod 29.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt[3]{{\left(t \cdot y\right)}^{2} \cdot \left(t \cdot y\right)}}}{\frac{1}{x}}\right)} - 1 \]

      6. unpow2 29.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt[3]{\color{blue}{\left(\left(t \cdot y\right) \cdot \left(t \cdot y\right)\right)} \cdot \left(t \cdot y\right)}}{\frac{1}{x}}\right)} - 1 \]

      7. add-cbrt-cube 38.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{t \cdot y}}{\frac{1}{x}}\right)} - 1 \]

   10. Applied egg-rr 38.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot y}{\frac{1}{x}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 48.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot y}{\frac{1}{x}}\right)\right)} \]

       2. expm1-log1p 91.8%

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

       3. associate-/r/ 91.8%

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

       4. /-rgt-identity 91.8%

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

       5. *-commutative 91.8%

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

   12. Simplified 91.8%

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

   if -3.3e50 < x < 130

   1. Initial program 94.5%

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

      1. distribute-rgt-out-- 94.5%

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

      2. associate-*l* 93.3%

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

      3. *-commutative 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in x around 0 79.8%

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

      1. mul-1-neg 79.8%

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

      2. associate-*r* 77.4%

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

      3. *-commutative 77.4%

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

      4. distribute-rgt-neg-out 77.4%

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

      5. associate-*l* 80.0%

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

   6. Simplified 80.0%

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

   if 130 < x

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 96.0%

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

      2. associate-*l* 90.6%

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

      3. *-commutative 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around inf 86.3%

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

      1. *-commutative 86.3%

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

   6. Simplified 86.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{elif}\;x \leq 130:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{+211}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.55e+211) (* y (* t (- x z))) (* t (* y (- z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.55e+211) {
		tmp = y * (t * (x - z));
	} else {
		tmp = t * (y * -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 (z <= 1.55d+211) then
        tmp = y * (t * (x - z))
    else
        tmp = t * (y * -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 (z <= 1.55e+211) {
		tmp = y * (t * (x - z));
	} else {
		tmp = t * (y * -z);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.55e+211:
		tmp = y * (t * (x - z))
	else:
		tmp = t * (y * -z)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.55e+211)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(t * Float64(y * Float64(-z)));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.55e+211)
		tmp = y * (t * (x - z));
	else
		tmp = t * (y * -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[z, 1.55e+211], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.5500000000000001e211

   1. Initial program 94.6%

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

      1. distribute-rgt-out-- 95.1%

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

      2. associate-*l* 91.5%

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

      3. *-commutative 91.5%

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

   3. Simplified 91.5%

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

   if 1.5500000000000001e211 < z

   1. Initial program 79.6%

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

      1. distribute-rgt-out-- 84.8%

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

      2. associate-*l* 85.6%

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

      3. *-commutative 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in x around 0 84.8%

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

      1. mul-1-neg 84.8%

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

      2. distribute-rgt-neg-in 84.8%

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

      3. *-commutative 84.8%

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

      4. distribute-lft-neg-out 84.8%

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

      5. *-commutative 84.8%

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

   6. Simplified 84.8%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{+211}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \]

Alternative 5: 53.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 4e-48) (* x (* t y)) (* t (* y x))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 4e-48) {
		tmp = x * (t * y);
	} else {
		tmp = t * (y * x);
	}
	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 <= 4d-48) then
        tmp = x * (t * y)
    else
        tmp = t * (y * x)
    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 <= 4e-48) {
		tmp = x * (t * y);
	} else {
		tmp = t * (y * x);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 4e-48:
		tmp = x * (t * y)
	else:
		tmp = t * (y * x)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 4e-48)
		tmp = Float64(x * Float64(t * y));
	else
		tmp = Float64(t * Float64(y * x));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 4e-48)
		tmp = x * (t * y);
	else
		tmp = t * (y * x);
	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, 4e-48], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999999e-48

   1. Initial program 93.5%

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

      1. distribute-rgt-out-- 94.0%

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

      2. associate-*l* 90.7%

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

      3. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around inf 47.2%

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

      1. associate-*r* 43.6%

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

      2. *-commutative 43.6%

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

   6. Simplified 43.6%

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

      1. associate-*r* 49.0%

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

      2. *-commutative 49.0%

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

      3. remove-double-div 49.0%

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

      4. div-inv 48.9%

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

      5. add-cube-cbrt 48.5%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{t \cdot y} \cdot \sqrt[3]{t \cdot y}\right) \cdot \sqrt[3]{t \cdot y}}}{\frac{1}{x}} \]

      6. associate-/l* 48.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{t \cdot y} \cdot \sqrt[3]{t \cdot y}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}} \]

      7. cbrt-unprod 44.8%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(t \cdot y\right) \cdot \left(t \cdot y\right)}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}} \]

      8. pow2 44.8%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(t \cdot y\right)}^{2}}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}} \]

   8. Applied egg-rr 44.8%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 30.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}\right)\right)} \]

      2. expm1-udef 25.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{\frac{1}{x}}{\sqrt[3]{t \cdot y}}}\right)} - 1} \]

      3. associate-/r/ 25.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}}}{\frac{1}{x}} \cdot \sqrt[3]{t \cdot y}}\right)} - 1 \]

      4. associate-*l/ 25.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt[3]{{\left(t \cdot y\right)}^{2}} \cdot \sqrt[3]{t \cdot y}}{\frac{1}{x}}}\right)} - 1 \]

      5. cbrt-prod 24.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt[3]{{\left(t \cdot y\right)}^{2} \cdot \left(t \cdot y\right)}}}{\frac{1}{x}}\right)} - 1 \]

      6. unpow2 24.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt[3]{\color{blue}{\left(\left(t \cdot y\right) \cdot \left(t \cdot y\right)\right)} \cdot \left(t \cdot y\right)}}{\frac{1}{x}}\right)} - 1 \]

      7. add-cbrt-cube 25.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{t \cdot y}}{\frac{1}{x}}\right)} - 1 \]

   10. Applied egg-rr 25.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot y}{\frac{1}{x}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 33.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot y}{\frac{1}{x}}\right)\right)} \]

       2. expm1-log1p 48.9%

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

       3. associate-/r/ 49.0%

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

       4. /-rgt-identity 49.0%

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

       5. *-commutative 49.0%

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

   12. Simplified 49.0%

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

   if 3.9999999999999999e-48 < x

   1. Initial program 93.5%

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

      1. distribute-rgt-out-- 95.1%

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

      2. associate-*l* 92.2%

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

      3. *-commutative 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in x around inf 77.7%

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

      1. *-commutative 77.7%

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

   6. Simplified 77.7%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 6: 52.9% 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 93.5%

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

   1. distribute-rgt-out-- 94.3%

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

   2. associate-*l* 91.0%

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

   3. *-commutative 91.0%

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

3. Simplified 91.0%

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

4. Taylor expanded in x around inf 54.6%

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

   1. *-commutative 54.6%

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

6. Simplified 54.6%

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

7. Final simplification 54.6%

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

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 2023298 
(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))