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

Percentage Accurate: 93.0% → 95.3%

Time: 6.6s

Alternatives: 6

Speedup: 1.0×

• 24.8% 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: 93.0% 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: 95.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-147}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - x\right) \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5e-147) (+ x (/ (* y (- z x)) t)) (+ x (* (- z x) (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5e-147) {
		tmp = x + ((y * (z - x)) / t);
	} else {
		tmp = x + ((z - x) * (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 (z <= (-5d-147)) then
        tmp = x + ((y * (z - x)) / t)
    else
        tmp = x + ((z - x) * (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5e-147) {
		tmp = x + ((y * (z - x)) / t);
	} else {
		tmp = x + ((z - x) * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5e-147:
		tmp = x + ((y * (z - x)) / t)
	else:
		tmp = x + ((z - x) * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5e-147)
		tmp = Float64(x + Float64(Float64(y * Float64(z - x)) / t));
	else
		tmp = Float64(x + Float64(Float64(z - x) * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5e-147)
		tmp = x + ((y * (z - x)) / t);
	else
		tmp = x + ((z - x) * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5e-147], N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z - x), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.00000000000000013e-147

   1. Initial program 99.0%

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

   if -5.00000000000000013e-147 < z

   1. Initial program 93.3%

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

      1. associate-*l/ 98.6%

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

   3. Simplified 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-147}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - x\right) \cdot \frac{y}{t}\\ \end{array} \]

Alternative 2: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-69}:\\ \;\;\;\;x - x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6.8e-37)
   (+ x (/ (* z y) t))
   (if (<= z 4e-69) (- x (* x (/ y t))) (+ x (* z (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.8e-37) {
		tmp = x + ((z * y) / t);
	} else if (z <= 4e-69) {
		tmp = x - (x * (y / t));
	} else {
		tmp = x + (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 (z <= (-6.8d-37)) then
        tmp = x + ((z * y) / t)
    else if (z <= 4d-69) then
        tmp = x - (x * (y / t))
    else
        tmp = x + (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 (z <= -6.8e-37) {
		tmp = x + ((z * y) / t);
	} else if (z <= 4e-69) {
		tmp = x - (x * (y / t));
	} else {
		tmp = x + (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -6.8e-37:
		tmp = x + ((z * y) / t)
	elif z <= 4e-69:
		tmp = x - (x * (y / t))
	else:
		tmp = x + (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6.8e-37)
		tmp = Float64(x + Float64(Float64(z * y) / t));
	elseif (z <= 4e-69)
		tmp = Float64(x - Float64(x * Float64(y / t)));
	else
		tmp = Float64(x + Float64(z * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -6.8e-37)
		tmp = x + ((z * y) / t);
	elseif (z <= 4e-69)
		tmp = x - (x * (y / t));
	else
		tmp = x + (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6.8e-37], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-69], N[(x - N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.80000000000000037e-37

   1. Initial program 98.8%

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

      1. associate-*l/ 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in z around inf 88.7%

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

   if -6.80000000000000037e-37 < z < 3.9999999999999999e-69

   1. Initial program 96.3%

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

      1. associate-/l* 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in z around 0 85.1%

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

      1. associate-*r/ 85.1%

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

      2. neg-mul-1 85.1%

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

   6. Simplified 85.1%

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

      1. clear-num 85.1%

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

      2. associate-/r/ 84.0%

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

      3. clear-num 84.0%

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

      4. add-sqr-sqrt 34.5%

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

      5. sqrt-unprod 58.2%

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

      6. sqr-neg 58.2%

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

      7. sqrt-unprod 29.1%

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

      8. add-sqr-sqrt 44.6%

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

   8. Applied egg-rr 44.6%

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

      1. frac-2neg 44.6%

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

      2. distribute-frac-neg 44.6%

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

      3. add-sqr-sqrt 19.7%

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

      4. sqrt-unprod 50.1%

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

      5. sqr-neg 50.1%

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

      6. sqrt-unprod 41.9%

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

      7. add-sqr-sqrt 84.0%

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

      8. frac-2neg 84.0%

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

      9. cancel-sign-sub-inv 84.0%

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

      10. associate-/r/ 88.4%

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

      11. div-inv 88.4%

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

      12. clear-num 88.4%

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

   10. Applied egg-rr 88.4%

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

   if 3.9999999999999999e-69 < z

   1. Initial program 90.0%

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

      1. associate-*l/ 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around inf 86.6%

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

      1. associate-*l/ 93.2%

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

      2. *-commutative 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-69}:\\ \;\;\;\;x - x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 3: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-64}:\\ \;\;\;\;x - \frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9.2e-39)
   (+ x (/ (* z y) t))
   (if (<= z 1.4e-64) (- x (/ x (/ t y))) (+ x (* z (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.2e-39) {
		tmp = x + ((z * y) / t);
	} else if (z <= 1.4e-64) {
		tmp = x - (x / (t / y));
	} else {
		tmp = x + (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 (z <= (-9.2d-39)) then
        tmp = x + ((z * y) / t)
    else if (z <= 1.4d-64) then
        tmp = x - (x / (t / y))
    else
        tmp = x + (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 (z <= -9.2e-39) {
		tmp = x + ((z * y) / t);
	} else if (z <= 1.4e-64) {
		tmp = x - (x / (t / y));
	} else {
		tmp = x + (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9.2e-39:
		tmp = x + ((z * y) / t)
	elif z <= 1.4e-64:
		tmp = x - (x / (t / y))
	else:
		tmp = x + (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9.2e-39)
		tmp = Float64(x + Float64(Float64(z * y) / t));
	elseif (z <= 1.4e-64)
		tmp = Float64(x - Float64(x / Float64(t / y)));
	else
		tmp = Float64(x + Float64(z * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -9.2e-39)
		tmp = x + ((z * y) / t);
	elseif (z <= 1.4e-64)
		tmp = x - (x / (t / y));
	else
		tmp = x + (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9.2e-39], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.4e-64], N[(x - N[(x / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.20000000000000033e-39

   1. Initial program 98.8%

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

      1. associate-*l/ 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in z around inf 88.7%

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

   if -9.20000000000000033e-39 < z < 1.40000000000000002e-64

   1. Initial program 96.3%

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

      1. associate-/l* 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in z around 0 85.1%

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

      1. associate-*r/ 85.1%

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

      2. neg-mul-1 85.1%

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

   6. Simplified 85.1%

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

      1. clear-num 85.1%

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

      2. associate-/r/ 84.0%

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

      3. clear-num 84.0%

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

      4. add-sqr-sqrt 34.5%

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

      5. sqrt-unprod 58.2%

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

      6. sqr-neg 58.2%

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

      7. sqrt-unprod 29.1%

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

      8. add-sqr-sqrt 44.6%

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

   8. Applied egg-rr 44.6%

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

      1. frac-2neg 44.6%

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

      2. distribute-frac-neg 44.6%

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

      3. add-sqr-sqrt 19.7%

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

      4. sqrt-unprod 50.1%

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

      5. sqr-neg 50.1%

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

      6. sqrt-unprod 41.9%

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

      7. add-sqr-sqrt 84.0%

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

      8. frac-2neg 84.0%

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

      9. cancel-sign-sub-inv 84.0%

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

      10. associate-/r/ 88.4%

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

      11. div-inv 88.4%

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

      12. clear-num 88.4%

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

   10. Applied egg-rr 88.4%

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

       1. clear-num 88.4%

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

       2. div-inv 88.4%

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

   12. Applied egg-rr 88.4%

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

   if 1.40000000000000002e-64 < z

   1. Initial program 90.0%

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

      1. associate-*l/ 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around inf 86.6%

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

      1. associate-*l/ 93.2%

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

      2. *-commutative 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-64}:\\ \;\;\;\;x - \frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. associate-*l/ 97.1%

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

3. Simplified 97.1%

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

4. Final simplification 97.1%

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

Alternative 5: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + z \cdot \frac{y}{t} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. associate-*l/ 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in z around inf 74.5%

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

   1. associate-*l/ 76.5%

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

   2. *-commutative 76.5%

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

6. Simplified 76.5%

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

7. Final simplification 76.5%

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

Alternative 6: 38.3% 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 95.7%

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

   1. associate-/l* 93.3%

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

3. Simplified 93.3%

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

4. Taylor expanded in z around 0 61.5%

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

   1. associate-*r/ 61.5%

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

   2. neg-mul-1 61.5%

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

6. Simplified 61.5%

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

   1. clear-num 61.4%

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

   2. associate-/r/ 61.0%

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

   3. clear-num 61.0%

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

   4. add-sqr-sqrt 31.2%

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

   5. sqrt-unprod 47.6%

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

   6. sqr-neg 47.6%

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

   7. sqrt-unprod 17.2%

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

   8. add-sqr-sqrt 37.9%

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

8. Applied egg-rr 37.9%

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

9. Taylor expanded in t around inf 37.5%

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

10. Final simplification 37.5%

    \[\leadsto x \]

Developer target: 90.7% 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 2023310 
(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)))