Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.5% → 95.5%

Time: 9.0s

Alternatives: 10

Speedup: 1.6×

• 32.6% 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: 92.5% 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: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, z, \mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 2e-121)
   (fma (* b a) z (fma a t (fma z y x)))
   (fma (fma z b 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-121) {
		tmp = fma((b * a), z, fma(a, t, fma(z, y, x)));
	} else {
		tmp = fma(fma(z, b, t), a, fma(y, z, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 2e-121)
		tmp = fma(Float64(b * a), z, fma(a, t, fma(z, y, x)));
	else
		tmp = fma(fma(z, b, t), a, fma(y, z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 2e-121], N[(N[(b * a), $MachinePrecision] * z + N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * b + t), $MachinePrecision] * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2e-121

   1. Initial program 95.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 97.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 97.8

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if 2e-121 < a

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 86.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.9

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 86.9

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

   4. Applied rewrites 86.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.9

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 99.9

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 2: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, y \cdot z\right)\\ \mathbf{if}\;a \leq -6.2 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (fma z b t) a (* y z))))
   (if (<= a -6.2e+35) t_1 (if (<= a 4e+14) (fma z y (fma t a x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(fma(z, b, t), a, (y * z));
	double tmp;
	if (a <= -6.2e+35) {
		tmp = t_1;
	} else if (a <= 4e+14) {
		tmp = fma(z, y, fma(t, a, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(fma(z, b, t), a, Float64(y * z))
	tmp = 0.0
	if (a <= -6.2e+35)
		tmp = t_1;
	elseif (a <= 4e+14)
		tmp = fma(z, y, fma(t, a, x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * b + t), $MachinePrecision] * a + N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.2e+35], t$95$1, If[LessEqual[a, 4e+14], N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.19999999999999973e35 or 4e14 < a

   1. Initial program 86.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 86.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 87.8

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.1

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 99.1

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 99.1

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 92.4

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

   9. Applied rewrites 92.4%

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

   if -6.19999999999999973e35 < a < 4e14

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + \left(a \cdot t + y \cdot z\right)} \]
   4. 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 83.8

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

   5. Applied rewrites 83.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, y \cdot z\right)\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -7.5e+49)
   (* (fma b a y) z)
   (if (<= b 3.5e+80) (fma z y (fma t a x)) (* (fma b z t) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -7.5e+49) {
		tmp = fma(b, a, y) * z;
	} else if (b <= 3.5e+80) {
		tmp = fma(z, y, fma(t, a, x));
	} else {
		tmp = fma(b, z, t) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -7.5e+49)
		tmp = Float64(fma(b, a, y) * z);
	elseif (b <= 3.5e+80)
		tmp = fma(z, y, fma(t, a, x));
	else
		tmp = Float64(fma(b, z, t) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -7.5e+49], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 3.5e+80], N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.4999999999999995e49

   1. Initial program 94.5%

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

   3. Taylor expanded in 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 77.7

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

   5. Applied rewrites 77.7%

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

   if -7.4999999999999995e49 < b < 3.49999999999999994e80

   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. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + \left(a \cdot t + y \cdot z\right)} \]
   4. 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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 3.49999999999999994e80 < b

   1. Initial program 95.8%

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

   3. Taylor expanded in 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 72.2

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

   5. Applied rewrites 72.2%

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

Alternative 4: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{if}\;a \leq -22000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b z t) a)))
   (if (<= a -22000000.0) t_1 (if (<= a 6.8e+28) (fma z y x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, z, t) * a;
	double tmp;
	if (a <= -22000000.0) {
		tmp = t_1;
	} else if (a <= 6.8e+28) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(b, z, t) * a)
	tmp = 0.0
	if (a <= -22000000.0)
		tmp = t_1;
	elseif (a <= 6.8e+28)
		tmp = fma(z, y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -22000000.0], t$95$1, If[LessEqual[a, 6.8e+28], N[(z * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2e7 or 6.8e28 < a

   1. Initial program 87.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

   if -2.2e7 < a < 6.8e28

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 78.2

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

   5. Applied rewrites 78.2%

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

Alternative 5: 73.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.22 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -5.4e-53) t_1 (if (<= z 2.22e-7) (fma t a x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, a, y) * z;
	double tmp;
	if (z <= -5.4e-53) {
		tmp = t_1;
	} else if (z <= 2.22e-7) {
		tmp = fma(t, a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(b, a, y) * z)
	tmp = 0.0
	if (z <= -5.4e-53)
		tmp = t_1;
	elseif (z <= 2.22e-7)
		tmp = fma(t, a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -5.4e-53], t$95$1, If[LessEqual[z, 2.22e-7], N[(t * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.3999999999999998e-53 or 2.22e-7 < z

   1. Initial program 89.2%

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

   3. Taylor expanded in 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 73.8

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

   5. Applied rewrites 73.8%

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

   if -5.3999999999999998e-53 < z < 2.22e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 73.7

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

   5. Applied rewrites 73.7%

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

Alternative 6: 60.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.4e+209) (fma t a x) (if (<= t 6.2e-30) (fma z y x) (fma t a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.4e+209) {
		tmp = fma(t, a, x);
	} else if (t <= 6.2e-30) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.4e+209)
		tmp = fma(t, a, x);
	elseif (t <= 6.2e-30)
		tmp = fma(z, y, x);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.4e+209], N[(t * a + x), $MachinePrecision], If[LessEqual[t, 6.2e-30], N[(z * y + x), $MachinePrecision], N[(t * a + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.39999999999999996e209 or 6.19999999999999982e-30 < t

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -2.39999999999999996e209 < t < 6.19999999999999982e-30

   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. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 63.4

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

   5. Applied rewrites 63.4%

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

Alternative 7: 57.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+69}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.4e+69) (* y z) (if (<= y 3.3e+139) (fma t a x) (* y z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.4e+69) {
		tmp = y * z;
	} else if (y <= 3.3e+139) {
		tmp = fma(t, a, x);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.4e+69)
		tmp = Float64(y * z);
	elseif (y <= 3.3e+139)
		tmp = fma(t, a, x);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.4e+69], N[(y * z), $MachinePrecision], If[LessEqual[y, 3.3e+139], N[(t * a + x), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.39999999999999991e69 or 3.3000000000000002e139 < y

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

   if -1.39999999999999991e69 < y < 3.3000000000000002e139

   1. Initial program 93.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 59.8

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

   5. Applied rewrites 59.8%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+69}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.8%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 94.0

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 94.4

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 94.4

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

4. Applied rewrites 94.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+r+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-*.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 96.2

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

   13. lift-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 96.2

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

   16. lift-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 96.2

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

6. Applied rewrites 96.2%

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

Alternative 9: 36.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+209}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.4e+209) (* t a) (if (<= t 6.2e-30) (* y z) (* t a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.4e+209) {
		tmp = t * a;
	} else if (t <= 6.2e-30) {
		tmp = y * z;
	} else {
		tmp = t * a;
	}
	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 (t <= (-2.4d+209)) then
        tmp = t * a
    else if (t <= 6.2d-30) then
        tmp = y * z
    else
        tmp = t * a
    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 (t <= -2.4e+209) {
		tmp = t * a;
	} else if (t <= 6.2e-30) {
		tmp = y * z;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -2.4e+209:
		tmp = t * a
	elif t <= 6.2e-30:
		tmp = y * z
	else:
		tmp = t * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.4e+209)
		tmp = Float64(t * a);
	elseif (t <= 6.2e-30)
		tmp = Float64(y * z);
	else
		tmp = Float64(t * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -2.4e+209)
		tmp = t * a;
	elseif (t <= 6.2e-30)
		tmp = y * z;
	else
		tmp = t * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.4e+209], N[(t * a), $MachinePrecision], If[LessEqual[t, 6.2e-30], N[(y * z), $MachinePrecision], N[(t * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.39999999999999996e209 or 6.19999999999999982e-30 < t

   1. Initial program 92.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   if -2.39999999999999996e209 < t < 6.19999999999999982e-30

   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. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+209}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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 = t * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 25.2

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

5. Applied rewrites 25.2%

   \[\leadsto \color{blue}{t \cdot a} \]
6. Add Preprocessing

Developer Target 1: 97.5% 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 2024235 
(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)))