Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.1% → 97.4%

Time: 6.3s

Alternatives: 8

Speedup: 1.3×

• 24.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-36}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.7e-36) (* 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 <= 1.7e-36) {
		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 <= 1.7d-36) 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 <= 1.7e-36) {
		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 <= 1.7e-36:
		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 <= 1.7e-36)
		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 <= 1.7e-36)
		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, 1.7e-36], 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 < 1.7000000000000001e-36

   1. Initial program 86.2%

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

      1. distribute-rgt-out-- 87.9%

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

      2. associate-*l* 97.1%

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

   3. Simplified 97.1%

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

   if 1.7000000000000001e-36 < t

   1. Initial program 93.6%

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

      1. distribute-rgt-out-- 93.6%

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

   3. Simplified 93.6%

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

      1. add-cube-cbrt 92.6%

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

      2. pow3 92.6%

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

   5. Applied egg-rr 92.6%

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

      1. unpow3 92.6%

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

      2. add-cube-cbrt 93.6%

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

      3. associate-*r* 86.1%

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

      4. *-commutative 86.1%

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

      5. flip-- 74.1%

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

      6. div-inv 74.1%

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

      7. associate-*l* 69.0%

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

      8. *-commutative 69.0%

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

      9. div-inv 69.0%

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

      10. clear-num 69.0%

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

      11. un-div-inv 69.1%

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

      12. clear-num 69.0%

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

      13. div-inv 69.0%

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

      14. *-commutative 69.0%

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

      15. associate-*l* 74.1%

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

      16. div-inv 74.1%

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

      17. flip-- 86.1%

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

      18. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

   8. Taylor expanded in y around 0 86.1%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

   10. Simplified 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 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-36}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 2: 70.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} t_1 := y \cdot \left(t \cdot x\right)\\ t_2 := y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* t x))) (t_2 (* y (* t (- z)))))
   (if (<= z -5.5e-55)
     t_2
     (if (<= z 4.2e-133)
       t_1
       (if (<= z 1.12e+14) t_2 (if (<= z 9.5e+51) t_1 (* t (* y (- z)))))))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double t_1 = y * (t * x);
	double t_2 = y * (t * -z);
	double tmp;
	if (z <= -5.5e-55) {
		tmp = t_2;
	} else if (z <= 4.2e-133) {
		tmp = t_1;
	} else if (z <= 1.12e+14) {
		tmp = t_2;
	} else if (z <= 9.5e+51) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y * (t * x)
    t_2 = y * (t * -z)
    if (z <= (-5.5d-55)) then
        tmp = t_2
    else if (z <= 4.2d-133) then
        tmp = t_1
    else if (z <= 1.12d+14) then
        tmp = t_2
    else if (z <= 9.5d+51) then
        tmp = t_1
    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 t_1 = y * (t * x);
	double t_2 = y * (t * -z);
	double tmp;
	if (z <= -5.5e-55) {
		tmp = t_2;
	} else if (z <= 4.2e-133) {
		tmp = t_1;
	} else if (z <= 1.12e+14) {
		tmp = t_2;
	} else if (z <= 9.5e+51) {
		tmp = t_1;
	} else {
		tmp = t * (y * -z);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	t_1 = y * (t * x)
	t_2 = y * (t * -z)
	tmp = 0
	if z <= -5.5e-55:
		tmp = t_2
	elif z <= 4.2e-133:
		tmp = t_1
	elif z <= 1.12e+14:
		tmp = t_2
	elif z <= 9.5e+51:
		tmp = t_1
	else:
		tmp = t * (y * -z)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	t_1 = Float64(y * Float64(t * x))
	t_2 = Float64(y * Float64(t * Float64(-z)))
	tmp = 0.0
	if (z <= -5.5e-55)
		tmp = t_2;
	elseif (z <= 4.2e-133)
		tmp = t_1;
	elseif (z <= 1.12e+14)
		tmp = t_2;
	elseif (z <= 9.5e+51)
		tmp = t_1;
	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)
	t_1 = y * (t * x);
	t_2 = y * (t * -z);
	tmp = 0.0;
	if (z <= -5.5e-55)
		tmp = t_2;
	elseif (z <= 4.2e-133)
		tmp = t_1;
	elseif (z <= 1.12e+14)
		tmp = t_2;
	elseif (z <= 9.5e+51)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e-55], t$95$2, If[LessEqual[z, 4.2e-133], t$95$1, If[LessEqual[z, 1.12e+14], t$95$2, If[LessEqual[z, 9.5e+51], t$95$1, N[(t * N[(y * (-z)), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4999999999999999e-55 or 4.2000000000000002e-133 < z < 1.12e14

   1. Initial program 87.4%

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

      1. distribute-rgt-out-- 88.5%

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

      2. associate-*l* 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in x around 0 76.8%

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

      1. mul-1-neg 76.8%

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

      2. *-commutative 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

   6. Simplified 76.8%

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

   if -5.4999999999999999e-55 < z < 4.2000000000000002e-133 or 1.12e14 < z < 9.4999999999999999e51

   1. Initial program 90.5%

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

      1. distribute-rgt-out-- 90.5%

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

      2. associate-*l* 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in x around inf 88.0%

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

   if 9.4999999999999999e51 < z

   1. Initial program 86.8%

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

      1. distribute-rgt-out-- 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in x around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. distribute-rgt-neg-out 77.0%

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

   6. Simplified 77.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-55}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+51}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \end{array} \]

Alternative 3: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+45}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+110}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.4e+45)
   (* t (* y x))
   (if (<= x 3.8e+110) (* y (* t (- z))) (* x (* t y)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.4e+45) {
		tmp = t * (y * x);
	} else if (x <= 3.8e+110) {
		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 <= (-1.4d+45)) then
        tmp = t * (y * x)
    else if (x <= 3.8d+110) 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 <= -1.4e+45) {
		tmp = t * (y * x);
	} else if (x <= 3.8e+110) {
		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 <= -1.4e+45:
		tmp = t * (y * x)
	elif x <= 3.8e+110:
		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 <= -1.4e+45)
		tmp = Float64(t * Float64(y * x));
	elseif (x <= 3.8e+110)
		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 <= -1.4e+45)
		tmp = t * (y * x);
	elseif (x <= 3.8e+110)
		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, -1.4e+45], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+110], N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e45

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in x around inf 78.9%

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

   if -1.4e45 < x < 3.79999999999999989e110

   1. Initial program 87.5%

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

      1. distribute-rgt-out-- 88.1%

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

      2. associate-*l* 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in x around 0 78.4%

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

      1. mul-1-neg 78.4%

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

      2. *-commutative 78.4%

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

      3. distribute-rgt-neg-in 78.4%

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

   6. Simplified 78.4%

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

   if 3.79999999999999989e110 < x

   1. Initial program 84.1%

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

      1. distribute-rgt-out-- 87.3%

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

   3. Simplified 87.3%

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

      1. add-cube-cbrt 86.2%

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

      2. pow3 86.1%

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

   5. Applied egg-rr 86.1%

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

      1. unpow3 86.2%

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

      2. add-cube-cbrt 87.3%

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

      3. associate-*r* 78.3%

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

      4. *-commutative 78.3%

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

      5. flip-- 27.7%

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

      6. div-inv 27.7%

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

      7. associate-*l* 21.7%

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

      8. *-commutative 21.7%

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

      9. div-inv 21.7%

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

      10. clear-num 21.7%

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

      11. un-div-inv 21.8%

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

      12. clear-num 21.7%

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

      13. div-inv 21.8%

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

      14. *-commutative 21.8%

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

      15. associate-*l* 27.6%

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

      16. div-inv 27.7%

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

      17. flip-- 78.4%

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

      18. *-commutative 78.4%

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

   7. Applied egg-rr 78.4%

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

   8. Taylor expanded in x around inf 68.9%

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

      1. associate-*r* 81.2%

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

      2. *-commutative 81.2%

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

   10. Simplified 81.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+45}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+110}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-71}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.65e-71) (* 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 (t <= 1.65e-71) {
		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 (t <= 1.65d-71) 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 (t <= 1.65e-71) {
		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 t <= 1.65e-71:
		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 (t <= 1.65e-71)
		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 (t <= 1.65e-71)
		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[t, 1.65e-71], 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 t < 1.6500000000000001e-71

   1. Initial program 86.2%

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

      1. distribute-rgt-out-- 87.5%

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

      2. associate-*l* 97.0%

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

   3. Simplified 97.0%

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

   if 1.6500000000000001e-71 < t

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 94.1%

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

   3. Simplified 94.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-71}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5: 55.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{-139}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.75e-139) (* y (* t x)) (* x (* t y))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.75e-139) {
		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 <= 1.75d-139) 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 <= 1.75e-139) {
		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 <= 1.75e-139:
		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 <= 1.75e-139)
		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 <= 1.75e-139)
		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, 1.75e-139], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.75000000000000001e-139

   1. Initial program 86.0%

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

      1. distribute-rgt-out-- 87.4%

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

      2. associate-*l* 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in x around inf 49.7%

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

   if 1.75000000000000001e-139 < t

   1. Initial program 92.1%

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

      1. distribute-rgt-out-- 93.1%

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

   3. Simplified 93.1%

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

      1. add-cube-cbrt 92.0%

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

      2. pow3 92.0%

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

   5. Applied egg-rr 92.0%

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

      1. unpow3 92.0%

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

      2. add-cube-cbrt 93.1%

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

      3. associate-*r* 89.3%

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

      4. *-commutative 89.3%

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

      5. flip-- 71.3%

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

      6. div-inv 71.2%

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

      7. associate-*l* 67.3%

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

      8. *-commutative 67.3%

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

      9. div-inv 67.3%

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

      10. clear-num 67.3%

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

      11. un-div-inv 67.4%

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

      12. clear-num 67.3%

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

      13. div-inv 67.3%

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

      14. *-commutative 67.3%

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

      15. associate-*l* 71.2%

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

      16. div-inv 71.3%

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

      17. flip-- 89.3%

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

      18. *-commutative 89.3%

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

   7. Applied egg-rr 89.3%

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

   8. Taylor expanded in x around inf 46.7%

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

      1. associate-*r* 51.5%

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

      2. *-commutative 51.5%

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

   10. Simplified 51.5%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{-139}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t \cdot y\right)\\ \end{array} \]

Alternative 6: 55.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.5e+55) (* y (* t x)) (* t (* y x))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.5e+55) {
		tmp = y * (t * x);
	} 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 (y <= (-9.5d+55)) then
        tmp = y * (t * x)
    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 (y <= -9.5e+55) {
		tmp = y * (t * x);
	} else {
		tmp = t * (y * x);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -9.5e+55:
		tmp = y * (t * x)
	else:
		tmp = t * (y * x)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.5e+55)
		tmp = Float64(y * Float64(t * x));
	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 (y <= -9.5e+55)
		tmp = y * (t * x);
	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[y, -9.5e+55], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.49999999999999989e55

   1. Initial program 75.2%

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

      1. distribute-rgt-out-- 79.5%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 51.2%

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

   if -9.49999999999999989e55 < 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-- 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in x around inf 49.2%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 7: 91.6% 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 88.4%

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

   1. distribute-rgt-out-- 89.6%

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

   2. associate-*l* 93.9%

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

3. Simplified 93.9%

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

4. Final simplification 93.9%

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

Alternative 8: 54.8% 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 88.4%

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

   1. distribute-rgt-out-- 89.6%

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

3. Simplified 89.6%

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

   1. add-cube-cbrt 88.6%

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

   2. pow3 88.6%

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

5. Applied egg-rr 88.6%

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

   1. unpow3 88.6%

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

   2. add-cube-cbrt 89.6%

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

   3. associate-*r* 93.9%

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

   4. *-commutative 93.9%

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

   5. flip-- 68.2%

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

   6. div-inv 68.2%

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

   7. associate-*l* 65.2%

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

   8. *-commutative 65.2%

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

   9. div-inv 65.2%

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

   10. clear-num 65.2%

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

   11. un-div-inv 65.2%

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

   12. clear-num 65.2%

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

   13. div-inv 65.1%

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

   14. *-commutative 65.1%

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

   15. associate-*l* 68.1%

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

   16. div-inv 68.2%

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

   17. flip-- 93.8%

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

   18. *-commutative 93.8%

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

7. Applied egg-rr 93.8%

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

8. Taylor expanded in x around inf 48.6%

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

   1. associate-*r* 46.0%

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

   2. *-commutative 46.0%

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

10. Simplified 46.0%

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

11. Final simplification 46.0%

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

Developer target: 95.7% 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 2023200 
(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))