Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.3% → 97.5%

Time: 8.5s

Alternatives: 14

Speedup: 1.4×

• 32.5% 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: 92.3% 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: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{if}\;t\_1 \leq 10^{+291}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x + a \cdot \mathsf{fma}\left(b, z, t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b))))
   (if (<= t_1 1e+291) t_1 (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 t_1 = ((x + (y * z)) + (t * a)) + ((a * z) * b);
	double tmp;
	if (t_1 <= 1e+291) {
		tmp = t_1;
	} else {
		tmp = fma(z, y, (x + (a * fma(b, z, t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b))
	tmp = 0.0
	if (t_1 <= 1e+291)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+291], t$95$1, N[(z * y + N[(x + N[(a * N[(b * z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < 9.9999999999999996e290

   1. Initial program 99.5%

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

   if 9.9999999999999996e290 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 71.7%

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

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

Alternative 2: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(b \cdot a, z, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.9e-12)
   (fma (fma b a y) z x)
   (if (<= z 1.28e-123) (fma (fma b z t) a x) (fma z y (fma (* b a) z x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.9e-12) {
		tmp = fma(fma(b, a, y), z, x);
	} else if (z <= 1.28e-123) {
		tmp = fma(fma(b, z, t), a, x);
	} else {
		tmp = fma(z, y, fma((b * a), z, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.9e-12)
		tmp = fma(fma(b, a, y), z, x);
	elseif (z <= 1.28e-123)
		tmp = fma(fma(b, z, t), a, x);
	else
		tmp = fma(z, y, fma(Float64(b * a), z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.9e-12], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 1.28e-123], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], N[(z * y + N[(N[(b * a), $MachinePrecision] * z + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.89999999999999972e-12

   1. Initial program 94.6%

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

   3. Taylor expanded in 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 98.2

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

   5. Applied rewrites 98.2%

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

   if -4.89999999999999972e-12 < z < 1.28000000000000002e-123

   1. Initial program 99.1%

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

   3. Taylor expanded in 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 91.0

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

   5. Applied rewrites 91.0%

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

   if 1.28000000000000002e-123 < z

   1. Initial program 88.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.7

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

   7. Applied rewrites 86.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(b \cdot a, z, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-12} \lor \neg \left(z \leq 1.28 \cdot 10^{-123}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.9e-12) (not (<= z 1.28e-123)))
   (fma (fma b a y) z x)
   (fma (fma b z t) a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.9e-12) || !(z <= 1.28e-123)) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(fma(b, z, t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -4.9e-12) || !(z <= 1.28e-123))
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(fma(b, z, t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -4.9e-12], N[Not[LessEqual[z, 1.28e-123]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.89999999999999972e-12 or 1.28000000000000002e-123 < z

   1. Initial program 91.0%

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

   3. Taylor expanded in 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 91.1

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

   5. Applied rewrites 91.1%

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

   if -4.89999999999999972e-12 < z < 1.28000000000000002e-123

   1. Initial program 99.1%

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

   3. Taylor expanded in 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 91.0

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

   5. Applied rewrites 91.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-12} \lor \neg \left(z \leq 1.28 \cdot 10^{-123}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-44} \lor \neg \left(z \leq 9.5 \cdot 10^{+117}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.1e-44) (not (<= z 9.5e+117)))
   (fma (fma b a y) z x)
   (fma t a (fma y z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.1e-44) || !(z <= 9.5e+117)) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(t, a, fma(y, z, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3.1e-44) || !(z <= 9.5e+117))
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(t, a, fma(y, z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3.1e-44], N[Not[LessEqual[z, 9.5e+117]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(t * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.09999999999999984e-44 or 9.50000000000000041e117 < z

   1. Initial program 88.7%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -3.09999999999999984e-44 < z < 9.50000000000000041e117

   1. Initial program 98.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 84.2

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

   10. Applied rewrites 84.2%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-44} \lor \neg \left(z \leq 9.5 \cdot 10^{+117}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{+193} \lor \neg \left(a \leq 3.9 \cdot 10^{+75}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -2.15e+193) (not (<= a 3.9e+75)))
   (* (fma b z t) a)
   (fma t a (fma y z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -2.15e+193) || !(a <= 3.9e+75)) {
		tmp = fma(b, z, t) * a;
	} else {
		tmp = fma(t, a, fma(y, z, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -2.15e+193) || !(a <= 3.9e+75))
		tmp = Float64(fma(b, z, t) * a);
	else
		tmp = fma(t, a, fma(y, z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -2.15e+193], N[Not[LessEqual[a, 3.9e+75]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], N[(t * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.1500000000000001e193 or 3.90000000000000038e75 < a

   1. Initial program 85.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 89.8

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

   5. Applied rewrites 89.8%

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

   if -2.1500000000000001e193 < a < 3.90000000000000038e75

   1. Initial program 97.8%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 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 t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 83.7

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

   10. Applied rewrites 83.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{+193} \lor \neg \left(a \leq 3.9 \cdot 10^{+75}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-40} \lor \neg \left(a \leq 1.3 \cdot 10^{-87}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -5e-40) (not (<= a 1.3e-87))) (* (fma b z t) a) (fma y z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -5e-40) || !(a <= 1.3e-87)) {
		tmp = fma(b, z, t) * a;
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -5e-40) || !(a <= 1.3e-87))
		tmp = Float64(fma(b, z, t) * a);
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -5e-40], N[Not[LessEqual[a, 1.3e-87]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], N[(y * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.99999999999999965e-40 or 1.30000000000000001e-87 < a

   1. Initial program 91.1%

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

   3. Taylor expanded in 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 73.5

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

   5. Applied rewrites 73.5%

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

   if -4.99999999999999965e-40 < a < 1.30000000000000001e-87

   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 94.0

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 92.9

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

   7. Applied rewrites 92.9%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 84.1

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

   10. Applied rewrites 84.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-40} \lor \neg \left(a \leq 1.3 \cdot 10^{-87}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-45} \lor \neg \left(z \leq 1.05 \cdot 10^{-105}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.7e-45) (not (<= z 1.05e-105)))
   (* (fma b a y) z)
   (fma t a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.7e-45) || !(z <= 1.05e-105)) {
		tmp = fma(b, a, y) * z;
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3.7e-45) || !(z <= 1.05e-105))
		tmp = Float64(fma(b, a, y) * z);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3.7e-45], N[Not[LessEqual[z, 1.05e-105]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], N[(t * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7e-45 or 1.05e-105 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in 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 74.4

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

   5. Applied rewrites 74.4%

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

   if -3.7e-45 < z < 1.05e-105

   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 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 t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 58.1

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

   7. Applied rewrites 58.1%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 78.4

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

   10. Applied rewrites 78.4%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-45} \lor \neg \left(z \leq 1.05 \cdot 10^{-105}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -5.2e-40)
   (fma t a x)
   (if (<= a 3.6e+75) (fma y z x) (* (* z b) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -5.2e-40) {
		tmp = fma(t, a, x);
	} else if (a <= 3.6e+75) {
		tmp = fma(y, z, x);
	} else {
		tmp = (z * b) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -5.2e-40)
		tmp = fma(t, a, x);
	elseif (a <= 3.6e+75)
		tmp = fma(y, z, x);
	else
		tmp = Float64(Float64(z * b) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -5.2e-40], N[(t * a + x), $MachinePrecision], If[LessEqual[a, 3.6e+75], N[(y * z + x), $MachinePrecision], N[(N[(z * b), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.2000000000000003e-40

   1. Initial program 93.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 63.6

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

   10. Applied rewrites 63.6%

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

   if -5.2000000000000003e-40 < a < 3.6e75

   1. Initial program 97.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing
   3. 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 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 88.1

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

   7. Applied rewrites 88.1%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 74.2

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

   10. Applied rewrites 74.2%

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

   if 3.6e75 < a

   1. Initial program 87.1%

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

   3. Taylor expanded in 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 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 71.0%

      \[\leadsto \left(z \cdot b\right) \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -5.2e-40)
   (fma t a x)
   (if (<= a 3.9e+75) (fma y z x) (* (* b a) z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -5.2e-40) {
		tmp = fma(t, a, x);
	} else if (a <= 3.9e+75) {
		tmp = fma(y, z, x);
	} else {
		tmp = (b * a) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -5.2e-40)
		tmp = fma(t, a, x);
	elseif (a <= 3.9e+75)
		tmp = fma(y, z, x);
	else
		tmp = Float64(Float64(b * a) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -5.2e-40], N[(t * a + x), $MachinePrecision], If[LessEqual[a, 3.9e+75], N[(y * z + x), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.2000000000000003e-40

   1. Initial program 93.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 63.6

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

   10. Applied rewrites 63.6%

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

   if -5.2000000000000003e-40 < a < 3.90000000000000038e75

   1. Initial program 97.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing
   3. 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 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 88.1

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

   7. Applied rewrites 88.1%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 74.2

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

   10. Applied rewrites 74.2%

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

   if 3.90000000000000038e75 < a

   1. Initial program 87.1%

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

   3. Taylor expanded in 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 65.3

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.9%

      \[\leadsto \left(b \cdot a\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
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 97.3

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

4. Applied rewrites 97.3%

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

Alternative 11: 63.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-20} \lor \neg \left(t \leq 2.05 \cdot 10^{+86}\right):\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2.6e-20) (not (<= t 2.05e+86))) (fma t a x) (fma y z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2.6e-20) || !(t <= 2.05e+86)) {
		tmp = fma(t, a, x);
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -2.6e-20) || !(t <= 2.05e+86))
		tmp = fma(t, a, x);
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2.6e-20], N[Not[LessEqual[t, 2.05e+86]], $MachinePrecision]], N[(t * a + x), $MachinePrecision], N[(y * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.59999999999999995e-20 or 2.05e86 < t

   1. Initial program 92.8%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 60.9

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

   7. Applied rewrites 60.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 70.4

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

   10. Applied rewrites 70.4%

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

   if -2.59999999999999995e-20 < t < 2.05e86

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 61.2

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

   10. Applied rewrites 61.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-20} \lor \neg \left(t \leq 2.05 \cdot 10^{+86}\right):\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+116} \lor \neg \left(y \leq 7 \cdot 10^{+159}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.5e+116) (not (<= y 7e+159))) (* y z) (fma t a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.5e+116) || !(y <= 7e+159)) {
		tmp = y * z;
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7.5e+116) || !(y <= 7e+159))
		tmp = Float64(y * z);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7.5e+116], N[Not[LessEqual[y, 7e+159]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(t * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5e116 or 6.9999999999999999e159 < y

   1. Initial program 95.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if -7.5e116 < y < 6.9999999999999999e159

   1. Initial program 94.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.1

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

   7. Applied rewrites 75.1%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 57.4

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

   10. Applied rewrites 57.4%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+116} \lor \neg \left(y \leq 7 \cdot 10^{+159}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-20} \lor \neg \left(t \leq 2.05 \cdot 10^{+86}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2.6e-20) (not (<= t 2.05e+86))) (* a t) (* y z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2.6e-20) || !(t <= 2.05e+86)) {
		tmp = a * t;
	} 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 ((t <= (-2.6d-20)) .or. (.not. (t <= 2.05d+86))) then
        tmp = a * t
    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 ((t <= -2.6e-20) || !(t <= 2.05e+86)) {
		tmp = a * t;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -2.6e-20) or not (t <= 2.05e+86):
		tmp = a * t
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -2.6e-20) || !(t <= 2.05e+86))
		tmp = Float64(a * t);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -2.6e-20) || ~((t <= 2.05e+86)))
		tmp = a * t;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2.6e-20], N[Not[LessEqual[t, 2.05e+86]], $MachinePrecision]], N[(a * t), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.59999999999999995e-20 or 2.05e86 < t

   1. Initial program 92.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 49.4

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

   5. Applied rewrites 49.4%

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

   if -2.59999999999999995e-20 < t < 2.05e86

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 31.6

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

   7. Applied rewrites 31.6%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-20} \lor \neg \left(t \leq 2.05 \cdot 10^{+86}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
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 97.3

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in y around inf

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

   1. lower-*.f64 26.2

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

7. Applied rewrites 26.2%

   \[\leadsto \color{blue}{y \cdot z} \]
8. Add Preprocessing

Developer Target 1: 97.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024313 
(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)))