Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D

Percentage Accurate: 92.9% → 97.9%

Time: 8.2s

Alternatives: 9

Speedup: 1.0×

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y * (z - x)) / 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 - x)) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y * (z - x)) / 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 - x)) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (fma (/ y t) (- z x) x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

   1. +-commutative 92.4%

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

   2. associate-*l/ 96.5%

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

   3. fma-def 96.6%

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

3. Simplified 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)} \]
4. Add Preprocessing

5. Final simplification 96.6%

   \[\leadsto \mathsf{fma}\left(\frac{y}{t}, z - x, x\right) \]
6. Add Preprocessing

Alternative 2: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-11} \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.5e-11) (not (<= x 0.0009)))
   (* x (- 1.0 (/ y t)))
   (+ x (* (/ y t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.5e-11) || !(x <= 0.0009)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + ((y / t) * z);
	}
	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 ((x <= (-1.5d-11)) .or. (.not. (x <= 0.0009d0))) then
        tmp = x * (1.0d0 - (y / t))
    else
        tmp = x + ((y / t) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.5e-11) || !(x <= 0.0009)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + ((y / t) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.5e-11) or not (x <= 0.0009):
		tmp = x * (1.0 - (y / t))
	else:
		tmp = x + ((y / t) * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.5e-11) || !(x <= 0.0009))
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = Float64(x + Float64(Float64(y / t) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.5e-11) || ~((x <= 0.0009)))
		tmp = x * (1.0 - (y / t));
	else
		tmp = x + ((y / t) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.5e-11], N[Not[LessEqual[x, 0.0009]], $MachinePrecision]], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-11 or 8.9999999999999998e-4 < x

   1. Initial program 89.6%

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

      1. associate-*l/ 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around inf 89.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y}{t}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

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

   7. Simplified 89.2%

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

   if -1.5e-11 < x < 8.9999999999999998e-4

   1. Initial program 94.9%

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

      1. associate-*l/ 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in z around inf 86.4%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
   6. Step-by-step derivation

      1. associate-*l/ 86.4%

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

      2. *-commutative 86.4%

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

   7. Simplified 86.4%

      \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-11} \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+57} \lor \neg \left(y \leq 3.9 \cdot 10^{+48}\right):\\ \;\;\;\;y \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.95e+57) (not (<= y 3.9e+48))) (* y (/ (- x) t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.95e+57) || !(y <= 3.9e+48)) {
		tmp = y * (-x / t);
	} else {
		tmp = x;
	}
	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 ((y <= (-1.95d+57)) .or. (.not. (y <= 3.9d+48))) then
        tmp = y * (-x / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.95e+57) || !(y <= 3.9e+48)) {
		tmp = y * (-x / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.95e+57) or not (y <= 3.9e+48):
		tmp = y * (-x / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.95e+57) || !(y <= 3.9e+48))
		tmp = Float64(y * Float64(Float64(-x) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.95e+57) || ~((y <= 3.9e+48)))
		tmp = y * (-x / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.95e+57], N[Not[LessEqual[y, 3.9e+48]], $MachinePrecision]], N[(y * N[((-x) / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.94999999999999984e57 or 3.9000000000000001e48 < y

   1. Initial program 86.6%

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

      1. associate-*l/ 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in x around inf 55.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y}{t}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 55.2%

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

      2. unsub-neg 55.2%

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

   7. Simplified 55.2%

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

      1. sub-neg 55.2%

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

      2. distribute-rgt-in 55.2%

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

      3. distribute-neg-frac 55.2%

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

      4. remove-double-neg 55.2%

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

      5. frac-2neg 55.2%

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

      6. associate-/r/ 55.9%

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

      7. *-un-lft-identity 55.9%

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

      8. frac-2neg 55.9%

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

      9. div-inv 55.9%

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

      10. distribute-frac-neg 55.9%

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

      11. remove-double-neg 55.9%

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

      12. clear-num 55.9%

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

      13. add-sqr-sqrt 32.3%

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

      14. sqrt-unprod 41.0%

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

      15. sqr-neg 41.0%

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

      16. sqrt-unprod 7.3%

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

      17. add-sqr-sqrt 19.3%

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

      18. cancel-sign-sub-inv 19.3%

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

      19. add-sqr-sqrt 7.3%

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

      20. sqrt-unprod 41.0%

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

   9. Applied egg-rr 55.9%

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

   10. Taylor expanded in y around inf 39.4%

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

       1. associate-*r/ 39.4%

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

       2. *-commutative 39.4%

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

       3. neg-mul-1 39.4%

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

       4. distribute-rgt-neg-in 39.4%

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

       5. associate-*r/ 42.5%

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

   12. Simplified 42.5%

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

   if -1.94999999999999984e57 < y < 3.9000000000000001e48

   1. Initial program 97.2%

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

      1. associate-*l/ 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in y around 0 61.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+57} \lor \neg \left(y \leq 3.9 \cdot 10^{+48}\right):\\ \;\;\;\;y \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.3e-76) x (if (<= t 4.8e-61) (* (/ y t) (- x)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.3e-76) {
		tmp = x;
	} else if (t <= 4.8e-61) {
		tmp = (y / t) * -x;
	} else {
		tmp = x;
	}
	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 <= (-2.3d-76)) then
        tmp = x
    else if (t <= 4.8d-61) then
        tmp = (y / t) * -x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.3e-76) {
		tmp = x;
	} else if (t <= 4.8e-61) {
		tmp = (y / t) * -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.3e-76:
		tmp = x
	elif t <= 4.8e-61:
		tmp = (y / t) * -x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.3e-76)
		tmp = x;
	elseif (t <= 4.8e-61)
		tmp = Float64(Float64(y / t) * Float64(-x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.3e-76)
		tmp = x;
	elseif (t <= 4.8e-61)
		tmp = (y / t) * -x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.3e-76], x, If[LessEqual[t, 4.8e-61], N[(N[(y / t), $MachinePrecision] * (-x)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -2.30000000000000006e-76 or 4.8000000000000002e-61 < t

   1. Initial program 89.2%

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

      1. associate-*l/ 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in y around 0 58.1%

      \[\leadsto \color{blue}{x} \]

   if -2.30000000000000006e-76 < t < 4.8000000000000002e-61

   1. Initial program 97.9%

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

      1. associate-*l/ 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in x around inf 53.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y}{t}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 53.2%

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

      2. unsub-neg 53.2%

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

   7. Simplified 53.2%

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

      1. sub-neg 53.2%

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

      2. distribute-rgt-in 53.2%

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

      3. distribute-neg-frac 53.2%

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

      4. remove-double-neg 53.2%

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

      5. frac-2neg 53.2%

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

      6. associate-/r/ 50.7%

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

      7. *-un-lft-identity 50.7%

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

      8. frac-2neg 50.7%

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

      9. div-inv 50.3%

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

      10. distribute-frac-neg 50.3%

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

      11. remove-double-neg 50.3%

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

      12. clear-num 50.4%

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

      13. add-sqr-sqrt 30.8%

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

      14. sqrt-unprod 39.1%

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

      15. sqr-neg 39.1%

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

      16. sqrt-unprod 3.6%

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

      17. add-sqr-sqrt 14.4%

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

      18. cancel-sign-sub-inv 14.4%

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

      19. add-sqr-sqrt 3.6%

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

      20. sqrt-unprod 39.1%

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

   9. Applied egg-rr 50.4%

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

   10. Taylor expanded in y around inf 43.9%

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

       1. associate-*r/ 43.9%

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

       2. *-commutative 43.9%

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

       3. neg-mul-1 43.9%

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

       4. distribute-rgt-neg-in 43.9%

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

       5. associate-*r/ 43.8%

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

   12. Simplified 43.8%

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

   13. Taylor expanded in y around 0 43.9%

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

       1. mul-1-neg 43.9%

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

       2. associate-*l/ 43.8%

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

       3. distribute-rgt-neg-in 43.8%

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

       4. associate-*l/ 43.9%

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

       5. associate-*r/ 43.6%

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

   15. Simplified 43.6%

       \[\leadsto \color{blue}{x \cdot \frac{-y}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-61}:\\ \;\;\;\;\frac{-y}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.8e-76) x (if (<= t 2.75e-61) (/ (- y) (/ t x)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.8e-76) {
		tmp = x;
	} else if (t <= 2.75e-61) {
		tmp = -y / (t / x);
	} else {
		tmp = x;
	}
	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 <= (-8.8d-76)) then
        tmp = x
    else if (t <= 2.75d-61) then
        tmp = -y / (t / x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.8e-76) {
		tmp = x;
	} else if (t <= 2.75e-61) {
		tmp = -y / (t / x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -8.8e-76:
		tmp = x
	elif t <= 2.75e-61:
		tmp = -y / (t / x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.8e-76)
		tmp = x;
	elseif (t <= 2.75e-61)
		tmp = Float64(Float64(-y) / Float64(t / x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8.8e-76)
		tmp = x;
	elseif (t <= 2.75e-61)
		tmp = -y / (t / x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -8.8e-76], x, If[LessEqual[t, 2.75e-61], N[((-y) / N[(t / x), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -8.79999999999999997e-76 or 2.7499999999999998e-61 < t

   1. Initial program 89.2%

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

      1. associate-*l/ 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in y around 0 58.1%

      \[\leadsto \color{blue}{x} \]

   if -8.79999999999999997e-76 < t < 2.7499999999999998e-61

   1. Initial program 97.9%

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

      1. associate-*l/ 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in x around inf 53.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y}{t}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 53.2%

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

      2. unsub-neg 53.2%

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

   7. Simplified 53.2%

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

      1. sub-neg 53.2%

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

      2. distribute-rgt-in 53.2%

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

      3. distribute-neg-frac 53.2%

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

      4. remove-double-neg 53.2%

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

      5. frac-2neg 53.2%

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

      6. associate-/r/ 50.7%

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

      7. *-un-lft-identity 50.7%

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

      8. frac-2neg 50.7%

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

      9. div-inv 50.3%

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

      10. distribute-frac-neg 50.3%

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

      11. remove-double-neg 50.3%

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

      12. clear-num 50.4%

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

      13. add-sqr-sqrt 30.8%

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

      14. sqrt-unprod 39.1%

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

      15. sqr-neg 39.1%

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

      16. sqrt-unprod 3.6%

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

      17. add-sqr-sqrt 14.4%

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

      18. cancel-sign-sub-inv 14.4%

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

      19. add-sqr-sqrt 3.6%

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

      20. sqrt-unprod 39.1%

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

   9. Applied egg-rr 50.4%

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

   10. Taylor expanded in y around inf 43.9%

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

       1. associate-*r/ 43.9%

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

       2. *-commutative 43.9%

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

       3. neg-mul-1 43.9%

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

       4. distribute-rgt-neg-in 43.9%

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

       5. associate-*r/ 43.8%

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

   12. Simplified 43.8%

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

       1. distribute-frac-neg 43.8%

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

       2. distribute-rgt-neg-in 43.8%

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

       3. clear-num 43.7%

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

       4. div-inv 44.0%

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

       5. distribute-neg-frac 44.0%

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

   14. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{\frac{-y}{\frac{t}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-61}:\\ \;\;\;\;\frac{-y}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.7e-129) x (if (<= t 3.7e-156) (/ x (/ t y)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.7e-129) {
		tmp = x;
	} else if (t <= 3.7e-156) {
		tmp = x / (t / y);
	} else {
		tmp = x;
	}
	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 <= (-4.7d-129)) then
        tmp = x
    else if (t <= 3.7d-156) then
        tmp = x / (t / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.7e-129) {
		tmp = x;
	} else if (t <= 3.7e-156) {
		tmp = x / (t / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.7e-129:
		tmp = x
	elif t <= 3.7e-156:
		tmp = x / (t / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.7e-129)
		tmp = x;
	elseif (t <= 3.7e-156)
		tmp = Float64(x / Float64(t / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -4.7e-129)
		tmp = x;
	elseif (t <= 3.7e-156)
		tmp = x / (t / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.7e-129], x, If[LessEqual[t, 3.7e-156], N[(x / N[(t / y), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -4.7000000000000002e-129 or 3.7e-156 < t

   1. Initial program 90.9%

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

      1. associate-*l/ 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in y around 0 52.5%

      \[\leadsto \color{blue}{x} \]

   if -4.7000000000000002e-129 < t < 3.7e-156

   1. Initial program 97.0%

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

      1. associate-*l/ 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around inf 49.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y}{t}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 49.2%

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

      2. unsub-neg 49.2%

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

   7. Simplified 49.2%

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

      1. sub-neg 49.2%

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

      2. distribute-rgt-in 49.2%

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

      3. distribute-neg-frac 49.2%

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

      4. remove-double-neg 49.2%

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

      5. frac-2neg 49.2%

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

      6. associate-/r/ 45.6%

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

      7. *-un-lft-identity 45.6%

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

      8. frac-2neg 45.6%

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

      9. div-inv 45.2%

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

      10. distribute-frac-neg 45.2%

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

      11. remove-double-neg 45.2%

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

      12. clear-num 45.3%

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

      13. add-sqr-sqrt 23.1%

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

      14. sqrt-unprod 34.3%

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

      15. sqr-neg 34.3%

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

      16. sqrt-unprod 4.4%

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

      17. add-sqr-sqrt 11.3%

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

      18. cancel-sign-sub-inv 11.3%

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

      19. add-sqr-sqrt 4.4%

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

      20. sqrt-unprod 34.3%

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

   9. Applied egg-rr 45.3%

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

   10. Taylor expanded in y around inf 44.4%

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

       1. associate-*r/ 44.4%

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

       2. *-commutative 44.4%

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

       3. neg-mul-1 44.4%

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

       4. distribute-rgt-neg-in 44.4%

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

       5. associate-*r/ 44.3%

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

   12. Simplified 44.3%

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

       1. frac-2neg 44.3%

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

       2. remove-double-neg 44.3%

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

       3. associate-*r/ 44.4%

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

       4. associate-*l/ 45.4%

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

       5. clear-num 45.4%

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

       6. associate-*l/ 43.9%

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

       7. *-un-lft-identity 43.9%

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

       8. add-sqr-sqrt 22.4%

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

       9. sqrt-unprod 31.7%

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

       10. sqr-neg 31.7%

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

       11. sqrt-unprod 6.7%

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

       12. add-sqr-sqrt 17.1%

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

   14. Applied egg-rr 17.1%

       \[\leadsto \color{blue}{\frac{x}{\frac{t}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / t) * (z - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

   1. associate-*l/ 96.5%

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

3. Simplified 96.5%

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

5. Final simplification 96.5%

   \[\leadsto x + \frac{y}{t} \cdot \left(z - x\right) \]
6. Add Preprocessing

Alternative 8: 65.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \frac{y}{t}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- 1.0 (/ y t))))

C

double code(double x, double y, double z, double t) {
	return x * (1.0 - (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 * (1.0d0 - (y / t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * (1.0 - (y / t));
}

Python

def code(x, y, z, t):
	return x * (1.0 - (y / t))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(1.0 - Float64(y / t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (1.0 - (y / t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.4%

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

   1. associate-*l/ 96.5%

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

3. Simplified 96.5%

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

5. Taylor expanded in x around inf 63.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y}{t}\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 63.6%

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

   2. unsub-neg 63.6%

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

7. Simplified 63.6%

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

8. Final simplification 63.6%

   \[\leadsto x \cdot \left(1 - \frac{y}{t}\right) \]
9. Add Preprocessing

Alternative 9: 37.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 92.4%

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

   1. associate-*l/ 96.5%

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

3. Simplified 96.5%

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

5. Taylor expanded in y around 0 40.4%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 40.4%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 90.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - ((x * (y / t)) + (-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 - ((x * (y / t)) + (-z * (y / t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024017 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))