Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.8% → 96.7%

Time: 7.6s

Alternatives: 8

Speedup: 1.3×

• 25.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -22000000000000:\\ \;\;\;\;{\left(\sqrt[3]{y \cdot \left(t \cdot \left(x - z\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -22000000000000.0)
   (pow (cbrt (* y (* t (- x z)))) 3.0)
   (* t (- (* y x) (* y z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -22000000000000.0) {
		tmp = pow(cbrt((y * (t * (x - z)))), 3.0);
	} else {
		tmp = t * ((y * x) - (y * z));
	}
	return tmp;
}

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -22000000000000.0) {
		tmp = Math.pow(Math.cbrt((y * (t * (x - z)))), 3.0);
	} else {
		tmp = t * ((y * x) - (y * z));
	}
	return tmp;
}

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -22000000000000.0)
		tmp = cbrt(Float64(y * Float64(t * Float64(x - z)))) ^ 3.0;
	else
		tmp = Float64(t * Float64(Float64(y * x) - Float64(y * z)));
	end
	return 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, -22000000000000.0], N[Power[N[Power[N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(t * N[(N[(y * x), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e13

   1. Initial program 69.4%

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

      1. *-commutative 69.4%

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

      2. distribute-rgt-out-- 73.1%

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

      3. associate-*r* 96.3%

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

      4. *-commutative 96.3%

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

   3. Simplified 96.3%

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

      1. add-cube-cbrt 95.5%

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

      2. pow3 95.6%

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

      3. associate-*l* 95.5%

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

   5. Applied egg-rr 95.5%

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

   if -2.2e13 < y

   1. Initial program 94.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -22000000000000:\\ \;\;\;\;{\left(\sqrt[3]{y \cdot \left(t \cdot \left(x - z\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \]

Alternative 2: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot x\right)\\ t_2 := y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

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 (* t (* y x))) (t_2 (* y (* t (- z)))))
   (if (<= z -2.1e-94)
     t_2
     (if (<= z 7.5e-237)
       t_1
       (if (<= z 5e+14) (* x (* y t)) (if (<= z 1.55e+29) t_1 t_2))))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double t_1 = t * (y * x);
	double t_2 = y * (t * -z);
	double tmp;
	if (z <= -2.1e-94) {
		tmp = t_2;
	} else if (z <= 7.5e-237) {
		tmp = t_1;
	} else if (z <= 5e+14) {
		tmp = x * (y * t);
	} else if (z <= 1.55e+29) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 = t * (y * x)
    t_2 = y * (t * -z)
    if (z <= (-2.1d-94)) then
        tmp = t_2
    else if (z <= 7.5d-237) then
        tmp = t_1
    else if (z <= 5d+14) then
        tmp = x * (y * t)
    else if (z <= 1.55d+29) then
        tmp = t_1
    else
        tmp = t_2
    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 = t * (y * x);
	double t_2 = y * (t * -z);
	double tmp;
	if (z <= -2.1e-94) {
		tmp = t_2;
	} else if (z <= 7.5e-237) {
		tmp = t_1;
	} else if (z <= 5e+14) {
		tmp = x * (y * t);
	} else if (z <= 1.55e+29) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	t_1 = t * (y * x)
	t_2 = y * (t * -z)
	tmp = 0
	if z <= -2.1e-94:
		tmp = t_2
	elif z <= 7.5e-237:
		tmp = t_1
	elif z <= 5e+14:
		tmp = x * (y * t)
	elif z <= 1.55e+29:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	t_1 = Float64(t * Float64(y * x))
	t_2 = Float64(y * Float64(t * Float64(-z)))
	tmp = 0.0
	if (z <= -2.1e-94)
		tmp = t_2;
	elseif (z <= 7.5e-237)
		tmp = t_1;
	elseif (z <= 5e+14)
		tmp = Float64(x * Float64(y * t));
	elseif (z <= 1.55e+29)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = t * (y * x);
	t_2 = y * (t * -z);
	tmp = 0.0;
	if (z <= -2.1e-94)
		tmp = t_2;
	elseif (z <= 7.5e-237)
		tmp = t_1;
	elseif (z <= 5e+14)
		tmp = x * (y * t);
	elseif (z <= 1.55e+29)
		tmp = t_1;
	else
		tmp = t_2;
	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[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e-94], t$95$2, If[LessEqual[z, 7.5e-237], t$95$1, If[LessEqual[z, 5e+14], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55e+29], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1000000000000001e-94 or 1.5499999999999999e29 < z

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. distribute-rgt-out-- 86.2%

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

      3. associate-*r* 90.6%

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

      4. *-commutative 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around 0 73.9%

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

      1. mul-1-neg 73.9%

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

      2. *-commutative 73.9%

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

      3. associate-*l* 76.4%

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

      4. distribute-lft-neg-in 76.4%

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

      5. *-commutative 76.4%

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

   6. Simplified 76.4%

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

   if -2.1000000000000001e-94 < z < 7.50000000000000034e-237 or 5e14 < z < 1.5499999999999999e29

   1. Initial program 95.6%

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

      1. distribute-rgt-out-- 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. *-commutative 88.2%

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

   6. Simplified 88.2%

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

   if 7.50000000000000034e-237 < z < 5e14

   1. Initial program 93.2%

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

      1. *-commutative 93.2%

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

      2. distribute-rgt-out-- 93.2%

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

      3. associate-*r* 96.5%

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

      4. *-commutative 96.5%

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

   3. Simplified 96.5%

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

      1. flip3-- 63.5%

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

      2. associate-*r/ 55.5%

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

      3. fma-def 55.5%

         \[\leadsto \frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{\color{blue}{\mathsf{fma}\left(x, x, z \cdot z + x \cdot z\right)}} \]

      4. distribute-rgt-out 55.5%

         \[\leadsto \frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{\mathsf{fma}\left(x, x, \color{blue}{z \cdot \left(z + x\right)}\right)} \]

      5. +-commutative 55.5%

         \[\leadsto \frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{\mathsf{fma}\left(x, x, z \cdot \color{blue}{\left(x + z\right)}\right)} \]

   5. Applied egg-rr 55.5%

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

      1. associate-/l* 63.4%

         \[\leadsto \color{blue}{\frac{y \cdot t}{\frac{\mathsf{fma}\left(x, x, z \cdot \left(x + z\right)\right)}{{x}^{3} - {z}^{3}}}} \]

      2. *-commutative 63.4%

         \[\leadsto \frac{\color{blue}{t \cdot y}}{\frac{\mathsf{fma}\left(x, x, z \cdot \left(x + z\right)\right)}{{x}^{3} - {z}^{3}}} \]

   7. Simplified 63.4%

      \[\leadsto \color{blue}{\frac{t \cdot y}{\frac{\mathsf{fma}\left(x, x, z \cdot \left(x + z\right)\right)}{{x}^{3} - {z}^{3}}}} \]

   8. Taylor expanded in x around inf 75.5%

      \[\leadsto \frac{t \cdot y}{\color{blue}{\frac{1}{x}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 75.5%

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

      2. /-rgt-identity 75.5%

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

   10. Applied egg-rr 75.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-237}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+29}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \end{array} \]

Alternative 3: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} t_1 := t \cdot \left(y \cdot x\right)\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

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 (* t (* y x))))
   (if (<= z -2.1e-94)
     (* y (* t (- z)))
     (if (<= z 3e-237)
       t_1
       (if (<= z 1.15e+18)
         (* x (* y t))
         (if (<= z 8.5e+27) t_1 (* z (* y (- t)))))))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double t_1 = t * (y * x);
	double tmp;
	if (z <= -2.1e-94) {
		tmp = y * (t * -z);
	} else if (z <= 3e-237) {
		tmp = t_1;
	} else if (z <= 1.15e+18) {
		tmp = x * (y * t);
	} else if (z <= 8.5e+27) {
		tmp = t_1;
	} else {
		tmp = z * (y * -t);
	}
	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) :: tmp
    t_1 = t * (y * x)
    if (z <= (-2.1d-94)) then
        tmp = y * (t * -z)
    else if (z <= 3d-237) then
        tmp = t_1
    else if (z <= 1.15d+18) then
        tmp = x * (y * t)
    else if (z <= 8.5d+27) then
        tmp = t_1
    else
        tmp = z * (y * -t)
    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 = t * (y * x);
	double tmp;
	if (z <= -2.1e-94) {
		tmp = y * (t * -z);
	} else if (z <= 3e-237) {
		tmp = t_1;
	} else if (z <= 1.15e+18) {
		tmp = x * (y * t);
	} else if (z <= 8.5e+27) {
		tmp = t_1;
	} else {
		tmp = z * (y * -t);
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	t_1 = t * (y * x)
	tmp = 0
	if z <= -2.1e-94:
		tmp = y * (t * -z)
	elif z <= 3e-237:
		tmp = t_1
	elif z <= 1.15e+18:
		tmp = x * (y * t)
	elif z <= 8.5e+27:
		tmp = t_1
	else:
		tmp = z * (y * -t)
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	t_1 = Float64(t * Float64(y * x))
	tmp = 0.0
	if (z <= -2.1e-94)
		tmp = Float64(y * Float64(t * Float64(-z)));
	elseif (z <= 3e-237)
		tmp = t_1;
	elseif (z <= 1.15e+18)
		tmp = Float64(x * Float64(y * t));
	elseif (z <= 8.5e+27)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(y * Float64(-t)));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = t * (y * x);
	tmp = 0.0;
	if (z <= -2.1e-94)
		tmp = y * (t * -z);
	elseif (z <= 3e-237)
		tmp = t_1;
	elseif (z <= 1.15e+18)
		tmp = x * (y * t);
	elseif (z <= 8.5e+27)
		tmp = t_1;
	else
		tmp = z * (y * -t);
	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[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e-94], N[(y * N[(t * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3e-237], t$95$1, If[LessEqual[z, 1.15e+18], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e+27], t$95$1, N[(z * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.1000000000000001e-94

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. distribute-rgt-out-- 86.5%

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

      3. associate-*r* 90.1%

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

      4. *-commutative 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in x around 0 74.3%

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

      1. mul-1-neg 74.3%

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

      2. *-commutative 74.3%

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

      3. associate-*l* 79.7%

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

      4. distribute-lft-neg-in 79.7%

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

      5. *-commutative 79.7%

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

   6. Simplified 79.7%

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

   if -2.1000000000000001e-94 < z < 3.00000000000000024e-237 or 1.15e18 < z < 8.5e27

   1. Initial program 95.6%

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

      1. distribute-rgt-out-- 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. *-commutative 88.2%

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

   6. Simplified 88.2%

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

   if 3.00000000000000024e-237 < z < 1.15e18

   1. Initial program 93.2%

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

      1. *-commutative 93.2%

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

      2. distribute-rgt-out-- 93.2%

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

      3. associate-*r* 96.5%

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

      4. *-commutative 96.5%

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

   3. Simplified 96.5%

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

      1. flip3-- 63.5%

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

      2. associate-*r/ 55.5%

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

      3. fma-def 55.5%

         \[\leadsto \frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{\color{blue}{\mathsf{fma}\left(x, x, z \cdot z + x \cdot z\right)}} \]

      4. distribute-rgt-out 55.5%

         \[\leadsto \frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{\mathsf{fma}\left(x, x, \color{blue}{z \cdot \left(z + x\right)}\right)} \]

      5. +-commutative 55.5%

         \[\leadsto \frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{\mathsf{fma}\left(x, x, z \cdot \color{blue}{\left(x + z\right)}\right)} \]

   5. Applied egg-rr 55.5%

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

      1. associate-/l* 63.4%

         \[\leadsto \color{blue}{\frac{y \cdot t}{\frac{\mathsf{fma}\left(x, x, z \cdot \left(x + z\right)\right)}{{x}^{3} - {z}^{3}}}} \]

      2. *-commutative 63.4%

         \[\leadsto \frac{\color{blue}{t \cdot y}}{\frac{\mathsf{fma}\left(x, x, z \cdot \left(x + z\right)\right)}{{x}^{3} - {z}^{3}}} \]

   7. Simplified 63.4%

      \[\leadsto \color{blue}{\frac{t \cdot y}{\frac{\mathsf{fma}\left(x, x, z \cdot \left(x + z\right)\right)}{{x}^{3} - {z}^{3}}}} \]

   8. Taylor expanded in x around inf 75.5%

      \[\leadsto \frac{t \cdot y}{\color{blue}{\frac{1}{x}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 75.5%

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

      2. /-rgt-identity 75.5%

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

   10. Applied egg-rr 75.5%

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

   if 8.5e27 < z

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. distribute-rgt-out-- 85.8%

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

      3. associate-*r* 91.1%

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

      4. *-commutative 91.1%

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

   3. Simplified 91.1%

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

      1. flip3-- 21.2%

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

      2. associate-*r/ 18.4%

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

      3. fma-def 18.4%

         \[\leadsto \frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{\color{blue}{\mathsf{fma}\left(x, x, z \cdot z + x \cdot z\right)}} \]

      4. distribute-rgt-out 18.4%

         \[\leadsto \frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{\mathsf{fma}\left(x, x, \color{blue}{z \cdot \left(z + x\right)}\right)} \]

      5. +-commutative 18.4%

         \[\leadsto \frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{\mathsf{fma}\left(x, x, z \cdot \color{blue}{\left(x + z\right)}\right)} \]

   5. Applied egg-rr 18.4%

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

      1. associate-/l* 21.2%

         \[\leadsto \color{blue}{\frac{y \cdot t}{\frac{\mathsf{fma}\left(x, x, z \cdot \left(x + z\right)\right)}{{x}^{3} - {z}^{3}}}} \]

      2. *-commutative 21.2%

         \[\leadsto \frac{\color{blue}{t \cdot y}}{\frac{\mathsf{fma}\left(x, x, z \cdot \left(x + z\right)\right)}{{x}^{3} - {z}^{3}}} \]

   7. Simplified 21.2%

      \[\leadsto \color{blue}{\frac{t \cdot y}{\frac{\mathsf{fma}\left(x, x, z \cdot \left(x + z\right)\right)}{{x}^{3} - {z}^{3}}}} \]

   8. Taylor expanded in x around 0 78.7%

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

      1. frac-2neg 78.7%

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

      2. div-inv 78.6%

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

      3. distribute-rgt-neg-in 78.6%

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

      4. distribute-neg-frac 78.6%

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

      5. add-sqr-sqrt 78.4%

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

      6. sqrt-unprod 42.4%

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

      7. sqr-neg 42.4%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 11.9%

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

      10. frac-2neg 11.9%

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

      11. frac-2neg 11.9%

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

      12. metadata-eval 11.9%

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

      13. remove-double-div 11.9%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 42.4%

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

      16. sqr-neg 42.4%

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

      17. sqrt-unprod 78.4%

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

      18. add-sqr-sqrt 78.7%

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

   10. Applied egg-rr 78.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-237}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 4: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1000000000:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -1000000000.0) (* (- x z) (* y t)) (* t (- (* y x) (* y z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1000000000.0) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t * ((y * x) - (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 (y <= (-1000000000.0d0)) then
        tmp = (x - z) * (y * t)
    else
        tmp = t * ((y * x) - (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 (y <= -1000000000.0) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t * ((y * x) - (y * z));
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -1000000000.0:
		tmp = (x - z) * (y * t)
	else:
		tmp = t * ((y * x) - (y * z))
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1000000000.0)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	else
		tmp = Float64(t * Float64(Float64(y * x) - Float64(y * 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 <= -1000000000.0)
		tmp = (x - z) * (y * t);
	else
		tmp = t * ((y * x) - (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[y, -1000000000.0], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(y * x), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e9

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

      2. distribute-rgt-out-- 74.0%

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

      3. associate-*r* 96.5%

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

      4. *-commutative 96.5%

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

   3. Simplified 96.5%

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

   if -1e9 < y

   1. Initial program 94.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1000000000:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \]

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -100000000:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -100000000.0) (* (- x z) (* y t)) (* t (* y (- x z)))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -100000000.0) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Fortran

NOTE: y and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-100000000.0d0)) then
        tmp = (x - z) * (y * t)
    else
        tmp = t * (y * (x - z))
    end if
    code = tmp
end function

Java

assert y < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -100000000.0) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Python

[y, t] = sort([y, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -100000000.0:
		tmp = (x - z) * (y * t)
	else:
		tmp = t * (y * (x - z))
	return tmp

Julia

y, t = sort([y, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -100000000.0)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	else
		tmp = Float64(t * Float64(y * Float64(x - z)));
	end
	return tmp
end

MATLAB

y, t = num2cell(sort([y, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -100000000.0)
		tmp = (x - z) * (y * t);
	else
		tmp = t * (y * (x - z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -100000000.0], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e8

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

      2. distribute-rgt-out-- 74.0%

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

      3. associate-*r* 96.5%

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

      4. *-commutative 96.5%

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

   3. Simplified 96.5%

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

   if -1e8 < y

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 94.6%

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

   3. Simplified 94.6%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -100000000:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6: 55.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [y, t] = \mathsf{sort}([y, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -20000000000000:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -20000000000000.0) (* y (* t x)) (* t (* y x))))

C

assert(y < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -20000000000000.0) {
		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 <= (-20000000000000.0d0)) 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 <= -20000000000000.0) {
		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 <= -20000000000000.0:
		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 <= -20000000000000.0)
		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 <= -20000000000000.0)
		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, -20000000000000.0], N[(y * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e13

   1. Initial program 69.4%

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

      1. *-commutative 69.4%

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

      2. distribute-rgt-out-- 73.1%

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

      3. associate-*r* 96.3%

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

      4. *-commutative 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in x around inf 45.6%

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

      1. associate-*r* 48.1%

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

      2. *-commutative 48.1%

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

   6. Simplified 48.1%

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

   if -2e13 < y

   1. Initial program 94.2%

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

      1. distribute-rgt-out-- 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in x around inf 52.4%

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

      1. *-commutative 52.4%

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

   6. Simplified 52.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -20000000000000:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 7: 92.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.8%

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

   1. *-commutative 88.8%

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

   2. distribute-rgt-out-- 90.0%

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

   3. associate-*r* 90.5%

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

   4. *-commutative 90.5%

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

3. Simplified 90.5%

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

4. Final simplification 90.5%

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

Alternative 8: 53.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 88.8%

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

   1. *-commutative 88.8%

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

   2. distribute-rgt-out-- 90.0%

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

   3. associate-*r* 90.5%

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

   4. *-commutative 90.5%

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

3. Simplified 90.5%

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

4. Taylor expanded in x around inf 50.9%

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

   1. associate-*r* 49.6%

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

   2. *-commutative 49.6%

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

6. Simplified 49.6%

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

7. Final simplification 49.6%

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

Developer target: 96.2% 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 2023297 
(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))