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

Percentage Accurate: 93.2% → 95.9%

Time: 9.3s

Alternatives: 14

Speedup: 1.1×

• 24.6% 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: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a 3e+36) (+ (/ (* (- z t) y) a) x) (fma (/ (- z t) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 3e+36) {
		tmp = (((z - t) * y) / a) + x;
	} else {
		tmp = fma(((z - t) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 3e+36)
		tmp = Float64(Float64(Float64(Float64(z - t) * y) / a) + x);
	else
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, 3e+36], N[(N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3e36

   1. Initial program 97.9%

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

   if 3e36 < a

   1. Initial program 82.3%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a} \cdot y\\ t_2 := \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+103}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+276}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) a) y)) (t_2 (/ (* (- z t) y) a)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -5e+103)
       t_2
       (if (<= t_2 2e+116) (fma (/ z a) y x) (if (<= t_2 1e+276) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / a) * y;
	double t_2 = ((z - t) * y) / a;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5e+103) {
		tmp = t_2;
	} else if (t_2 <= 2e+116) {
		tmp = fma((z / a), y, x);
	} else if (t_2 <= 1e+276) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) / a) * y)
	t_2 = Float64(Float64(Float64(z - t) * y) / a)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5e+103)
		tmp = t_2;
	elseif (t_2 <= 2e+116)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_2 <= 1e+276)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5e+103], t$95$2, If[LessEqual[t$95$2, 2e+116], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$2, 1e+276], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -inf.0 or 1.0000000000000001e276 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 83.0%

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

      1. lift-+.f64 N/A

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

      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 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a}} \]
   6. 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 83.0

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

   7. Applied rewrites 83.0%

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

   9. Applied rewrites 96.5%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a) < -5e103 or 2.00000000000000003e116 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.0000000000000001e276

   1. Initial program 99.8%

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

   3. Taylor expanded in a 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 80.9

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

   5. Applied rewrites 80.9%

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

   if -5e103 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2.00000000000000003e116

   1. Initial program 99.9%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -\infty:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq -5 \cdot 10^{+103}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 10^{+276}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)))
   (if (<= t_1 -5e+103) t_1 (if (<= t_1 2e+116) (fma (/ z a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double tmp;
	if (t_1 <= -5e+103) {
		tmp = t_1;
	} else if (t_1 <= 2e+116) {
		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(z - t) * y) / a)
	tmp = 0.0
	if (t_1 <= -5e+103)
		tmp = t_1;
	elseif (t_1 <= 2e+116)
		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[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+103], t$95$1, If[LessEqual[t$95$1, 2e+116], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5e103 or 2.00000000000000003e116 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 89.1%

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

   3. Taylor expanded in a 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 82.2

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

   5. Applied rewrites 82.2%

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

   if -5e103 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2.00000000000000003e116

   1. Initial program 99.9%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -5 \cdot 10^{+103}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 33.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -\infty:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x, y, z, t, a):
	tmp = 0
	if (((z - t) * y) / a) <= -math.inf:
		tmp = (y / a) * z
	else:
		tmp = (z * y) / a
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -inf.0

   1. Initial program 86.6%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 67.8

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

   7. Applied rewrites 67.8%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 95.7%

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

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

      3. lower-*.f64 33.4

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

   5. Applied rewrites 33.4%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -\infty:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, -t, x\right)\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.85:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) (- t) x)))
   (if (<= t -2.05e+94) t_1 (if (<= t 0.85) (fma (/ y a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), -t, x);
	double tmp;
	if (t <= -2.05e+94) {
		tmp = t_1;
	} else if (t <= 0.85) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), Float64(-t), x)
	tmp = 0.0
	if (t <= -2.05e+94)
		tmp = t_1;
	elseif (t <= 0.85)
		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[(N[(y / a), $MachinePrecision] * (-t) + x), $MachinePrecision]}, If[LessEqual[t, -2.05e+94], t$95$1, If[LessEqual[t, 0.85], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.05000000000000015e94 or 0.849999999999999978 < t

   1. Initial program 90.1%

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

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -2.05000000000000015e94 < t < 0.849999999999999978

   1. Initial program 96.7%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{a}} \]
   6. 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 89.4

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

   7. Applied rewrites 89.4%

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

Alternative 6: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - t \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- z t) 2e-83) (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 ((z - t) <= 2e-83) {
		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(z - t) <= 2e-83)
		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[(z - t), $MachinePrecision], 2e-83], 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 z t) < 2.0000000000000001e-83

   1. Initial program 92.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 95.2

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

   4. Applied rewrites 95.2%

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

   if 2.0000000000000001e-83 < (-.f64 z t)

   1. Initial program 95.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 98.2

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

   4. Applied rewrites 98.2%

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

Alternative 7: 69.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - t \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- z t) 5e-22) (fma (/ z a) y x) (fma (/ y a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z - t) <= 5e-22) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z - t) <= 5e-22)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z - t), $MachinePrecision], 5e-22], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z t) < 4.99999999999999954e-22

   1. Initial program 93.0%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 73.9

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

   5. Applied rewrites 73.9%

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

   if 4.99999999999999954e-22 < (-.f64 z t)

   1. Initial program 95.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{a}} \]
   6. 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 74.5

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

   7. Applied rewrites 74.5%

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

Alternative 8: 72.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.5e+247) (/ (* (- t) y) a) (fma (/ y a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.5e+247) {
		tmp = (-t * y) / a;
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.5e+247)
		tmp = Float64(Float64(Float64(-t) * y) / a);
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.5e+247], N[(N[((-t) * y), $MachinePrecision] / a), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.50000000000000011e247

   1. Initial program 94.3%

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

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

      3. distribute-lft-neg-out N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 82.9%

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

   if -2.50000000000000011e247 < 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. *-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 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{a}} \]
   6. 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 75.8

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

   7. Applied rewrites 75.8%

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

Alternative 9: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+94}:\\ \;\;\;\;\frac{-y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4e+94) (* (/ (- y) a) t) (fma (/ y a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4e+94) {
		tmp = (-y / a) * t;
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4e+94)
		tmp = Float64(Float64(Float64(-y) / a) * t);
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4e+94], N[(N[((-y) / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.0000000000000001e94

   1. Initial program 95.4%

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

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

      3. distribute-lft-neg-out N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 65.0%

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

   if -4.0000000000000001e94 < t

   1. Initial program 93.7%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{a}} \]
   6. 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 78.1

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

   7. Applied rewrites 78.1%

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

Alternative 10: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+245}:\\ \;\;\;\;\frac{-t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.2e+245) (* (/ (- t) a) y) (fma (/ y a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.2e+245) {
		tmp = (-t / a) * y;
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.2e+245)
		tmp = Float64(Float64(Float64(-t) / a) * y);
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.2e+245], N[(N[((-t) / a), $MachinePrecision] * y), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2000000000000001e245

   1. Initial program 94.3%

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

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

      3. distribute-lft-neg-out N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if -2.2000000000000001e245 < 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. *-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 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{a}} \]
   6. 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 75.8

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

   7. Applied rewrites 75.8%

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

Alternative 11: 97.0% 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 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. *-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 94.3

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

4. Applied rewrites 94.3%

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

Alternative 12: 68.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 70.5

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

5. Applied rewrites 70.5%

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

Alternative 13: 31.9% accurate, 1.4× speedup

Math

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

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(Float64(z * 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[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]

Derivation

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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 37.8

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

5. Applied rewrites 37.8%

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

Alternative 14: 31.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 / a) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 37.8

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

5. Applied rewrites 37.8%

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

7. Applied rewrites 36.4%

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

8. Final simplification 36.4%

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

Developer Target 1: 99.1% 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 2024267 
(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)))