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

Percentage Accurate: 93.2% → 99.5%

Time: 9.1s

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: 93.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+234}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+237}:\\ \;\;\;\;x + \frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))) (t_2 (fma (/ (- z t) a) y x)))
   (if (<= t_1 -5e+234) t_2 (if (<= t_1 1e+237) (+ x (/ t_1 a)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double t_2 = fma(((z - t) / a), y, x);
	double tmp;
	if (t_1 <= -5e+234) {
		tmp = t_2;
	} else if (t_1 <= 1e+237) {
		tmp = x + (t_1 / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	t_2 = fma(Float64(Float64(z - t) / a), y, x)
	tmp = 0.0
	if (t_1 <= -5e+234)
		tmp = t_2;
	elseif (t_1 <= 1e+237)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+234], t$95$2, If[LessEqual[t$95$1, 1e+237], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -5.0000000000000003e234 or 9.9999999999999994e236 < (*.f64 y (-.f64 z t))

   1. Initial program 79.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -5.0000000000000003e234 < (*.f64 y (-.f64 z t)) < 9.9999999999999994e236

   1. Initial program 99.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (* y (- z t)) a) 1e-36)
   (fma (/ (- z t) a) y x)
   (fma (/ y a) (- z t) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((y * (z - t)) / a) <= 1e-36) {
		tmp = fma(((z - t) / a), y, x);
	} else {
		tmp = fma((y / a), (z - t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(y * Float64(z - t)) / a) <= 1e-36)
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	else
		tmp = fma(Float64(y / a), Float64(z - t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], 1e-36], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < 9.9999999999999994e-37

   1. Initial program 96.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 96.5

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

   4. Applied rewrites 96.5%

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

   if 9.9999999999999994e-37 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 98.7

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

   4. Applied rewrites 98.7%

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

Alternative 3: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{-a}, y, x\right)\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+35}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t (- a)) y x)))
   (if (<= t -5.6e-37) t_1 (if (<= t 9.8e+35) (+ x (/ (* y z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / -a), y, x);
	double tmp;
	if (t <= -5.6e-37) {
		tmp = t_1;
	} else if (t <= 9.8e+35) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / Float64(-a)), y, x)
	tmp = 0.0
	if (t <= -5.6e-37)
		tmp = t_1;
	elseif (t <= 9.8e+35)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	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 + x), $MachinePrecision]}, If[LessEqual[t, -5.6e-37], t$95$1, If[LessEqual[t, 9.8e+35], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.6000000000000002e-37 or 9.8000000000000005e35 < t

   1. Initial program 90.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if -5.6000000000000002e-37 < t < 9.8000000000000005e35

   1. Initial program 96.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-/.f64 N/A

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

      2. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{-a}, y, x\right)\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+35}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{-a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y \cdot t}{a}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+23}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* y t) a))))
   (if (<= t -5.6e-37) t_1 (if (<= t 3e+23) (+ x (/ (* y z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y * t) / a);
	double tmp;
	if (t <= -5.6e-37) {
		tmp = t_1;
	} else if (t <= 3e+23) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = 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 = x - ((y * t) / a)
    if (t <= (-5.6d-37)) then
        tmp = t_1
    else if (t <= 3d+23) then
        tmp = x + ((y * z) / a)
    else
        tmp = 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 = x - ((y * t) / a);
	double tmp;
	if (t <= -5.6e-37) {
		tmp = t_1;
	} else if (t <= 3e+23) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - ((y * t) / a)
	tmp = 0
	if t <= -5.6e-37:
		tmp = t_1
	elif t <= 3e+23:
		tmp = x + ((y * z) / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y * t) / a))
	tmp = 0.0
	if (t <= -5.6e-37)
		tmp = t_1;
	elseif (t <= 3e+23)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - ((y * t) / a);
	tmp = 0.0;
	if (t <= -5.6e-37)
		tmp = t_1;
	elseif (t <= 3e+23)
		tmp = x + ((y * z) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.6e-37], t$95$1, If[LessEqual[t, 3e+23], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.6000000000000002e-37 or 3.0000000000000001e23 < t

   1. Initial program 90.3%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if -5.6000000000000002e-37 < t < 3.0000000000000001e23

   1. Initial program 97.1%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-/.f64 N/A

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

      2. lower-*.f64 94.7

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

   5. Applied rewrites 94.7%

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

Alternative 5: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+119}:\\ \;\;\;\;y \cdot \frac{t}{-a}\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.8e+119)
   (* y (/ t (- a)))
   (if (<= t 1.66e+196) (fma (/ y a) z x) (- (* t (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.8e+119) {
		tmp = y * (t / -a);
	} else if (t <= 1.66e+196) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = -(t * (y / a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.8e+119)
		tmp = Float64(y * Float64(t / Float64(-a)));
	elseif (t <= 1.66e+196)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(-Float64(t * Float64(y / a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.8e+119], N[(y * N[(t / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.66e+196], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], (-N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if t < -1.80000000000000001e119

   1. Initial program 93.2%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 62.3

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

   5. Applied rewrites 62.3%

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

      1. lift-neg.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.0

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

   7. Applied rewrites 67.0%

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

   if -1.80000000000000001e119 < t < 1.65999999999999994e196

   1. Initial program 94.2%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-/.f64 N/A

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

      2. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 85.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if 1.65999999999999994e196 < t

   1. Initial program 87.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+119}:\\ \;\;\;\;y \cdot \frac{t}{-a}\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -t \cdot \frac{y}{a}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* t (/ y a)))))
   (if (<= t -3.1e+119) t_1 (if (<= t 1.66e+196) (fma (/ y a) z 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 <= -3.1e+119) {
		tmp = t_1;
	} else if (t <= 1.66e+196) {
		tmp = fma((y / a), z, 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 <= -3.1e+119)
		tmp = t_1;
	elseif (t <= 1.66e+196)
		tmp = fma(Float64(y / a), z, 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, -3.1e+119], t$95$1, If[LessEqual[t, 1.66e+196], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.09999999999999995e119 or 1.65999999999999994e196 < t

   1. Initial program 91.2%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -3.09999999999999995e119 < t < 1.65999999999999994e196

   1. Initial program 94.2%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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. lower-/.f64 N/A

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

      2. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 85.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+119}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a 4.5e-173) (/ (* y z) a) (* y (/ z a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 4.5e-173) {
		tmp = (y * z) / a;
	} else {
		tmp = y * (z / a);
	}
	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 (a <= 4.5d-173) then
        tmp = (y * z) / a
    else
        tmp = y * (z / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 4.5e-173) {
		tmp = (y * z) / a;
	} else {
		tmp = y * (z / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= 4.5e-173:
		tmp = (y * z) / a
	else:
		tmp = y * (z / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 4.5e-173)
		tmp = Float64(Float64(y * z) / a);
	else
		tmp = Float64(y * Float64(z / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= 4.5e-173)
		tmp = (y * z) / a;
	else
		tmp = y * (z / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, 4.5e-173], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.50000000000000018e-173

   1. Initial program 94.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 38.5

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

   5. Applied rewrites 38.5%

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

   if 4.50000000000000018e-173 < a

   1. Initial program 90.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 23.8

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

   5. Applied rewrites 23.8%

      \[\leadsto \color{blue}{\frac{y \cdot z}{a}} \]
   6. 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 29.9

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

   7. Applied rewrites 29.9%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 93.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 95.7

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

4. Applied rewrites 95.7%

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

Alternative 9: 71.7% 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 93.4%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
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. lower-/.f64 N/A

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

   2. lower-*.f64 69.5

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

5. Applied rewrites 69.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 70.5

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-/.f64 N/A

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

   15. div-inv N/A

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

   16. lower-/.f64 70.5

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

7. Applied rewrites 70.5%

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

Alternative 10: 68.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 68.4

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

5. Applied rewrites 68.4%

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

Alternative 11: 35.0% 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 93.4%

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

3. Taylor expanded in z around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 33.1

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

5. Applied rewrites 33.1%

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

   1. lift-*.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. associate-/r/ N/A

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

   7. clear-num N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 33.4

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

7. Applied rewrites 33.4%

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

8. Final simplification 33.4%

   \[\leadsto z \cdot \frac{y}{a} \]
9. 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 2024214 
(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)))