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

Percentage Accurate: 92.9% → 98.8%

Time: 8.0s

Alternatives: 11

Speedup: 1.1×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.9% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+216}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+270}:\\ \;\;\;\;t\_1 + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)) (t_2 (fma (/ y a) (- z t) x)))
   (if (<= t_1 -1e+216) t_2 (if (<= t_1 2e+270) (+ t_1 x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double t_2 = fma((y / a), (z - t), x);
	double tmp;
	if (t_1 <= -1e+216) {
		tmp = t_2;
	} else if (t_1 <= 2e+270) {
		tmp = t_1 + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) * y) / a)
	t_2 = fma(Float64(y / a), Float64(z - t), x)
	tmp = 0.0
	if (t_1 <= -1e+216)
		tmp = t_2;
	elseif (t_1 <= 2e+270)
		tmp = Float64(t_1 + x);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+216], t$95$2, If[LessEqual[t$95$1, 2e+270], N[(t$95$1 + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1e216 or 2.0000000000000001e270 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 87.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -1e216 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2.0000000000000001e270

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -1 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{z \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)))
   (if (<= t_1 -4e+101) t_1 (if (<= t_1 5e+123) (+ (/ (* z y) a) 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 <= -4e+101) {
		tmp = t_1;
	} else if (t_1 <= 5e+123) {
		tmp = ((z * y) / a) + x;
	} 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 = ((z - t) * y) / a
    if (t_1 <= (-4d+101)) then
        tmp = t_1
    else if (t_1 <= 5d+123) then
        tmp = ((z * y) / a) + x
    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 = ((z - t) * y) / a;
	double tmp;
	if (t_1 <= -4e+101) {
		tmp = t_1;
	} else if (t_1 <= 5e+123) {
		tmp = ((z * y) / a) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((z - t) * y) / a
	tmp = 0
	if t_1 <= -4e+101:
		tmp = t_1
	elif t_1 <= 5e+123:
		tmp = ((z * y) / a) + 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 <= -4e+101)
		tmp = t_1;
	elseif (t_1 <= 5e+123)
		tmp = Float64(Float64(Float64(z * y) / a) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z - t) * y) / a;
	tmp = 0.0;
	if (t_1 <= -4e+101)
		tmp = t_1;
	elseif (t_1 <= 5e+123)
		tmp = ((z * y) / a) + x;
	else
		tmp = t_1;
	end
	tmp_2 = 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, -4e+101], t$95$1, If[LessEqual[t$95$1, 5e+123], N[(N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -3.9999999999999999e101 or 4.99999999999999974e123 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 90.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -3.9999999999999999e101 < (/.f64 (*.f64 y (-.f64 z t)) a) < 4.99999999999999974e123

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{z \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)))
   (if (<= t_1 -4e+101) t_1 (if (<= t_1 1e+129) (fma (/ y a) z 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 <= -4e+101) {
		tmp = t_1;
	} else if (t_1 <= 1e+129) {
		tmp = fma((y / a), z, 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 <= -4e+101)
		tmp = t_1;
	elseif (t_1 <= 1e+129)
		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[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+101], t$95$1, If[LessEqual[t$95$1, 1e+129], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -3.9999999999999999e101 or 1e129 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 90.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -3.9999999999999999e101 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1e129

   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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5000000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{-1}{a} \cdot \left(z - t\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{a}}{\frac{\frac{-1}{z - t}}{y}} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5000000.0)
   (fma (- y) (* (/ -1.0 a) (- z t)) x)
   (+ (/ (/ -1.0 a) (/ (/ -1.0 (- z t)) y)) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5000000.0) {
		tmp = fma(-y, ((-1.0 / a) * (z - t)), x);
	} else {
		tmp = ((-1.0 / a) / ((-1.0 / (z - t)) / y)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5000000.0)
		tmp = fma(Float64(-y), Float64(Float64(-1.0 / a) * Float64(z - t)), x);
	else
		tmp = Float64(Float64(Float64(-1.0 / a) / Float64(Float64(-1.0 / Float64(z - t)) / y)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5000000.0], N[((-y) * N[(N[(-1.0 / a), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(-1.0 / a), $MachinePrecision] / N[(N[(-1.0 / N[(z - t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5e6

   1. Initial program 82.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -5e6 < a

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 98.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-1 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

   6. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.2

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

   8. Applied rewrites 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5000000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{-1}{a} \cdot \left(z - t\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{a}}{\frac{\frac{-1}{z - t}}{y}} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -300000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{-1}{a} \cdot \left(z - t\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -300000.0)
   (fma (- y) (* (/ -1.0 a) (- z t)) x)
   (+ (/ (* (- z t) y) a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -300000.0) {
		tmp = fma(-y, ((-1.0 / a) * (z - t)), x);
	} else {
		tmp = (((z - t) * y) / a) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -300000.0)
		tmp = fma(Float64(-y), Float64(Float64(-1.0 / a) * Float64(z - t)), x);
	else
		tmp = Float64(Float64(Float64(Float64(z - t) * y) / a) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -300000.0], N[((-y) * N[(N[(-1.0 / a), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3e5

   1. Initial program 82.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -3e5 < a

   1. Initial program 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -300000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{-1}{a} \cdot \left(z - t\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) z x)))
   (if (<= z -1.4e-6) t_1 (if (<= z 1.55e-6) (- x (* (/ t a) y)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), z, x);
	double tmp;
	if (z <= -1.4e-6) {
		tmp = t_1;
	} else if (z <= 1.55e-6) {
		tmp = x - ((t / a) * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), z, x)
	tmp = 0.0
	if (z <= -1.4e-6)
		tmp = t_1;
	elseif (z <= 1.55e-6)
		tmp = Float64(x - Float64(Float64(t / a) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -1.4e-6], t$95$1, If[LessEqual[z, 1.55e-6], N[(x - N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999994e-6 or 1.55e-6 < z

   1. Initial program 94.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if -1.39999999999999994e-6 < z < 1.55e-6

   1. Initial program 96.5%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

Alternative 7: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+149}:\\ \;\;\;\;\frac{-y}{a} \cdot t\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8e+149)
   (* (/ (- y) a) t)
   (if (<= t 6.2e+139) (fma (/ y a) z x) (* (/ (- t) a) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8e+149) {
		tmp = (-y / a) * t;
	} else if (t <= 6.2e+139) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = (-t / a) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8e+149)
		tmp = Float64(Float64(Float64(-y) / a) * t);
	elseif (t <= 6.2e+139)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(Float64(Float64(-t) / a) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8e+149], N[(N[((-y) / a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t, 6.2e+139], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[((-t) / a), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.00000000000000039e149

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.6%

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

   10. Applied rewrites 64.9%

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

   if -8.00000000000000039e149 < t < 6.2e139

   1. Initial program 98.9%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

   if 6.2e139 < t

   1. Initial program 76.9%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+149}:\\ \;\;\;\;\frac{-y}{a} \cdot t\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-y}{a} \cdot t\\ \mathbf{if}\;t \leq -8 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- y) a) t)))
   (if (<= t -8e+149) t_1 (if (<= t 4.6e+139) (fma (/ y a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-y / a) * t;
	double tmp;
	if (t <= -8e+149) {
		tmp = t_1;
	} else if (t <= 4.6e+139) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-y) / a) * t)
	tmp = 0.0
	if (t <= -8e+149)
		tmp = t_1;
	elseif (t <= 4.6e+139)
		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), $MachinePrecision]}, If[LessEqual[t, -8e+149], t$95$1, If[LessEqual[t, 4.6e+139], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.00000000000000039e149 or 4.6e139 < t

   1. Initial program 84.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.6%

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

   10. Applied rewrites 65.1%

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

   if -8.00000000000000039e149 < t < 4.6e139

   1. Initial program 98.9%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+149}:\\ \;\;\;\;\frac{-y}{a} \cdot t\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-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 96.2

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

4. Applied rewrites 96.2%

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

Alternative 10: 71.9% 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 95.2%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 75.6

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

5. Applied rewrites 75.6%

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

Alternative 11: 34.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 = (y / a) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in z around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 35.9

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

5. Applied rewrites 35.9%

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

7. Applied rewrites 38.6%

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

8. Final simplification 38.6%

   \[\leadsto \frac{y}{a} \cdot z \]
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 2024294 
(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)))