Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 91.8% → 98.0%

Time: 8.7s

Alternatives: 14

Speedup: 1.4×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(z, \frac{y}{a} + b, \frac{x}{a}\right) + t\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b))))
   (if (<= t_1 2e+307) t_1 (* (+ (fma z (+ (/ y a) b) (/ x a)) t) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x + (y * z)) + (t * a)) + ((a * z) * b);
	double tmp;
	if (t_1 <= 2e+307) {
		tmp = t_1;
	} else {
		tmp = (fma(z, ((y / a) + b), (x / a)) + t) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b))
	tmp = 0.0
	if (t_1 <= 2e+307)
		tmp = t_1;
	else
		tmp = Float64(Float64(fma(z, Float64(Float64(y / a) + b), Float64(x / a)) + t) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+307], t$95$1, N[(N[(N[(z * N[(N[(y / a), $MachinePrecision] + b), $MachinePrecision] + N[(x / a), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < 1.99999999999999997e307

   1. Initial program 98.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   if 1.99999999999999997e307 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 82.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-out N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 98.2

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

   8. Applied rewrites 98.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(z, \frac{y}{a} + b, \frac{x}{a}\right) + t\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.75 \cdot 10^{-89}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x + a \cdot \mathsf{fma}\left(b, z, t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 2.75e-89)
   (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b))
   (fma z y (+ x (* a (fma b z t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 2.75e-89) {
		tmp = ((x + (y * z)) + (t * a)) + ((a * z) * b);
	} else {
		tmp = fma(z, y, (x + (a * fma(b, z, t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 2.75e-89)
		tmp = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b));
	else
		tmp = fma(z, y, Float64(x + Float64(a * fma(b, z, t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 2.75e-89], N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(z * y + N[(x + N[(a * N[(b * z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.75000000000000006e-89

   1. Initial program 96.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   if 2.75000000000000006e-89 < a

   1. Initial program 91.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 3: 87.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+33} \lor \neg \left(a \leq 260\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -8.2e+33) (not (<= a 260.0)))
   (fma (fma b z t) a x)
   (fma (fma b a y) z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -8.2e+33) || !(a <= 260.0)) {
		tmp = fma(fma(b, z, t), a, x);
	} else {
		tmp = fma(fma(b, a, y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -8.2e+33) || !(a <= 260.0))
		tmp = fma(fma(b, z, t), a, x);
	else
		tmp = fma(fma(b, a, y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -8.2e+33], N[Not[LessEqual[a, 260.0]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.1999999999999999e33 or 260 < a

   1. Initial program 91.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -8.1999999999999999e33 < a < 260

   1. Initial program 97.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+33} \lor \neg \left(a \leq 260\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-25} \lor \neg \left(z \leq 1.75 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.45e-25) (not (<= z 1.75e-23)))
   (fma (fma b a y) z x)
   (fma t a (fma y z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.45e-25) || !(z <= 1.75e-23)) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(t, a, fma(y, z, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.45e-25) || !(z <= 1.75e-23))
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(t, a, fma(y, z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.45e-25], N[Not[LessEqual[z, 1.75e-23]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(t * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e-25 or 1.74999999999999997e-23 < z

   1. Initial program 90.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if -1.45e-25 < z < 1.74999999999999997e-23

   1. Initial program 99.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 92.4

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

   8. Applied rewrites 92.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-25} \lor \neg \left(z \leq 1.75 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+22} \lor \neg \left(z \leq 5.8 \cdot 10^{+171}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.1e+22) (not (<= z 5.8e+171)))
   (* (fma b a y) z)
   (fma t a (fma y z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.1e+22) || !(z <= 5.8e+171)) {
		tmp = fma(b, a, y) * z;
	} else {
		tmp = fma(t, a, fma(y, z, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.1e+22) || !(z <= 5.8e+171))
		tmp = Float64(fma(b, a, y) * z);
	else
		tmp = fma(t, a, fma(y, z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.1e+22], N[Not[LessEqual[z, 5.8e+171]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], N[(t * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e22 or 5.79999999999999969e171 < z

   1. Initial program 86.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if -1.1e22 < z < 5.79999999999999969e171

   1. Initial program 98.8%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 88.5

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

   8. Applied rewrites 88.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+22} \lor \neg \left(z \leq 5.8 \cdot 10^{+171}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(z, b, t\right) \cdot a\right)\\ \mathbf{elif}\;a \leq 260:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.2e+37)
   (fma z y (* (fma z b t) a))
   (if (<= a 260.0) (fma (fma b a y) z x) (fma (fma b z t) a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.2e+37) {
		tmp = fma(z, y, (fma(z, b, t) * a));
	} else if (a <= 260.0) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(fma(b, z, t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.2e+37)
		tmp = fma(z, y, Float64(fma(z, b, t) * a));
	elseif (a <= 260.0)
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(fma(b, z, t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.2e+37], N[(z * y + N[(N[(z * b + t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 260.0], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.2000000000000001e37

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -2.2000000000000001e37 < a < 260

   1. Initial program 97.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 260 < a

   1. Initial program 90.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 93.5

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

   5. Applied rewrites 93.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(z, b, t\right) \cdot a\right)\\ \mathbf{elif}\;a \leq 260:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{elif}\;a \leq 260:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.4e+37)
   (fma (fma b z t) a (* z y))
   (if (<= a 260.0) (fma (fma b a y) z x) (fma (fma b z t) a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.4e+37) {
		tmp = fma(fma(b, z, t), a, (z * y));
	} else if (a <= 260.0) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(fma(b, z, t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.4e+37)
		tmp = fma(fma(b, z, t), a, Float64(z * y));
	elseif (a <= 260.0)
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(fma(b, z, t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.4e+37], N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 260.0], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.4e37

   1. Initial program 91.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -2.4e37 < a < 260

   1. Initial program 97.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 260 < a

   1. Initial program 90.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 93.5

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

   5. Applied rewrites 93.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{elif}\;a \leq 260:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+30} \lor \neg \left(a \leq 3400\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -2e+30) (not (<= a 3400.0))) (* (fma b z t) a) (fma y z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -2e+30) || !(a <= 3400.0)) {
		tmp = fma(b, z, t) * a;
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -2e+30) || !(a <= 3400.0))
		tmp = Float64(fma(b, z, t) * a);
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -2e+30], N[Not[LessEqual[a, 3400.0]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], N[(y * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e30 or 3400 < a

   1. Initial program 91.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot z + t\right)} \cdot a \]

      4. lower-fma.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -2e30 < a < 3400

   1. Initial program 97.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 77.2

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

   8. Applied rewrites 77.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+30} \lor \neg \left(a \leq 3400\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-37} \lor \neg \left(z \leq 5 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.1e-37) (not (<= z 5e-20))) (* (fma b a y) z) (fma t a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.1e-37) || !(z <= 5e-20)) {
		tmp = fma(b, a, y) * z;
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.1e-37) || !(z <= 5e-20))
		tmp = Float64(fma(b, a, y) * z);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.1e-37], N[Not[LessEqual[z, 5e-20]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], N[(t * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000001e-37 or 4.9999999999999999e-20 < z

   1. Initial program 91.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if -2.1000000000000001e-37 < z < 4.9999999999999999e-20

   1. Initial program 99.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{a \cdot t} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-37} \lor \neg \left(z \leq 5 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (fma z y (+ x (* a (fma b z t)))))

C

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

Julia

function code(x, y, z, t, a, b)
	return fma(z, y, Float64(x + Float64(a * fma(b, z, t))))
end

Wolfram

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

Derivation

1. Initial program 94.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. distribute-lft-out N/A

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

   18. lower-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 95.7

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

4. Applied rewrites 95.7%

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

Alternative 11: 62.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+39} \lor \neg \left(y \leq 7 \cdot 10^{+71}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.32e+39) (not (<= y 7e+71))) (fma y z x) (fma t a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.32e+39) || !(y <= 7e+71)) {
		tmp = fma(y, z, x);
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.32e+39) || !(y <= 7e+71))
		tmp = fma(y, z, x);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.32e+39], N[Not[LessEqual[y, 7e+71]], $MachinePrecision]], N[(y * z + x), $MachinePrecision], N[(t * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32e39 or 6.9999999999999998e71 < y

   1. Initial program 93.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 71.8

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

   8. Applied rewrites 71.8%

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

   if -1.32e39 < y < 6.9999999999999998e71

   1. Initial program 95.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{a \cdot t} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+39} \lor \neg \left(y \leq 7 \cdot 10^{+71}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+157} \lor \neg \left(y \leq 5.5 \cdot 10^{+72}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.8e+157) (not (<= y 5.5e+72))) (* y z) (fma t a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.8e+157) || !(y <= 5.5e+72)) {
		tmp = y * z;
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7.8e+157) || !(y <= 5.5e+72))
		tmp = Float64(y * z);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7.8e+157], N[Not[LessEqual[y, 5.5e+72]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(t * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.79999999999999941e157 or 5.5e72 < y

   1. Initial program 91.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 68.1

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

   7. Applied rewrites 68.1%

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

   if -7.79999999999999941e157 < y < 5.5e72

   1. Initial program 96.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{a \cdot t} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.0%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+157} \lor \neg \left(y \leq 5.5 \cdot 10^{+72}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+60} \lor \neg \left(y \leq 10^{+72}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3e+60) (not (<= y 1e+72))) (* y z) (* a t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3e+60) || !(y <= 1e+72)) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-3d+60)) .or. (.not. (y <= 1d+72))) then
        tmp = y * z
    else
        tmp = a * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3e+60) || !(y <= 1e+72)) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3e+60) or not (y <= 1e+72):
		tmp = y * z
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3e+60) || !(y <= 1e+72))
		tmp = Float64(y * z);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3e+60) || ~((y <= 1e+72)))
		tmp = y * z;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3e+60], N[Not[LessEqual[y, 1e+72]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(a * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9999999999999998e60 or 9.99999999999999944e71 < y

   1. Initial program 93.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 93.1

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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 61.7

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

   7. Applied rewrites 61.7%

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

   if -2.9999999999999998e60 < y < 9.99999999999999944e71

   1. Initial program 95.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 34.2

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

   5. Applied rewrites 34.2%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+60} \lor \neg \left(y \leq 10^{+72}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. distribute-lft-out N/A

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

   18. lower-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 95.7

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in y around inf

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

   1. lower-*.f64 29.9

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

7. Applied rewrites 29.9%

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

8. Final simplification 29.9%

   \[\leadsto y \cdot z \]
9. Add Preprocessing

Developer Target 1: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* z (+ (* b a) y)) (+ x (* t a)))))
   (if (< z -11820553527347888000.0)
     t_1
     (if (< z 4.7589743188364287e-122)
       (+ (* (+ (* b z) t) a) (+ (* z y) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * ((b * a) + y)) + (x + (t * a))
    if (z < (-11820553527347888000.0d0)) then
        tmp = t_1
    else if (z < 4.7589743188364287d-122) then
        tmp = (((b * z) + t) * a) + ((z * y) + x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z * ((b * a) + y)) + (x + (t * a))
	tmp = 0
	if z < -11820553527347888000.0:
		tmp = t_1
	elif z < 4.7589743188364287e-122:
		tmp = (((b * z) + t) * a) + ((z * y) + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * Float64(Float64(b * a) + y)) + Float64(x + Float64(t * a)))
	tmp = 0.0
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = Float64(Float64(Float64(Float64(b * z) + t) * a) + Float64(Float64(z * y) + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z * ((b * a) + y)) + (x + (t * a));
	tmp = 0.0;
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(N[(b * a), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -11820553527347888000.0], t$95$1, If[Less[z, 4.7589743188364287e-122], N[(N[(N[(N[(b * z), $MachinePrecision] + t), $MachinePrecision] * a), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024299 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -11820553527347888000) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 47589743188364287/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a))))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))