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

Percentage Accurate: 92.9% → 97.7%

Time: 6.9s

Alternatives: 7

Speedup: 1.1×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 97.3

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

4. Applied rewrites 97.3%

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

Alternative 2: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+17} \lor \neg \left(t \leq 9.4 \cdot 10^{-101}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4e+17) (not (<= t 9.4e-101)))
   (fma (/ z t) y x)
   (/ (* (- z x) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4e+17) || !(t <= 9.4e-101)) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = ((z - x) * y) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4e+17) || !(t <= 9.4e-101))
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(Float64(Float64(z - x) * y) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4e+17], N[Not[LessEqual[t, 9.4e-101]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4e17 or 9.3999999999999999e-101 < t

   1. Initial program 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 87.4

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

   7. Applied rewrites 87.4%

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

   if -4e17 < t < 9.3999999999999999e-101

   1. Initial program 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 84.5

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

   7. Applied rewrites 84.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+17} \lor \neg \left(t \leq 9.4 \cdot 10^{-101}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+72} \lor \neg \left(z \leq 2.65 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.8e+72) (not (<= z 2.65e-5)))
   (fma (/ z t) y x)
   (* (- 1.0 (/ y t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.8e+72) || !(z <= 2.65e-5)) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = (1.0 - (y / t)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.8e+72) || !(z <= 2.65e-5))
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(Float64(1.0 - Float64(y / t)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.8e+72], N[Not[LessEqual[z, 2.65e-5]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.80000000000000034e72 or 2.65e-5 < z

   1. Initial program 87.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.4

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

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

   if -5.80000000000000034e72 < z < 2.65e-5

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\left(1 - \frac{y}{t}\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+72} \lor \neg \left(z \leq 2.65 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.8e+72)
   (+ x (/ (* z y) t))
   (if (<= z 2.65e-5) (* (- 1.0 (/ y t)) x) (fma (/ z t) y x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.8e+72) {
		tmp = x + ((z * y) / t);
	} else if (z <= 2.65e-5) {
		tmp = (1.0 - (y / t)) * x;
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.8e+72)
		tmp = Float64(x + Float64(Float64(z * y) / t));
	elseif (z <= 2.65e-5)
		tmp = Float64(Float64(1.0 - Float64(y / t)) * x);
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.8e+72], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.65e-5], N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.80000000000000034e72

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -5.80000000000000034e72 < z < 2.65e-5

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if 2.65e-5 < z

   1. Initial program 84.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 82.7

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

   7. Applied rewrites 82.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-249} \lor \neg \left(z \leq -1.15 \cdot 10^{-299}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.8e-249) (not (<= z -1.15e-299)))
   (fma (/ z t) y x)
   (* (- x) (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.8e-249) || !(z <= -1.15e-299)) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = -x * (y / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.8e-249) || !(z <= -1.15e-299))
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(Float64(-x) * Float64(y / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.8e-249], N[Not[LessEqual[z, -1.15e-299]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[((-x) * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.79999999999999997e-249 or -1.15e-299 < z

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 91.2

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if -1.79999999999999997e-249 < z < -1.15e-299

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-249} \lor \neg \left(z \leq -1.15 \cdot 10^{-299}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 90.8

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

4. Applied rewrites 90.8%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 70.9

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

7. Applied rewrites 70.9%

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

Alternative 7: 40.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in x around 0

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 34.9

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

5. Applied rewrites 34.9%

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

7. Applied rewrites 38.6%

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

8. Final simplification 38.6%

   \[\leadsto z \cdot \frac{y}{t} \]
9. Add Preprocessing

Developer Target 1: 90.6% 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 2024337 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (+ (* x (/ y t)) (* (- z) (/ y t)))))

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