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

Percentage Accurate: 93.6% → 95.0%

Time: 4.7s

Alternatives: 11

Speedup: 1.1×

• 25.3% 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: 93.6% 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: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{1}{z - t} \cdot a} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -4.8e-117)
   (fma (/ y a) (- z t) x)
   (+ (/ y (* (/ 1.0 (- z t)) a)) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -4.8e-117) {
		tmp = fma((y / a), (z - t), x);
	} else {
		tmp = (y / ((1.0 / (z - t)) * a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -4.8e-117)
		tmp = fma(Float64(y / a), Float64(z - t), x);
	else
		tmp = Float64(Float64(y / Float64(Float64(1.0 / Float64(z - t)) * a)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -4.8e-117], N[(N[(y / a), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(1.0 / N[(z - t), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000028e-117

   1. Initial program 90.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if -4.80000000000000028e-117 < x

   1. Initial program 92.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto x + \color{blue}{\frac{1 \cdot y}{\frac{1}{z - t} \cdot a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{1}{z - t} \cdot a} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-220}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y 5e-220) (+ (/ (- z t) (/ a y)) x) (fma (/ (- z t) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 5e-220) {
		tmp = ((z - t) / (a / y)) + x;
	} else {
		tmp = fma(((z - t) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 5e-220)
		tmp = Float64(Float64(Float64(z - t) / Float64(a / y)) + x);
	else
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, 5e-220], N[(N[(N[(z - t), $MachinePrecision] / N[(a / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-220

   1. Initial program 91.6%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 94.4

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

   4. Applied rewrites 94.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. remove-double-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 97.6

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

   6. Applied rewrites 97.6%

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

   if 5.0000000000000002e-220 < y

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-220}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+36}:\\ \;\;\;\;x - \frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.4e-21)
   (fma (/ y a) z x)
   (if (<= z 4.9e+36) (- x (* (/ t a) y)) (fma (/ z a) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.4e-21) {
		tmp = fma((y / a), z, x);
	} else if (z <= 4.9e+36) {
		tmp = x - ((t / a) * y);
	} else {
		tmp = fma((z / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.4e-21)
		tmp = fma(Float64(y / a), z, x);
	elseif (z <= 4.9e+36)
		tmp = Float64(x - Float64(Float64(t / a) * y));
	else
		tmp = fma(Float64(z / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.4e-21], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 4.9e+36], N[(x - N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.4000000000000003e-21

   1. Initial program 89.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -6.4000000000000003e-21 < z < 4.89999999999999981e36

   1. Initial program 92.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if 4.89999999999999981e36 < z

   1. Initial program 93.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 93.5

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

   7. Applied rewrites 93.5%

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

Alternative 4: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- t) (/ y a))))
   (if (<= t -1.9e+188) t_1 (if (<= t 1.1e+137) (fma (/ z a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -t * (y / a);
	double tmp;
	if (t <= -1.9e+188) {
		tmp = t_1;
	} else if (t <= 1.1e+137) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-t) * Float64(y / a))
	tmp = 0.0
	if (t <= -1.9e+188)
		tmp = t_1;
	elseif (t <= 1.1e+137)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.9e+188], t$95$1, If[LessEqual[t, 1.1e+137], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8999999999999999e188 or 1.10000000000000008e137 < t

   1. Initial program 85.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.7%

      \[\leadsto \frac{\left(-t\right) \cdot y}{a} \]
   9. Step-by-step derivation

   10. Applied rewrites 75.3%

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

   if -1.8999999999999999e188 < t < 1.10000000000000008e137

   1. Initial program 93.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 81.0

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

   7. Applied rewrites 81.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+188}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t}{a} \cdot y\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- t) a) y)))
   (if (<= t -1.9e+188) t_1 (if (<= t 7e+133) (fma (/ z a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-t / a) * y;
	double tmp;
	if (t <= -1.9e+188) {
		tmp = t_1;
	} else if (t <= 7e+133) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-t) / a) * y)
	tmp = 0.0
	if (t <= -1.9e+188)
		tmp = t_1;
	elseif (t <= 7e+133)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-t) / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t, -1.9e+188], t$95$1, If[LessEqual[t, 7e+133], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8999999999999999e188 or 6.9999999999999997e133 < t

   1. Initial program 85.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.7%

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

   if -1.8999999999999999e188 < t < 6.9999999999999997e133

   1. Initial program 93.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 81.0

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

   7. Applied rewrites 81.0%

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

Alternative 6: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - t \leq -1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- z t) -1e+70) (fma (/ y a) (- z t) x) (fma (/ (- z t) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z - t) <= -1e+70) {
		tmp = fma((y / a), (z - t), x);
	} else {
		tmp = fma(((z - t) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z - t) <= -1e+70)
		tmp = fma(Float64(y / a), Float64(z - t), x);
	else
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z - t), $MachinePrecision], -1e+70], N[(N[(y / a), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z t) < -1.00000000000000007e70

   1. Initial program 86.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -1.00000000000000007e70 < (-.f64 z t)

   1. Initial program 94.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

Alternative 7: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - t \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- z t) -2e-44) (fma (/ y a) z x) (fma (/ z a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z - t) <= -2e-44) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = fma((z / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z - t) <= -2e-44)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = fma(Float64(z / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z - t), $MachinePrecision], -2e-44], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z t) < -1.99999999999999991e-44

   1. Initial program 89.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if -1.99999999999999991e-44 < (-.f64 z t)

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 73.7

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

   7. Applied rewrites 73.7%

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

Alternative 8: 32.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - t \leq -2 \cdot 10^{-44}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- z t) -2e-44) (* z (/ y a)) (* (/ z a) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z - t) <= -2e-44) {
		tmp = z * (y / a);
	} else {
		tmp = (z / a) * y;
	}
	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) :: tmp
    if ((z - t) <= (-2d-44)) then
        tmp = z * (y / a)
    else
        tmp = (z / a) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z - t) <= -2e-44) {
		tmp = z * (y / a);
	} else {
		tmp = (z / a) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z - t) <= -2e-44:
		tmp = z * (y / a)
	else:
		tmp = (z / a) * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z - t) <= -2e-44)
		tmp = Float64(z * Float64(y / a));
	else
		tmp = Float64(Float64(z / a) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z - t) <= -2e-44)
		tmp = z * (y / a);
	else
		tmp = (z / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z - t), $MachinePrecision], -2e-44], N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z t) < -1.99999999999999991e-44

   1. Initial program 89.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 40.5%

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

   if -1.99999999999999991e-44 < (-.f64 z t)

   1. Initial program 94.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 29.6

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

   5. Applied rewrites 29.6%

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

Alternative 9: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 95.3

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

4. Applied rewrites 95.3%

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

Alternative 10: 72.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 72.1

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

5. Applied rewrites 72.1%

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

Alternative 11: 33.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in z around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 33.4

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

5. Applied rewrites 33.4%

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

7. Applied rewrites 33.2%

   \[\leadsto z \cdot \color{blue}{\frac{y}{a}} \]
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 2024304 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))