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

Percentage Accurate: 93.4% → 99.1%

Time: 9.5s

Alternatives: 11

Speedup: 1.1×

• 25.6% 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: 93.4% 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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t\_1 \leq 10^{+80}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(-t, y, y \cdot z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 (- INFINITY))
     (+ x (/ y (/ a (- z t))))
     (if (<= t_1 1e+80)
       (+ x (/ (fma (- t) y (* y z)) a))
       (fma (/ y a) (- z t) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + (y / (a / (z - t)));
	} else if (t_1 <= 1e+80) {
		tmp = x + (fma(-t, y, (y * z)) / a);
	} else {
		tmp = fma((y / a), (z - t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	elseif (t_1 <= 1e+80)
		tmp = Float64(x + Float64(fma(Float64(-t), y, Float64(y * z)) / a));
	else
		tmp = fma(Float64(y / a), Float64(z - t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+80], N[(x + N[(N[((-t) * y + N[(y * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 63.1%

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

      1. lift--.f64 N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1e80

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   if 1e80 < (*.f64 y (-.f64 z t))

   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

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

Alternative 2: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+172}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+85}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (- z t) (/ y a))))
   (if (<= t_1 -2e+172) t_2 (if (<= t_1 1e+85) (- x (/ (* y t) a)) 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 = (z - t) * (y / a);
	double tmp;
	if (t_1 <= -2e+172) {
		tmp = t_2;
	} else if (t_1 <= 1e+85) {
		tmp = x - ((y * t) / a);
	} 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 = (z - t) * (y / a)
    if (t_1 <= (-2d+172)) then
        tmp = t_2
    else if (t_1 <= 1d+85) then
        tmp = x - ((y * t) / a)
    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 = (z - t) * (y / a);
	double tmp;
	if (t_1 <= -2e+172) {
		tmp = t_2;
	} else if (t_1 <= 1e+85) {
		tmp = x - ((y * t) / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = (z - t) * (y / a)
	tmp = 0
	if t_1 <= -2e+172:
		tmp = t_2
	elif t_1 <= 1e+85:
		tmp = x - ((y * t) / a)
	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(z - t) * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -2e+172)
		tmp = t_2;
	elseif (t_1 <= 1e+85)
		tmp = Float64(x - Float64(Float64(y * t) / a));
	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 = (z - t) * (y / a);
	tmp = 0.0;
	if (t_1 <= -2e+172)
		tmp = t_2;
	elseif (t_1 <= 1e+85)
		tmp = x - ((y * t) / a);
	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[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+172], t$95$2, If[LessEqual[t$95$1, 1e+85], N[(x - N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -2.0000000000000002e172 or 1e85 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 84.4%

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

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

      3. lower--.f64 82.3

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

   5. Simplified 82.3%

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

      1. sub-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lift-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. lift-neg.f64 N/A

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

      16. sub-neg N/A

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

      17. lift--.f64 N/A

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

      18. associate-*r* N/A

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

      19. associate-/r/ N/A

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

      20. clear-num N/A

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

      21. lift-/.f64 N/A

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

      22. lower-*.f64 92.1

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

   7. Applied egg-rr 92.1%

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

   if -2.0000000000000002e172 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1e85

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.6

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

   5. Simplified 87.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2 \cdot 10^{+172}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 10^{+85}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+86}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (- z t) (/ y a))))
   (if (<= t_1 -2e+86) t_2 (if (<= t_1 5e+112) (fma (/ y a) z 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 = (z - t) * (y / a);
	double tmp;
	if (t_1 <= -2e+86) {
		tmp = t_2;
	} else if (t_1 <= 5e+112) {
		tmp = fma((y / a), z, 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(z - t) * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -2e+86)
		tmp = t_2;
	elseif (t_1 <= 5e+112)
		tmp = fma(Float64(y / a), z, 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[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+86], t$95$2, If[LessEqual[t$95$1, 5e+112], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -2e86 or 5e112 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 85.1%

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

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

      3. lower--.f64 82.3

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

   5. Simplified 82.3%

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

      1. sub-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lift-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. lift-neg.f64 N/A

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

      16. sub-neg N/A

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

      17. lift--.f64 N/A

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

      18. associate-*r* N/A

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

      19. associate-/r/ N/A

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

      20. clear-num N/A

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

      21. lift-/.f64 N/A

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

      22. lower-*.f64 91.0

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

   7. Applied egg-rr 91.0%

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

   if -2e86 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5e112

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 98.2

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 85.9

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

   7. Simplified 85.9%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-fma.f64 87.5

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

   9. Applied egg-rr 87.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2 \cdot 10^{+86}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+80}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(-t, y, y \cdot z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 (- INFINITY))
     (fma (/ (- z t) a) y x)
     (if (<= t_1 1e+80)
       (+ x (/ (fma (- t) y (* y z)) a))
       (fma (/ y a) (- z t) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((z - t) / a), y, x);
	} else if (t_1 <= 1e+80) {
		tmp = x + (fma(-t, y, (y * z)) / a);
	} else {
		tmp = fma((y / a), (z - t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	elseif (t_1 <= 1e+80)
		tmp = Float64(x + Float64(fma(Float64(-t), y, Float64(y * z)) / a));
	else
		tmp = fma(Float64(y / a), Float64(z - t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+80], N[(x + N[(N[((-t) * y + N[(y * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 63.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1e80

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   if 1e80 < (*.f64 y (-.f64 z t))

   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

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

Alternative 5: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+80}:\\ \;\;\;\;x + \frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 (- INFINITY))
     (fma (/ (- z t) a) y x)
     (if (<= t_1 1e+80) (+ x (/ t_1 a)) (fma (/ y a) (- z t) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((z - t) / a), y, x);
	} else if (t_1 <= 1e+80) {
		tmp = x + (t_1 / a);
	} else {
		tmp = fma((y / a), (z - t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	elseif (t_1 <= 1e+80)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = fma(Float64(y / a), Float64(z - t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+80], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 63.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1e80

   1. Initial program 99.7%

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

   if 1e80 < (*.f64 y (-.f64 z t))

   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

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

Alternative 6: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -y \cdot \frac{t}{a}\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{+216}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* y (/ t a)))))
   (if (<= t -4.3e+216) t_1 (if (<= t 1.12e+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 * (t / a));
	double tmp;
	if (t <= -4.3e+216) {
		tmp = t_1;
	} else if (t <= 1.12e+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(y * Float64(t / a)))
	tmp = 0.0
	if (t <= -4.3e+216)
		tmp = t_1;
	elseif (t <= 1.12e+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[(y * N[(t / a), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t, -4.3e+216], t$95$1, If[LessEqual[t, 1.12e+139], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.29999999999999969e216 or 1.12e139 < t

   1. Initial program 82.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 72.2

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

   5. Simplified 72.2%

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

      1. lift-neg.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 77.0

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

   7. Applied egg-rr 77.0%

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

   if -4.29999999999999969e216 < t < 1.12e139

   1. Initial program 94.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 95.7

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

   4. Applied egg-rr 95.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 80.1

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

   7. Simplified 80.1%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-fma.f64 85.2

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

   9. Applied egg-rr 85.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+216}:\\ \;\;\;\;-y \cdot \frac{t}{a}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -t \cdot \frac{y}{a}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* t (/ y a)))))
   (if (<= t -3.2e+255) t_1 (if (<= t 2e+136) (fma (/ y a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -(t * (y / a));
	double tmp;
	if (t <= -3.2e+255) {
		tmp = t_1;
	} else if (t <= 2e+136) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(-Float64(t * Float64(y / a)))
	tmp = 0.0
	if (t <= -3.2e+255)
		tmp = t_1;
	elseif (t <= 2e+136)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = (-N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t, -3.2e+255], t$95$1, If[LessEqual[t, 2e+136], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1999999999999998e255 or 2.00000000000000012e136 < t

   1. Initial program 81.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 74.3

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

   5. Simplified 74.3%

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

   if -3.1999999999999998e255 < t < 2.00000000000000012e136

   1. Initial program 94.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 96.2

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

   4. Applied egg-rr 96.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 79.4

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

   7. Simplified 79.4%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-fma.f64 84.5

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

   9. Applied egg-rr 84.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+255}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 96.2

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

4. Applied egg-rr 96.2%

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

Alternative 9: 71.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 95.5

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

4. Applied egg-rr 95.5%

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

5. Taylor expanded in z around inf

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

   1. lower-/.f64 69.0

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

7. Simplified 69.0%

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

   4. lift-/.f64 N/A

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

   5. lower-fma.f64 73.4

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

9. Applied egg-rr 73.4%

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

Alternative 10: 68.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 69.0

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

5. Simplified 69.0%

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

Alternative 11: 34.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
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. lower-*.f64 33.3

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

5. Simplified 33.3%

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

   1. lift-*.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. associate-/r/ N/A

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

   7. clear-num N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 38.0

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

7. Applied egg-rr 38.0%

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

8. Final simplification 38.0%

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

Developer Target 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{t\_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (+ x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (+ x (/ (* y (- z t)) a))
       (+ x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x + (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) / a)
    else
        tmp = x + (y / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x + (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) / a)
	else:
		tmp = x + (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x + Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x + Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x + (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) / a);
	else
		tmp = x + (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x + N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024212 
(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)))