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

Percentage Accurate: 92.8% → 97.1%

Time: 6.7s

Alternatives: 10

Speedup: 1.1×

• 24.4% 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: 92.8% 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: 97.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+191}:\\ \;\;\;\;x - \frac{y}{{\left(z - t\right)}^{-1} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1e+191)
   (- x (/ y (* (pow (- z t) -1.0) a)))
   (fma (- t z) (/ y a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1e+191) {
		tmp = x - (y / (pow((z - t), -1.0) * a));
	} else {
		tmp = fma((t - z), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1e+191)
		tmp = Float64(x - Float64(y / Float64((Float64(z - t) ^ -1.0) * a)));
	else
		tmp = fma(Float64(t - z), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1e+191], N[(x - N[(y / N[(N[Power[N[(z - t), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000007e191

   1. Initial program 83.0%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -1.00000000000000007e191 < y

   1. Initial program 92.9%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - t\right)}, \frac{y}{a}, x\right) \]

      10. associate-+l- N/A

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

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+191}:\\ \;\;\;\;x - \frac{y}{{\left(z - t\right)}^{-1} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+63} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+64}\right):\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (or (<= t_1 -2e+63) (not (<= t_1 2e+64)))
     (* (- t z) (/ y a))
     (fma (/ t a) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if ((t_1 <= -2e+63) || !(t_1 <= 2e+64)) {
		tmp = (t - z) * (y / a);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if ((t_1 <= -2e+63) || !(t_1 <= 2e+64))
		tmp = Float64(Float64(t - z) * Float64(y / a));
	else
		tmp = fma(Float64(t / a), y, x);
	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]}, If[Or[LessEqual[t$95$1, -2e+63], N[Not[LessEqual[t$95$1, 2e+64]], $MachinePrecision]], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -2.00000000000000012e63 or 2.00000000000000004e64 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 87.5%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if -2.00000000000000012e63 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2.00000000000000004e64

   1. Initial program 97.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.4

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

   5. Applied rewrites 81.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2 \cdot 10^{+63} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 2 \cdot 10^{+64}\right):\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.3 \cdot 10^{+190}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -6.3e+190) (- x (/ y (/ a (- z t)))) (fma (- t z) (/ y a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -6.3e+190) {
		tmp = x - (y / (a / (z - t)));
	} else {
		tmp = fma((t - z), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -6.3e+190)
		tmp = Float64(x - Float64(y / Float64(a / Float64(z - t))));
	else
		tmp = fma(Float64(t - z), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -6.3e+190], N[(x - N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.3000000000000002e190

   1. Initial program 83.0%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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 -6.3000000000000002e190 < y

   1. Initial program 92.9%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - t\right)}, \frac{y}{a}, x\right) \]

      10. associate-+l- N/A

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

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 4: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+43}:\\ \;\;\;\;x - \frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.5e+65)
   (fma (/ y a) t x)
   (if (<= t 2.2e+43) (- x (* (/ y a) z)) (* (- t z) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.5e+65) {
		tmp = fma((y / a), t, x);
	} else if (t <= 2.2e+43) {
		tmp = x - ((y / a) * z);
	} else {
		tmp = (t - z) * (y / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.5e+65)
		tmp = fma(Float64(y / a), t, x);
	elseif (t <= 2.2e+43)
		tmp = Float64(x - Float64(Float64(y / a) * z));
	else
		tmp = Float64(Float64(t - z) * Float64(y / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.5e+65], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[t, 2.2e+43], N[(x - N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.5e65

   1. Initial program 90.3%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - t\right)}, \frac{y}{a}, x\right) \]

      10. associate-+l- N/A

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

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity 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

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 85.0

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

   8. Applied rewrites 85.0%

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

   if -4.5e65 < t < 2.20000000000000001e43

   1. Initial program 93.3%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 88.5

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

   5. Applied rewrites 88.5%

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

   7. Applied rewrites 91.4%

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

   if 2.20000000000000001e43 < t

   1. Initial program 89.7%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 88.0

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

   5. Applied rewrites 88.0%

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

Alternative 5: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+219} \lor \neg \left(z \leq 3.45 \cdot 10^{+134}\right):\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3e+219) (not (<= z 3.45e+134)))
   (* (- z) (/ y a))
   (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3e+219) || !(z <= 3.45e+134)) {
		tmp = -z * (y / a);
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3e+219) || !(z <= 3.45e+134))
		tmp = Float64(Float64(-z) * Float64(y / a));
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3e+219], N[Not[LessEqual[z, 3.45e+134]], $MachinePrecision]], N[((-z) * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9999999999999997e219 or 3.4500000000000001e134 < z

   1. Initial program 90.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if -2.9999999999999997e219 < z < 3.4500000000000001e134

   1. Initial program 92.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - t\right)}, \frac{y}{a}, x\right) \]

      10. associate-+l- N/A

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

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity 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

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 77.8

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

   8. Applied rewrites 77.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+219} \lor \neg \left(z \leq 3.45 \cdot 10^{+134}\right):\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+219} \lor \neg \left(z \leq 1.5 \cdot 10^{+263}\right):\\ \;\;\;\;\frac{-z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3e+219) (not (<= z 1.5e+263)))
   (* (/ (- z) a) y)
   (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3e+219) || !(z <= 1.5e+263)) {
		tmp = (-z / a) * y;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3e+219) || !(z <= 1.5e+263))
		tmp = Float64(Float64(Float64(-z) / a) * y);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3e+219], N[Not[LessEqual[z, 1.5e+263]], $MachinePrecision]], N[(N[((-z) / a), $MachinePrecision] * y), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9999999999999997e219 or 1.49999999999999995e263 < z

   1. Initial program 91.5%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 72.7%

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

   if -2.9999999999999997e219 < z < 1.49999999999999995e263

   1. Initial program 91.8%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - t\right)}, \frac{y}{a}, x\right) \]

      10. associate-+l- N/A

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

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity 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

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 74.9

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

   8. Applied rewrites 74.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+219} \lor \neg \left(z \leq 1.5 \cdot 10^{+263}\right):\\ \;\;\;\;\frac{-z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
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. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - t\right)}, \frac{y}{a}, x\right) \]

   10. associate-+l- N/A

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

   11. neg-sub0 N/A

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

   12. mul-1-neg N/A

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

   13. +-commutative N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 N/A

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

   17. lower-/.f64 97.1

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

5. Applied rewrites 97.1%

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

Alternative 8: 71.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
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. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - t\right)}, \frac{y}{a}, x\right) \]

   10. associate-+l- N/A

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

   11. neg-sub0 N/A

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

   12. mul-1-neg N/A

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

   13. +-commutative N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 N/A

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

   17. lower-/.f64 97.1

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in z around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity 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

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 70.7

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

8. Applied rewrites 70.7%

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

Alternative 9: 68.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]
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. associate-*l/ N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 68.1

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

5. Applied rewrites 68.1%

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

Alternative 10: 34.1% 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 91.8%

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

3. Taylor expanded in t around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 34.4

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

5. Applied rewrites 34.4%

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

7. Applied rewrites 36.4%

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

Developer Target 1: 99.2% 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 2024313 
(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)))