Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.4% → 95.6%

Time: 9.6s

Alternatives: 15

Speedup: 1.4×

• 32.0% 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: 92.4% 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: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x + a \cdot \mathsf{fma}\left(b, z, t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 5e-97)
   (+ (+ (+ 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 <= 5e-97) {
		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 <= 5e-97)
		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, 5e-97], 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 < 4.9999999999999995e-97

   1. Initial program 97.6%

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

   if 4.9999999999999995e-97 < a

   1. Initial program 82.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 2: 53.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.72 \cdot 10^{+75} \lor \neg \left(t \leq 2.9 \cdot 10^{+50} \lor \neg \left(t \leq 1.14 \cdot 10^{+108} \lor \neg \left(t \leq 7.9 \cdot 10^{+151}\right)\right)\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.72e+75)
         (not
          (or (<= t 2.9e+50)
              (not (or (<= t 1.14e+108) (not (<= t 7.9e+151)))))))
   (* a t)
   (* (fma b a y) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.72e+75) || !((t <= 2.9e+50) || !((t <= 1.14e+108) || !(t <= 7.9e+151)))) {
		tmp = a * t;
	} else {
		tmp = fma(b, a, y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.72e+75) || !((t <= 2.9e+50) || !((t <= 1.14e+108) || !(t <= 7.9e+151))))
		tmp = Float64(a * t);
	else
		tmp = Float64(fma(b, a, y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.72e+75], N[Not[Or[LessEqual[t, 2.9e+50], N[Not[Or[LessEqual[t, 1.14e+108], N[Not[LessEqual[t, 7.9e+151]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(a * t), $MachinePrecision], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.72e75 or 2.9e50 < t < 1.13999999999999994e108 or 7.9e151 < t

   1. Initial program 89.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if -1.72e75 < t < 2.9e50 or 1.13999999999999994e108 < t < 7.9e151

   1. Initial program 93.7%

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

   3. Taylor expanded in 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 60.0

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

   5. Applied rewrites 60.0%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.72 \cdot 10^{+75} \lor \neg \left(t \leq 2.9 \cdot 10^{+50} \lor \neg \left(t \leq 1.14 \cdot 10^{+108} \lor \neg \left(t \leq 7.9 \cdot 10^{+151}\right)\right)\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot b, a, x\right)\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-277}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot b, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b z t) a)))
   (if (<= a -6.6e+47)
     t_1
     (if (<= a -4.2e-15)
       (fma (* z b) a x)
       (if (<= a -4.2e-277)
         (* (fma b a y) z)
         (if (<= a 6e-98) (fma (* a b) z 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 <= -6.6e+47) {
		tmp = t_1;
	} else if (a <= -4.2e-15) {
		tmp = fma((z * b), a, x);
	} else if (a <= -4.2e-277) {
		tmp = fma(b, a, y) * z;
	} else if (a <= 6e-98) {
		tmp = fma((a * b), z, 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 <= -6.6e+47)
		tmp = t_1;
	elseif (a <= -4.2e-15)
		tmp = fma(Float64(z * b), a, x);
	elseif (a <= -4.2e-277)
		tmp = Float64(fma(b, a, y) * z);
	elseif (a <= 6e-98)
		tmp = fma(Float64(a * b), z, 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, -6.6e+47], t$95$1, If[LessEqual[a, -4.2e-15], N[(N[(z * b), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[a, -4.2e-277], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[a, 6e-98], N[(N[(a * b), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.5999999999999998e47 or 6e-98 < a

   1. Initial program 86.0%

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

   3. Taylor expanded in 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 80.5

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

   5. Applied rewrites 80.5%

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

   if -6.5999999999999998e47 < a < -4.19999999999999962e-15

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 78.0%

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

   if -4.19999999999999962e-15 < a < -4.1999999999999999e-277

   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 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 61.5

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

   5. Applied rewrites 61.5%

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

   if -4.1999999999999999e-277 < a < 6e-98

   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 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 94.9

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.0%

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

Alternative 4: 62.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a \cdot b, z, x\right)\\ t_2 := \mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{+47}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-277}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-98}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* a b) z x)) (t_2 (* (fma b z t) a)))
   (if (<= a -6.6e+47)
     t_2
     (if (<= a -8e-14)
       t_1
       (if (<= a -4.2e-277) (* (fma b a y) z) (if (<= a 6e-98) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((a * b), z, x);
	double t_2 = fma(b, z, t) * a;
	double tmp;
	if (a <= -6.6e+47) {
		tmp = t_2;
	} else if (a <= -8e-14) {
		tmp = t_1;
	} else if (a <= -4.2e-277) {
		tmp = fma(b, a, y) * z;
	} else if (a <= 6e-98) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(a * b), z, x)
	t_2 = Float64(fma(b, z, t) * a)
	tmp = 0.0
	if (a <= -6.6e+47)
		tmp = t_2;
	elseif (a <= -8e-14)
		tmp = t_1;
	elseif (a <= -4.2e-277)
		tmp = Float64(fma(b, a, y) * z);
	elseif (a <= 6e-98)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * z + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -6.6e+47], t$95$2, If[LessEqual[a, -8e-14], t$95$1, If[LessEqual[a, -4.2e-277], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[a, 6e-98], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.5999999999999998e47 or 6e-98 < a

   1. Initial program 86.0%

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

   3. Taylor expanded in 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 80.5

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

   5. Applied rewrites 80.5%

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

   if -6.5999999999999998e47 < a < -7.99999999999999999e-14 or -4.1999999999999999e-277 < a < 6e-98

   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 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 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 64.8%

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

   if -7.99999999999999999e-14 < a < -4.1999999999999999e-277

   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 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 62.4

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

   5. Applied rewrites 62.4%

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

Alternative 5: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+236}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(1, x, a \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -4.5e+236)
     t_1
     (if (<= z -6.2e-163)
       (* (fma b z t) a)
       (if (<= z 9.6e-12) (fma 1.0 x (* a t)) 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 <= -4.5e+236) {
		tmp = t_1;
	} else if (z <= -6.2e-163) {
		tmp = fma(b, z, t) * a;
	} else if (z <= 9.6e-12) {
		tmp = fma(1.0, x, (a * t));
	} 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 <= -4.5e+236)
		tmp = t_1;
	elseif (z <= -6.2e-163)
		tmp = Float64(fma(b, z, t) * a);
	elseif (z <= 9.6e-12)
		tmp = fma(1.0, x, Float64(a * t));
	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, -4.5e+236], t$95$1, If[LessEqual[z, -6.2e-163], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 9.6e-12], N[(1.0 * x + N[(a * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.50000000000000018e236 or 9.59999999999999948e-12 < z

   1. Initial program 86.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   if -4.50000000000000018e236 < z < -6.19999999999999949e-163

   1. Initial program 89.7%

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

   3. Taylor expanded in 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 69.6

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

   5. Applied rewrites 69.6%

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

   if -6.19999999999999949e-163 < z < 9.59999999999999948e-12

   1. Initial program 99.7%

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

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. times-frac N/A

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

      8. /-rgt-identity N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 82.3%

      \[\leadsto \mathsf{fma}\left(1, x, a \cdot t\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{if}\;a \leq -3.1 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-277}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-114}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b z t) a)))
   (if (<= a -3.1e-15)
     t_1
     (if (<= a -3.7e-277)
       (* (fma b a y) z)
       (if (<= a 9.5e-114) (* 1.0 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 <= -3.1e-15) {
		tmp = t_1;
	} else if (a <= -3.7e-277) {
		tmp = fma(b, a, y) * z;
	} else if (a <= 9.5e-114) {
		tmp = 1.0 * 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 <= -3.1e-15)
		tmp = t_1;
	elseif (a <= -3.7e-277)
		tmp = Float64(fma(b, a, y) * z);
	elseif (a <= 9.5e-114)
		tmp = Float64(1.0 * 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, -3.1e-15], t$95$1, If[LessEqual[a, -3.7e-277], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[a, 9.5e-114], N[(1.0 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.0999999999999999e-15 or 9.49999999999999958e-114 < a

   1. Initial program 87.7%

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

   3. Taylor expanded in 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 76.6

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

   5. Applied rewrites 76.6%

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

   if -3.0999999999999999e-15 < a < -3.69999999999999985e-277

   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 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 61.5

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

   5. Applied rewrites 61.5%

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

   if -3.69999999999999985e-277 < a < 9.49999999999999958e-114

   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. 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 91.3

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 54.8%

       \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-277}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-114}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(z, b, t\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -3.1e-15)
   (fma (fma b z t) a x)
   (if (<= a 7.8e-97) (fma (fma b a y) z x) (fma z y (* (fma z b t) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.1e-15) {
		tmp = fma(fma(b, z, t), a, x);
	} else if (a <= 7.8e-97) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(z, y, (fma(z, b, t) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -3.1e-15)
		tmp = fma(fma(b, z, t), a, x);
	elseif (a <= 7.8e-97)
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(z, y, Float64(fma(z, b, t) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -3.1e-15], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[a, 7.8e-97], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(z * y + N[(N[(z * b + t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.0999999999999999e-15

   1. Initial program 94.1%

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

   3. Taylor expanded in 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 92.5

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

   5. Applied rewrites 92.5%

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

   if -3.0999999999999999e-15 < a < 7.7999999999999997e-97

   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 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 92.2

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

   5. Applied rewrites 92.2%

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

   if 7.7999999999999997e-97 < a

   1. Initial program 82.1%

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

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

   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 92.3

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

   7. Applied rewrites 92.3%

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

Alternative 8: 39.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot z\right) \cdot b\\ \mathbf{if}\;b \leq -4.4 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-299}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+44}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a z) b)))
   (if (<= b -4.4e+59)
     t_1
     (if (<= b 4.5e-299) (* y z) (if (<= b 3.4e+44) (* a t) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * z) * b;
	double tmp;
	if (b <= -4.4e+59) {
		tmp = t_1;
	} else if (b <= 4.5e-299) {
		tmp = y * z;
	} else if (b <= 3.4e+44) {
		tmp = a * t;
	} 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 = (a * z) * b
    if (b <= (-4.4d+59)) then
        tmp = t_1
    else if (b <= 4.5d-299) then
        tmp = y * z
    else if (b <= 3.4d+44) then
        tmp = a * t
    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 = (a * z) * b;
	double tmp;
	if (b <= -4.4e+59) {
		tmp = t_1;
	} else if (b <= 4.5e-299) {
		tmp = y * z;
	} else if (b <= 3.4e+44) {
		tmp = a * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * z) * b
	tmp = 0
	if b <= -4.4e+59:
		tmp = t_1
	elif b <= 4.5e-299:
		tmp = y * z
	elif b <= 3.4e+44:
		tmp = a * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * z) * b)
	tmp = 0.0
	if (b <= -4.4e+59)
		tmp = t_1;
	elseif (b <= 4.5e-299)
		tmp = Float64(y * z);
	elseif (b <= 3.4e+44)
		tmp = Float64(a * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * z) * b;
	tmp = 0.0;
	if (b <= -4.4e+59)
		tmp = t_1;
	elseif (b <= 4.5e-299)
		tmp = y * z;
	elseif (b <= 3.4e+44)
		tmp = a * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -4.4e+59], t$95$1, If[LessEqual[b, 4.5e-299], N[(y * z), $MachinePrecision], If[LessEqual[b, 3.4e+44], N[(a * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.3999999999999999e59 or 3.4e44 < b

   1. Initial program 94.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

      \[\leadsto \left(a \cdot b\right) \cdot z \]

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 58.0%

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

   if -4.3999999999999999e59 < b < 4.50000000000000003e-299

   1. Initial program 91.8%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 41.7

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

   7. Applied rewrites 41.7%

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

   if 4.50000000000000003e-299 < b < 3.4e44

   1. Initial program 90.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+59}:\\ \;\;\;\;\left(a \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-299}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+44}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-15} \lor \neg \left(a \leq 3.6 \cdot 10^{-66}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -3.1e-15) (not (<= a 3.6e-66)))
   (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 <= -3.1e-15) || !(a <= 3.6e-66)) {
		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 <= -3.1e-15) || !(a <= 3.6e-66))
		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, -3.1e-15], N[Not[LessEqual[a, 3.6e-66]], $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 < -3.0999999999999999e-15 or 3.60000000000000012e-66 < a

   1. Initial program 86.9%

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

   3. Taylor expanded in 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 91.8

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

   5. Applied rewrites 91.8%

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

   if -3.0999999999999999e-15 < a < 3.60000000000000012e-66

   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 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 91.4

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

   5. Applied rewrites 91.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-15} \lor \neg \left(a \leq 3.6 \cdot 10^{-66}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+49} \lor \neg \left(a \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1e+49) (not (<= a 4e-12)))
   (* (fma b z t) a)
   (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 <= -1e+49) || !(a <= 4e-12)) {
		tmp = fma(b, z, t) * a;
	} 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 <= -1e+49) || !(a <= 4e-12))
		tmp = Float64(fma(b, z, t) * a);
	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, -1e+49], N[Not[LessEqual[a, 4e-12]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.99999999999999946e48 or 3.99999999999999992e-12 < a

   1. Initial program 84.4%

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

   3. Taylor expanded in 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 84.0

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

   5. Applied rewrites 84.0%

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

   if -9.99999999999999946e48 < a < 3.99999999999999992e-12

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 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 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+49} \lor \neg \left(a \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-54}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-177}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+49}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.9e-54)
   (* a t)
   (if (<= t 6.5e-177) (* 1.0 x) (if (<= t 4.5e+49) (* y z) (* a t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.9e-54) {
		tmp = a * t;
	} else if (t <= 6.5e-177) {
		tmp = 1.0 * x;
	} else if (t <= 4.5e+49) {
		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 (t <= (-1.9d-54)) then
        tmp = a * t
    else if (t <= 6.5d-177) then
        tmp = 1.0d0 * x
    else if (t <= 4.5d+49) 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 (t <= -1.9e-54) {
		tmp = a * t;
	} else if (t <= 6.5e-177) {
		tmp = 1.0 * x;
	} else if (t <= 4.5e+49) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.9e-54:
		tmp = a * t
	elif t <= 6.5e-177:
		tmp = 1.0 * x
	elif t <= 4.5e+49:
		tmp = y * z
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.9e-54)
		tmp = Float64(a * t);
	elseif (t <= 6.5e-177)
		tmp = Float64(1.0 * x);
	elseif (t <= 4.5e+49)
		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 (t <= -1.9e-54)
		tmp = a * t;
	elseif (t <= 6.5e-177)
		tmp = 1.0 * x;
	elseif (t <= 4.5e+49)
		tmp = y * z;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.9e-54], N[(a * t), $MachinePrecision], If[LessEqual[t, 6.5e-177], N[(1.0 * x), $MachinePrecision], If[LessEqual[t, 4.5e+49], N[(y * z), $MachinePrecision], N[(a * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9000000000000001e-54 or 4.49999999999999982e49 < t

   1. Initial program 90.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

   if -1.9000000000000001e-54 < t < 6.4999999999999998e-177

   1. Initial program 93.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 80.2

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

   7. Applied rewrites 80.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 41.6%

       \[\leadsto 1 \cdot x \]

   if 6.4999999999999998e-177 < t < 4.49999999999999982e49

   1. Initial program 96.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 41.4

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

   7. Applied rewrites 41.4%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-54}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-177}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+49}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.55 \cdot 10^{+33}:\\ \;\;\;\;\left(z \cdot b\right) \cdot a\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+44}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -4.55e+33) (* (* z b) a) (if (<= b 3.4e+44) (* a t) (* (* a z) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4.55e+33) {
		tmp = (z * b) * a;
	} else if (b <= 3.4e+44) {
		tmp = a * t;
	} else {
		tmp = (a * z) * b;
	}
	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 (b <= (-4.55d+33)) then
        tmp = (z * b) * a
    else if (b <= 3.4d+44) then
        tmp = a * t
    else
        tmp = (a * z) * b
    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 (b <= -4.55e+33) {
		tmp = (z * b) * a;
	} else if (b <= 3.4e+44) {
		tmp = a * t;
	} else {
		tmp = (a * z) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -4.55e+33:
		tmp = (z * b) * a
	elif b <= 3.4e+44:
		tmp = a * t
	else:
		tmp = (a * z) * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -4.55e+33)
		tmp = Float64(Float64(z * b) * a);
	elseif (b <= 3.4e+44)
		tmp = Float64(a * t);
	else
		tmp = Float64(Float64(a * z) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -4.55e+33)
		tmp = (z * b) * a;
	elseif (b <= 3.4e+44)
		tmp = a * t;
	else
		tmp = (a * z) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -4.55e+33], N[(N[(z * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[b, 3.4e+44], N[(a * t), $MachinePrecision], N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5499999999999998e33

   1. Initial program 95.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(b \cdot z\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 58.0%

      \[\leadsto \left(z \cdot b\right) \cdot a \]

   if -4.5499999999999998e33 < b < 3.4e44

   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 t around inf

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

      1. lower-*.f64 43.1

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

   5. Applied rewrites 43.1%

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

   if 3.4e44 < b

   1. Initial program 93.7%

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

   3. Taylor expanded in 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 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 53.3%

      \[\leadsto \left(a \cdot b\right) \cdot z \]

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 54.7%

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

Alternative 13: 95.7% 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 92.2%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
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 96.3

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

4. Applied rewrites 96.3%

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

Alternative 14: 39.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+97} \lor \neg \left(x \leq 2.9 \cdot 10^{-46}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= x -2.7e+97) (not (<= x 2.9e-46))) (* 1.0 x) (* y z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x <= -2.7e+97) || !(x <= 2.9e-46)) {
		tmp = 1.0 * x;
	} else {
		tmp = y * z;
	}
	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 ((x <= (-2.7d+97)) .or. (.not. (x <= 2.9d-46))) then
        tmp = 1.0d0 * x
    else
        tmp = y * z
    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 ((x <= -2.7e+97) || !(x <= 2.9e-46)) {
		tmp = 1.0 * x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x <= -2.7e+97) or not (x <= 2.9e-46):
		tmp = 1.0 * x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((x <= -2.7e+97) || !(x <= 2.9e-46))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x <= -2.7e+97) || ~((x <= 2.9e-46)))
		tmp = 1.0 * x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[x, -2.7e+97], N[Not[LessEqual[x, 2.9e-46]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999993e97 or 2.90000000000000005e-46 < x

   1. Initial program 94.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 94.7

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 51.4%

       \[\leadsto 1 \cdot x \]

   if -2.69999999999999993e97 < x < 2.90000000000000005e-46

   1. Initial program 90.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 33.5

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

   7. Applied rewrites 33.5%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+97} \lor \neg \left(x \leq 2.9 \cdot 10^{-46}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.6% 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 92.2%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
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 96.3

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

4. Applied rewrites 96.3%

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

5. Taylor expanded in y around inf

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

   1. lower-*.f64 25.4

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

7. Applied rewrites 25.4%

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

8. Final simplification 25.4%

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

Developer Target 1: 97.3% 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 2024318 
(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)))