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

Percentage Accurate: 93.2% → 96.8%

Time: 9.1s

Alternatives: 8

Speedup: 1.1×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-162}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t - z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.45e-162) (+ x (/ y (/ a (- t z)))) (fma (/ y a) (- t z) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.45e-162) {
		tmp = x + (y / (a / (t - z)));
	} else {
		tmp = fma((y / a), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.45e-162)
		tmp = Float64(x + Float64(y / Float64(a / Float64(t - z))));
	else
		tmp = fma(Float64(y / a), Float64(t - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.45e-162], N[(x + N[(y / N[(a / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4500000000000001e-162

   1. Initial program 86.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -1.4500000000000001e-162 < a

   1. Initial program 94.7%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-162}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t - z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+38}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (/ (* y (- t z)) a)))
   (if (<= t_1 -1e+38) t_2 (if (<= t_1 5e+57) (fma y (/ t a) x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (y * (t - z)) / a;
	double tmp;
	if (t_1 <= -1e+38) {
		tmp = t_2;
	} else if (t_1 <= 5e+57) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(Float64(y * Float64(t - z)) / a)
	tmp = 0.0
	if (t_1 <= -1e+38)
		tmp = t_2;
	elseif (t_1 <= 5e+57)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+38], t$95$2, If[LessEqual[t$95$1, 5e+57], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.99999999999999977e37 or 4.99999999999999972e57 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 87.1%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(0 - \left(z - t\right)\right)}}{a} \]

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + t\right)}{a} \]

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{y \cdot \left(t + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)}{a} \]

      14. sub-neg N/A

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

      15. lower--.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if -9.99999999999999977e37 < (/.f64 (*.f64 y (-.f64 z t)) a) < 4.99999999999999972e57

   1. Initial program 98.5%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.5

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

   5. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, -z, x\right)\\ \mathbf{if}\;z \leq -1.26 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 30000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) (- z) x)))
   (if (<= z -1.26e+101)
     t_1
     (if (<= z 30000000000000.0) (fma y (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), -z, x);
	double tmp;
	if (z <= -1.26e+101) {
		tmp = t_1;
	} else if (z <= 30000000000000.0) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), Float64(-z), x)
	tmp = 0.0
	if (z <= -1.26e+101)
		tmp = t_1;
	elseif (z <= 30000000000000.0)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * (-z) + x), $MachinePrecision]}, If[LessEqual[z, -1.26e+101], t$95$1, If[LessEqual[z, 30000000000000.0], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2600000000000001e101 or 3e13 < z

   1. Initial program 87.7%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, -1 \cdot \color{blue}{z}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.5%

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

   if -1.2600000000000001e101 < z < 3e13

   1. Initial program 94.6%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 83.9

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

   5. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{-a}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ y (- a)))))
   (if (<= z -9.5e+111) t_1 (if (<= z 6.5e+72) (fma y (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (y / -a);
	double tmp;
	if (z <= -9.5e+111) {
		tmp = t_1;
	} else if (z <= 6.5e+72) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(y / Float64(-a)))
	tmp = 0.0
	if (z <= -9.5e+111)
		tmp = t_1;
	elseif (z <= 6.5e+72)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.5e+111], t$95$1, If[LessEqual[z, 6.5e+72], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.50000000000000019e111 or 6.5000000000000001e72 < z

   1. Initial program 88.4%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y \cdot z}{a}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y \cdot z}{a}\right)} \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot \frac{z}{a}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot \frac{z}{a}}\right) \]

      5. lower-/.f64 60.5

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

   5. Applied rewrites 60.5%

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

   7. Applied rewrites 68.0%

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

   if -9.50000000000000019e111 < z < 6.5000000000000001e72

   1. Initial program 93.8%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 82.6

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

   5. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{a}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+111}:\\ \;\;\;\;z \cdot \frac{y}{-a}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{-a}\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+200}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ z (- a)))))
   (if (<= z -1.02e+200) t_1 (if (<= z 1.4e+73) (fma y (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / -a);
	double tmp;
	if (z <= -1.02e+200) {
		tmp = t_1;
	} else if (z <= 1.4e+73) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z / Float64(-a)))
	tmp = 0.0
	if (z <= -1.02e+200)
		tmp = t_1;
	elseif (z <= 1.4e+73)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.02e+200], t$95$1, If[LessEqual[z, 1.4e+73], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02000000000000001e200 or 1.40000000000000004e73 < z

   1. Initial program 85.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y \cdot z}{a}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y \cdot z}{a}\right)} \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot \frac{z}{a}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot \frac{z}{a}}\right) \]

      5. lower-/.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -1.02000000000000001e200 < z < 1.40000000000000004e73

   1. Initial program 94.4%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{a}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+200}:\\ \;\;\;\;y \cdot \frac{z}{-a}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.9%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*l/ N/A

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

   2. distribute-lft-out-- N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. associate-+l- N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate-+r+ N/A

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

5. Applied rewrites 96.6%

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

Alternative 7: 67.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.9%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 67.2

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

5. Applied rewrites 67.2%

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

Alternative 8: 33.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ t \cdot \frac{y}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.9%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 30.7

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

5. Applied rewrites 30.7%

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

7. Applied rewrites 31.9%

   \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{t\_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (- x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (- x (/ (* y (- z t)) a))
       (- x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x - (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x - ((y * (z - t)) / a)
    else
        tmp = x - (y / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x - (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x - ((y * (z - t)) / a)
	else:
		tmp = x - (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x - Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x - Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x - Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x - (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x - ((y * (z - t)) / a);
	else
		tmp = x - (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x - N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -430450648655599/4000000000000000000000000) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t)))))))

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