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

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 8

Speedup: 1.0×

• 25.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

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) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

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) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{z - \left(x + y\right)}{t} \cdot -0.5 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ (- z (+ x y)) t) -0.5))

C

double code(double x, double y, double z, double t) {
	return ((z - (x + y)) / t) * -0.5;
}

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 = ((z - (x + y)) / t) * (-0.5d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((z - (x + y)) / t) * -0.5;
}

Python

def code(x, y, z, t):
	return ((z - (x + y)) / t) * -0.5

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(z - Float64(x + y)) / t) * -0.5)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((z - (x + y)) / t) * -0.5;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(z - N[(x + y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-/r* 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

   2. remove-double-neg 100.0%

      \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

   3. distribute-frac-neg2 100.0%

      \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

   4. neg-mul-1 100.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

   5. *-commutative 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

   6. associate-/l* 100.0%

      \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

   7. distribute-frac-neg2 100.0%

      \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

   8. distribute-neg-frac 100.0%

      \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

   9. neg-sub0 100.0%

      \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

   10. associate--r- 100.0%

       \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

   11. neg-sub0 100.0%

       \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

   12. +-commutative 100.0%

       \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

   13. sub-neg 100.0%

       \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

   14. metadata-eval 100.0%

       \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 47.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-212}:\\ \;\;\;\;-0.5 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.3e+53)
   (* 0.5 (/ x t))
   (if (<= x -2e-212) (* -0.5 (/ z t)) (* 0.5 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.3e+53) {
		tmp = 0.5 * (x / t);
	} else if (x <= -2e-212) {
		tmp = -0.5 * (z / t);
	} else {
		tmp = 0.5 * (y / 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 (x <= (-1.3d+53)) then
        tmp = 0.5d0 * (x / t)
    else if (x <= (-2d-212)) then
        tmp = (-0.5d0) * (z / t)
    else
        tmp = 0.5d0 * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.3e+53) {
		tmp = 0.5 * (x / t);
	} else if (x <= -2e-212) {
		tmp = -0.5 * (z / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.3e+53:
		tmp = 0.5 * (x / t)
	elif x <= -2e-212:
		tmp = -0.5 * (z / t)
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.3e+53)
		tmp = Float64(0.5 * Float64(x / t));
	elseif (x <= -2e-212)
		tmp = Float64(-0.5 * Float64(z / t));
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.3e+53)
		tmp = 0.5 * (x / t);
	elseif (x <= -2e-212)
		tmp = -0.5 * (z / t);
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.3e+53], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-212], N[(-0.5 * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.29999999999999999e53

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 68.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

   if -1.29999999999999999e53 < x < -1.99999999999999991e-212

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 60.2%

      \[\leadsto \color{blue}{\frac{z}{t}} \cdot -0.5 \]

   if -1.99999999999999991e-212 < x

   1. Initial program 99.2%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 46.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-212}:\\ \;\;\;\;-0.5 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-180}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.3e+51)
   (* 0.5 (/ x t))
   (if (<= x -5.2e-180) (* z (/ -0.5 t)) (* 0.5 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.3e+51) {
		tmp = 0.5 * (x / t);
	} else if (x <= -5.2e-180) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = 0.5 * (y / 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 (x <= (-2.3d+51)) then
        tmp = 0.5d0 * (x / t)
    else if (x <= (-5.2d-180)) then
        tmp = z * ((-0.5d0) / t)
    else
        tmp = 0.5d0 * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.3e+51) {
		tmp = 0.5 * (x / t);
	} else if (x <= -5.2e-180) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.3e+51:
		tmp = 0.5 * (x / t)
	elif x <= -5.2e-180:
		tmp = z * (-0.5 / t)
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.3e+51)
		tmp = Float64(0.5 * Float64(x / t));
	elseif (x <= -5.2e-180)
		tmp = Float64(z * Float64(-0.5 / t));
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.3e+51)
		tmp = 0.5 * (x / t);
	elseif (x <= -5.2e-180)
		tmp = z * (-0.5 / t);
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.3e+51], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-180], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.30000000000000005e51

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 68.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

   if -2.30000000000000005e51 < x < -5.1999999999999998e-180

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 58.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   6. Step-by-step derivation

      1. metadata-eval 58.2%

         \[\leadsto \color{blue}{\left(-0.5\right)} \cdot \frac{z}{t} \]

      2. distribute-lft-neg-in 58.2%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]

      3. *-lft-identity 58.2%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      4. associate-*l/ 58.1%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      5. associate-*l* 58.1%

         \[\leadsto -\color{blue}{\left(0.5 \cdot \frac{1}{t}\right) \cdot z} \]

      6. *-commutative 58.1%

         \[\leadsto -\color{blue}{z \cdot \left(0.5 \cdot \frac{1}{t}\right)} \]

      7. distribute-rgt-neg-in 58.1%

         \[\leadsto \color{blue}{z \cdot \left(-0.5 \cdot \frac{1}{t}\right)} \]

      8. associate-*r/ 58.1%

         \[\leadsto z \cdot \left(-\color{blue}{\frac{0.5 \cdot 1}{t}}\right) \]

      9. metadata-eval 58.1%

         \[\leadsto z \cdot \left(-\frac{\color{blue}{0.5}}{t}\right) \]

      10. distribute-neg-frac 58.1%

          \[\leadsto z \cdot \color{blue}{\frac{-0.5}{t}} \]

      11. metadata-eval 58.1%

          \[\leadsto z \cdot \frac{\color{blue}{-0.5}}{t} \]

   7. Simplified 58.1%

      \[\leadsto \color{blue}{z \cdot \frac{-0.5}{t}} \]

   if -5.1999999999999998e-180 < x

   1. Initial program 99.3%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 46.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-222}:\\ \;\;\;\;\frac{-0.5 \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \left(z - y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-222) (/ (* -0.5 (- z x)) t) (/ (* -0.5 (- z y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-222) {
		tmp = (-0.5 * (z - x)) / t;
	} else {
		tmp = (-0.5 * (z - y)) / 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 ((x + y) <= (-2d-222)) then
        tmp = ((-0.5d0) * (z - x)) / t
    else
        tmp = ((-0.5d0) * (z - y)) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-222) {
		tmp = (-0.5 * (z - x)) / t;
	} else {
		tmp = (-0.5 * (z - y)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-222:
		tmp = (-0.5 * (z - x)) / t
	else:
		tmp = (-0.5 * (z - y)) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-222)
		tmp = Float64(Float64(-0.5 * Float64(z - x)) / t);
	else
		tmp = Float64(Float64(-0.5 * Float64(z - y)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-222)
		tmp = (-0.5 * (z - x)) / t;
	else
		tmp = (-0.5 * (z - y)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-222], N[(N[(-0.5 * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(-0.5 * N[(z - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.0000000000000001e-222

   1. Initial program 99.3%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 70.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z - x}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 70.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(z - x\right)}{t}} \]

   7. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(z - x\right)}{t}} \]

   if -2.0000000000000001e-222 < (+.f64 x y)

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z - y}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 70.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(z - y\right)}{t}} \]

   7. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(z - y\right)}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{-0.5 \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.6e+161) (/ (* -0.5 (- z x)) t) (* 0.5 (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+161) {
		tmp = (-0.5 * (z - x)) / t;
	} else {
		tmp = 0.5 * (y / 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 (y <= 2.6d+161) then
        tmp = ((-0.5d0) * (z - x)) / t
    else
        tmp = 0.5d0 * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+161) {
		tmp = (-0.5 * (z - x)) / t;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.6e+161:
		tmp = (-0.5 * (z - x)) / t
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.6e+161)
		tmp = Float64(Float64(-0.5 * Float64(z - x)) / t);
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.6e+161)
		tmp = (-0.5 * (z - x)) / t;
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.6e+161], N[(N[(-0.5 * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5999999999999998e161

   1. Initial program 99.6%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 75.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z - x}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 75.7%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(z - x\right)}{t}} \]

   7. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(z - x\right)}{t}} \]

   if 2.5999999999999998e161 < y

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 79.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+161}:\\ \;\;\;\;\frac{-0.5}{\frac{t}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.1e+161) (/ -0.5 (/ t (- z x))) (* 0.5 (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.1e+161) {
		tmp = -0.5 / (t / (z - x));
	} else {
		tmp = 0.5 * (y / 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 (y <= 4.1d+161) then
        tmp = (-0.5d0) / (t / (z - x))
    else
        tmp = 0.5d0 * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.1e+161) {
		tmp = -0.5 / (t / (z - x));
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.1e+161:
		tmp = -0.5 / (t / (z - x))
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.1e+161)
		tmp = Float64(-0.5 / Float64(t / Float64(z - x)));
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.1e+161)
		tmp = -0.5 / (t / (z - x));
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.1e+161], N[(-0.5 / N[(t / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.1000000000000001e161

   1. Initial program 99.6%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{z - \left(x + y\right)}{t}} \]

      2. clear-num 99.7%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{t}{z - \left(x + y\right)}}} \]

      3. un-div-inv 99.7%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{t}{z - \left(x + y\right)}}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{t}{z - \left(x + y\right)}}} \]

   7. Taylor expanded in y around 0 75.5%

      \[\leadsto \frac{-0.5}{\color{blue}{\frac{t}{z - x}}} \]

   if 4.1000000000000001e161 < y

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 79.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 46.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.7e-25) (* 0.5 (/ x t)) (* 0.5 (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.7e-25) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = 0.5 * (y / 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 (x <= (-3.7d-25)) then
        tmp = 0.5d0 * (x / t)
    else
        tmp = 0.5d0 * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.7e-25) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.7e-25:
		tmp = 0.5 * (x / t)
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.7e-25)
		tmp = Float64(0.5 * Float64(x / t));
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.7e-25)
		tmp = 0.5 * (x / t);
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.7e-25], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.70000000000000009e-25

   1. Initial program 100.0%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 61.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

   if -3.70000000000000009e-25 < x

   1. Initial program 99.5%

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

      1. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

      3. distribute-frac-neg2 100.0%

         \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

      4. neg-mul-1 100.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

      6. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

      7. distribute-frac-neg2 100.0%

         \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

      9. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

      10. associate--r- 100.0%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

      11. neg-sub0 100.0%

          \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

      13. sub-neg 100.0%

          \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 44.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 37.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{x}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 0.5 (/ x t)))

C

double code(double x, double y, double z, double t) {
	return 0.5 * (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 = 0.5d0 * (x / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.5 * (x / t);
}

Python

def code(x, y, z, t):
	return 0.5 * (x / t)

Julia

function code(x, y, z, t)
	return Float64(0.5 * Float64(x / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.5 * (x / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-/r* 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(x + y\right) - z}{t}}{2}} \]

   2. remove-double-neg 100.0%

      \[\leadsto \frac{\color{blue}{-\left(-\frac{\left(x + y\right) - z}{t}\right)}}{2} \]

   3. distribute-frac-neg2 100.0%

      \[\leadsto \frac{-\color{blue}{\frac{\left(x + y\right) - z}{-t}}}{2} \]

   4. neg-mul-1 100.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(x + y\right) - z}{-t}}}{2} \]

   5. *-commutative 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot -1}}{2} \]

   6. associate-/l* 100.0%

      \[\leadsto \color{blue}{\frac{\left(x + y\right) - z}{-t} \cdot \frac{-1}{2}} \]

   7. distribute-frac-neg2 100.0%

      \[\leadsto \color{blue}{\left(-\frac{\left(x + y\right) - z}{t}\right)} \cdot \frac{-1}{2} \]

   8. distribute-neg-frac 100.0%

      \[\leadsto \color{blue}{\frac{-\left(\left(x + y\right) - z\right)}{t}} \cdot \frac{-1}{2} \]

   9. neg-sub0 100.0%

      \[\leadsto \frac{\color{blue}{0 - \left(\left(x + y\right) - z\right)}}{t} \cdot \frac{-1}{2} \]

   10. associate--r- 100.0%

       \[\leadsto \frac{\color{blue}{\left(0 - \left(x + y\right)\right) + z}}{t} \cdot \frac{-1}{2} \]

   11. neg-sub0 100.0%

       \[\leadsto \frac{\color{blue}{\left(-\left(x + y\right)\right)} + z}{t} \cdot \frac{-1}{2} \]

   12. +-commutative 100.0%

       \[\leadsto \frac{\color{blue}{z + \left(-\left(x + y\right)\right)}}{t} \cdot \frac{-1}{2} \]

   13. sub-neg 100.0%

       \[\leadsto \frac{\color{blue}{z - \left(x + y\right)}}{t} \cdot \frac{-1}{2} \]

   14. metadata-eval 100.0%

       \[\leadsto \frac{z - \left(x + y\right)}{t} \cdot \color{blue}{-0.5} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot -0.5} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 36.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))