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

Percentage Accurate: 93.6% → 99.5%

Time: 7.9s

Alternatives: 10

Speedup: 0.9×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \mathsf{fma}\left(\frac{z - t}{a}, -y, x\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+209}:\\ \;\;\;\;x - \frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))) (t_2 (fma (/ (- z t) a) (- y) x)))
   (if (<= t_1 (- INFINITY)) t_2 (if (<= t_1 2e+209) (- x (/ t_1 a)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double t_2 = fma(((z - t) / a), -y, x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 2e+209) {
		tmp = x - (t_1 / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	t_2 = fma(Float64(Float64(z - t) / a), Float64(-y), x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 2e+209)
		tmp = Float64(x - Float64(t_1 / a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * (-y) + x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 2e+209], N[(x - N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -inf.0 or 2.0000000000000001e209 < (*.f64 y (-.f64 z t))

   1. Initial program 78.0%

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

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 y (-.f64 z t)) < 2.0000000000000001e209

   1. Initial program 98.8%

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

Alternative 2: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;x - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (- t z) (/ y a))))
   (if (<= t_1 -1e+308) t_2 (if (<= t_1 2e+284) (- x t_1) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (t - z) * (y / a);
	double tmp;
	if (t_1 <= -1e+308) {
		tmp = t_2;
	} else if (t_1 <= 2e+284) {
		tmp = x - t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    t_2 = (t - z) * (y / a)
    if (t_1 <= (-1d+308)) then
        tmp = t_2
    else if (t_1 <= 2d+284) then
        tmp = x - t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (t - z) * (y / a);
	double tmp;
	if (t_1 <= -1e+308) {
		tmp = t_2;
	} else if (t_1 <= 2e+284) {
		tmp = x - t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = (t - z) * (y / a)
	tmp = 0
	if t_1 <= -1e+308:
		tmp = t_2
	elif t_1 <= 2e+284:
		tmp = x - t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(Float64(t - z) * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -1e+308)
		tmp = t_2;
	elseif (t_1 <= 2e+284)
		tmp = Float64(x - t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	t_2 = (t - z) * (y / a);
	tmp = 0.0;
	if (t_1 <= -1e+308)
		tmp = t_2;
	elseif (t_1 <= 2e+284)
		tmp = x - t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+308], t$95$2, If[LessEqual[t$95$1, 2e+284], N[(x - t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1e308 or 2.00000000000000016e284 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 79.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(z - t\right)\right)} \cdot \frac{y}{a} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(0 - z\right) + t\right)} \cdot \frac{y}{a} \]

      8. neg-sub0 N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if -1e308 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2.00000000000000016e284

   1. Initial program 98.7%

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

Alternative 3: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -3 \cdot 10^{+122}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (- t z) (/ y a))))
   (if (<= t_1 -3e+122) t_2 (if (<= t_1 1e+62) (fma (/ y a) t x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (t - z) * (y / a);
	double tmp;
	if (t_1 <= -3e+122) {
		tmp = t_2;
	} else if (t_1 <= 1e+62) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(Float64(t - z) * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -3e+122)
		tmp = t_2;
	elseif (t_1 <= 1e+62)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -3e+122], t$95$2, If[LessEqual[t$95$1, 1e+62], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -2.99999999999999986e122 or 1.00000000000000004e62 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(z - t\right)\right)} \cdot \frac{y}{a} \]

      7. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(0 - z\right) + t\right)} \cdot \frac{y}{a} \]

      8. neg-sub0 N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -2.99999999999999986e122 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.00000000000000004e62

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 81.5

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

   5. Applied rewrites 81.5%

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

Alternative 4: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+60}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) t x)))
   (if (<= t -5.2e-18) t_1 (if (<= t 2.1e+60) (- x (/ (* y z) a)) 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 <= -5.2e-18) {
		tmp = t_1;
	} else if (t <= 2.1e+60) {
		tmp = x - ((y * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), t, x)
	tmp = 0.0
	if (t <= -5.2e-18)
		tmp = t_1;
	elseif (t <= 2.1e+60)
		tmp = Float64(x - Float64(Float64(y * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[t, -5.2e-18], t$95$1, If[LessEqual[t, 2.1e+60], N[(x - N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2000000000000001e-18 or 2.1000000000000001e60 < t

   1. Initial program 91.3%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -5.2000000000000001e-18 < t < 2.1000000000000001e60

   1. Initial program 95.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+60}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+219}:\\ \;\;\;\;\frac{-z}{a} \cdot y\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.2e+219)
   (* (/ (- z) a) y)
   (if (<= z 7.6e+168) (fma (/ y a) t x) (* (/ (- y) a) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.2e+219) {
		tmp = (-z / a) * y;
	} else if (z <= 7.6e+168) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = (-y / a) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.2e+219)
		tmp = Float64(Float64(Float64(-z) / a) * y);
	elseif (z <= 7.6e+168)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(Float64(Float64(-y) / a) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.2e+219], N[(N[((-z) / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 7.6e+168], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(N[((-y) / a), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.2000000000000004e219

   1. Initial program 85.4%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 16.6

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

   5. Applied rewrites 16.6%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. neg-mul-1 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 81.9

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

   8. Applied rewrites 81.9%

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

   if -9.2000000000000004e219 < z < 7.6000000000000005e168

   1. Initial program 94.2%

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

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if 7.6000000000000005e168 < z

   1. Initial program 94.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 77.6

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

   5. Applied rewrites 77.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+219}:\\ \;\;\;\;\frac{-z}{a} \cdot y\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-y}{a} \cdot z\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+219}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- y) a) z)))
   (if (<= z -9.2e+219) t_1 (if (<= z 7.6e+168) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-y / a) * z;
	double tmp;
	if (z <= -9.2e+219) {
		tmp = t_1;
	} else if (z <= 7.6e+168) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-y) / a) * z)
	tmp = 0.0
	if (z <= -9.2e+219)
		tmp = t_1;
	elseif (z <= 7.6e+168)
		tmp = fma(Float64(y / a), t, 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] * z), $MachinePrecision]}, If[LessEqual[z, -9.2e+219], t$95$1, If[LessEqual[z, 7.6e+168], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.2000000000000004e219 or 7.6000000000000005e168 < z

   1. Initial program 91.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -9.2000000000000004e219 < z < 7.6000000000000005e168

   1. Initial program 94.2%

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

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+219}:\\ \;\;\;\;\frac{-y}{a} \cdot z\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z - t, \frac{-1}{a} \cdot y, x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma (- z t) (* (/ -1.0 a) y) x))

C

double code(double x, double y, double z, double t, double a) {
	return fma((z - t), ((-1.0 / a) * y), x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(z - t), Float64(Float64(-1.0 / a) * y), x)
end

Wolfram

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

Derivation

1. Initial program 93.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. div-inv N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 96.3

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

4. Applied rewrites 96.3%

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

5. Final simplification 96.3%

   \[\leadsto \mathsf{fma}\left(z - t, \frac{-1}{a} \cdot y, x\right) \]
6. Add Preprocessing

Alternative 8: 71.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l/ N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 68.4

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

5. Applied rewrites 68.4%

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

Alternative 9: 34.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 31.4

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

5. Applied rewrites 31.4%

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

7. Applied rewrites 33.9%

   \[\leadsto \frac{y}{a} \cdot \color{blue}{t} \]
8. Add Preprocessing

Alternative 10: 32.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 31.4

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

5. Applied rewrites 31.4%

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

7. Applied rewrites 32.4%

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

8. Final simplification 32.4%

   \[\leadsto \frac{t}{a} \cdot y \]
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 2024268 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))