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

Percentage Accurate: 93.2% → 98.7%

Time: 9.7s

Alternatives: 10

Speedup: 1.1×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{\frac{a}{t - z}}\\ \mathbf{if}\;y \leq -6 \cdot 10^{-113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-61}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ y (/ a (- t z))))))
   (if (<= y -6e-113) t_1 (if (<= y 5e-61) (+ x (/ (* y (- t z)) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y / (a / (t - z)));
	double tmp;
	if (y <= -6e-113) {
		tmp = t_1;
	} else if (y <= 5e-61) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y / (a / (t - z)))
    if (y <= (-6d-113)) then
        tmp = t_1
    else if (y <= 5d-61) then
        tmp = x + ((y * (t - z)) / a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y / (a / (t - z)));
	double tmp;
	if (y <= -6e-113) {
		tmp = t_1;
	} else if (y <= 5e-61) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y / (a / (t - z)))
	tmp = 0
	if y <= -6e-113:
		tmp = t_1
	elif y <= 5e-61:
		tmp = x + ((y * (t - z)) / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y / Float64(a / Float64(t - z))))
	tmp = 0.0
	if (y <= -6e-113)
		tmp = t_1;
	elseif (y <= 5e-61)
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y / (a / (t - z)));
	tmp = 0.0;
	if (y <= -6e-113)
		tmp = t_1;
	elseif (y <= 5e-61)
		tmp = x + ((y * (t - z)) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y / N[(a / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e-113], t$95$1, If[LessEqual[y, 5e-61], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.0000000000000002e-113 or 4.9999999999999999e-61 < y

   1. Initial program 89.8%

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

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

      6. associate-/l* N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -6.0000000000000002e-113 < y < 4.9999999999999999e-61

   1. Initial program 99.3%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-113}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t - z}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-61}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t - z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+276}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+36}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))) (t_2 (fma (/ y a) (- t z) x)))
   (if (<= t_1 -4e+276)
     t_2
     (if (<= t_1 2e+36) (+ x (/ (* y (- t z)) 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((y / a), (t - z), x);
	double tmp;
	if (t_1 <= -4e+276) {
		tmp = t_2;
	} else if (t_1 <= 2e+36) {
		tmp = x + ((y * (t - z)) / 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(y / a), Float64(t - z), x)
	tmp = 0.0
	if (t_1 <= -4e+276)
		tmp = t_2;
	elseif (t_1 <= 2e+36)
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / 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[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+276], t$95$2, If[LessEqual[t$95$1, 2e+36], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -4.0000000000000002e276 or 2.00000000000000008e36 < (*.f64 y (-.f64 z t))

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Applied rewrites 99.8%

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

   if -4.0000000000000002e276 < (*.f64 y (-.f64 z t)) < 2.00000000000000008e36

   1. Initial program 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -4 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+36}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.4e-37)
   (fma y (/ t a) x)
   (if (<= t 3e+87) (- x (/ (* y z) a)) (fma (/ y a) t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.4e-37) {
		tmp = fma(y, (t / a), x);
	} else if (t <= 3e+87) {
		tmp = x - ((y * z) / a);
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.4e-37)
		tmp = fma(y, Float64(t / a), x);
	elseif (t <= 3e+87)
		tmp = Float64(x - Float64(Float64(y * z) / a));
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.4e-37], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 3e+87], N[(x - N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.40000000000000004e-37

   1. Initial program 89.6%

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

   3. Taylor expanded in z around 0

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

      1. 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

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if -4.40000000000000004e-37 < t < 2.9999999999999999e87

   1. Initial program 96.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 2.9999999999999999e87 < t

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Applied rewrites 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 90.5

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.4

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

   10. Applied rewrites 81.4%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-/.f64 N/A

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

       5. div-inv N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l/ N/A

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

       8. lift-/.f64 N/A

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

       9. lower-fma.f64 85.4

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

   12. Applied rewrites 85.4%

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

Alternative 4: 74.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.38 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 1.38e+160) (fma (/ y a) t x) (* z (/ (- y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 1.38e+160) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = z * (-y / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 1.38e+160)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(z * Float64(Float64(-y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 1.38e+160], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(z * N[((-y) / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.38e160

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Applied rewrites 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 93.8

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.4

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

   10. Applied rewrites 73.4%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-/.f64 N/A

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

       5. div-inv N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l/ N/A

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

       8. lift-/.f64 N/A

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

       9. lower-fma.f64 76.6

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

   12. Applied rewrites 76.6%

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

   if 1.38e160 < z

   1. Initial program 89.1%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

      1. lift-/.f64 N/A

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

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 84.5

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. un-div-inv N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 84.5

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

   7. Applied rewrites 84.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.38 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 1.8e+160) (fma (/ y a) t x) (- (* y (/ z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 1.8e+160) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = -(y * (z / a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 1.8e+160)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(-Float64(y * Float64(z / a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 1.8e+160], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], (-N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < 1.80000000000000011e160

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-+l- N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

   5. Applied rewrites 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 93.8

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.4

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

   10. Applied rewrites 73.4%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-/.f64 N/A

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

       5. div-inv N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l/ N/A

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

       8. lift-/.f64 N/A

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

       9. lower-fma.f64 76.6

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

   12. Applied rewrites 76.6%

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

   if 1.80000000000000011e160 < z

   1. Initial program 89.1%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

Alternative 6: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in x around 0

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

   1. associate-*l/ N/A

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

   2. distribute-lft-out-- N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. associate-+l- N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate-+r+ N/A

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

5. Applied rewrites 95.7%

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

Alternative 7: 71.1% 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 93.3%

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

3. Taylor expanded in x around 0

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

   1. associate-*l/ N/A

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

   2. distribute-lft-out-- N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. associate-+l- N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate-+r+ N/A

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

5. Applied rewrites 95.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 93.3

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

7. Applied rewrites 93.3%

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

8. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 67.9

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

10. Applied rewrites 67.9%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lift-/.f64 N/A

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

    5. div-inv N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*l/ N/A

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

    8. lift-/.f64 N/A

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

    9. lower-fma.f64 71.5

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

12. Applied rewrites 71.5%

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

Alternative 8: 67.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 68.9

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

5. Applied rewrites 68.9%

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

Alternative 9: 34.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

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

   3. lower-*.f64 32.0

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

5. Applied rewrites 32.0%

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

   1. lift-*.f64 N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 34.5

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

7. Applied rewrites 34.5%

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

8. Final simplification 34.5%

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

Alternative 10: 31.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

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

   3. lower-*.f64 32.0

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

5. Applied rewrites 32.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 32.4

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

7. Applied rewrites 32.4%

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

8. Final simplification 32.4%

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

Developer Target 1: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, 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)))