Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.5% → 95.2%

Time: 8.1s

Alternatives: 12

Speedup: 1.6×

• 31.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\left(a \cdot z\right) \cdot b + \left(a \cdot t + \left(z \cdot y + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5e+138)
   (+ (* (* a z) b) (+ (* a t) (+ (* z y) x)))
   (fma a (fma b z t) (fma z y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5e+138) {
		tmp = ((a * z) * b) + ((a * t) + ((z * y) + x));
	} else {
		tmp = fma(a, fma(b, z, t), fma(z, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5e+138)
		tmp = Float64(Float64(Float64(a * z) * b) + Float64(Float64(a * t) + Float64(Float64(z * y) + x)));
	else
		tmp = fma(a, fma(b, z, t), fma(z, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5e+138], N[(N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision] + N[(N[(a * t), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(b * z + t), $MachinePrecision] + N[(z * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000000016e138

   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

   if -5.00000000000000016e138 < b

   1. Initial program 90.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 96.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 96.9

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

   4. Applied rewrites 96.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\left(a \cdot z\right) \cdot b + \left(a \cdot t + \left(z \cdot y + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.4e-37)
   (fma t a (fma z y x))
   (if (<= y 7.5e+83) (fma (fma b z t) a x) (fma z y (fma t a x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.4e-37) {
		tmp = fma(t, a, fma(z, y, x));
	} else if (y <= 7.5e+83) {
		tmp = fma(fma(b, z, t), a, x);
	} else {
		tmp = fma(z, y, fma(t, a, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.4e-37)
		tmp = fma(t, a, fma(z, y, x));
	elseif (y <= 7.5e+83)
		tmp = fma(fma(b, z, t), a, x);
	else
		tmp = fma(z, y, fma(t, a, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.4e-37], N[(t * a + N[(z * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+83], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4000000000000001e-37

   1. Initial program 91.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 88.3

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

   5. Applied rewrites 88.3%

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

   7. Applied rewrites 89.8%

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

   if -1.4000000000000001e-37 < y < 7.49999999999999989e83

   1. Initial program 91.5%

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

   3. Taylor expanded in y around 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 91.2

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

   5. Applied rewrites 91.2%

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

   if 7.49999999999999989e83 < y

   1. Initial program 91.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 3: 73.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{if}\;a \leq -5.8 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b z t) a)))
   (if (<= a -5.8e+77) t_1 (if (<= a 7e-31) (fma z y x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, z, t) * a;
	double tmp;
	if (a <= -5.8e+77) {
		tmp = t_1;
	} else if (a <= 7e-31) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(b, z, t) * a)
	tmp = 0.0
	if (a <= -5.8e+77)
		tmp = t_1;
	elseif (a <= 7e-31)
		tmp = fma(z, y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -5.8e+77], t$95$1, If[LessEqual[a, 7e-31], N[(z * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8000000000000003e77 or 6.99999999999999971e-31 < a

   1. Initial program 85.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if -5.8000000000000003e77 < a < 6.99999999999999971e-31

   1. Initial program 96.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 78.9

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

   5. Applied rewrites 78.9%

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

Alternative 4: 74.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -1.95e-22) t_1 (if (<= z 2.8e+18) (fma t a x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, a, y) * z;
	double tmp;
	if (z <= -1.95e-22) {
		tmp = t_1;
	} else if (z <= 2.8e+18) {
		tmp = fma(t, a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(b, a, y) * z)
	tmp = 0.0
	if (z <= -1.95e-22)
		tmp = t_1;
	elseif (z <= 2.8e+18)
		tmp = fma(t, a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.95e-22], t$95$1, If[LessEqual[z, 2.8e+18], N[(t * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.94999999999999999e-22 or 2.8e18 < z

   1. Initial program 86.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -1.94999999999999999e-22 < z < 2.8e18

   1. Initial program 97.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 79.3

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

   5. Applied rewrites 79.3%

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

Alternative 5: 61.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+81}:\\ \;\;\;\;\left(z \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -4e+81) (* (* z b) a) (if (<= a 1.6e-57) (fma z y x) (fma t a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -4e+81) {
		tmp = (z * b) * a;
	} else if (a <= 1.6e-57) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -4e+81)
		tmp = Float64(Float64(z * b) * a);
	elseif (a <= 1.6e-57)
		tmp = fma(z, y, x);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -4e+81], N[(N[(z * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 1.6e-57], N[(z * y + x), $MachinePrecision], N[(t * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.99999999999999969e81

   1. Initial program 80.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

   if -3.99999999999999969e81 < a < 1.6e-57

   1. Initial program 96.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if 1.6e-57 < a

   1. Initial program 89.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 58.5

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

   5. Applied rewrites 58.5%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+81}:\\ \;\;\;\;\left(z \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.65 \cdot 10^{+81}:\\ \;\;\;\;\left(a \cdot b\right) \cdot z\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -3.65e+81)
   (* (* a b) z)
   (if (<= a 1.6e-57) (fma z y x) (fma t a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.65e+81) {
		tmp = (a * b) * z;
	} else if (a <= 1.6e-57) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -3.65e+81)
		tmp = Float64(Float64(a * b) * z);
	elseif (a <= 1.6e-57)
		tmp = fma(z, y, x);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -3.65e+81], N[(N[(a * b), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[a, 1.6e-57], N[(z * y + x), $MachinePrecision], N[(t * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.6499999999999998e81

   1. Initial program 80.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

   7. Applied rewrites 52.4%

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

   if -3.6499999999999998e81 < a < 1.6e-57

   1. Initial program 96.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if 1.6e-57 < a

   1. Initial program 89.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 58.5

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

   5. Applied rewrites 58.5%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.65 \cdot 10^{+81}:\\ \;\;\;\;\left(a \cdot b\right) \cdot z\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.9e+24) (fma z y x) (if (<= y 2.2e+82) (fma t a x) (fma z y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.9e+24) {
		tmp = fma(z, y, x);
	} else if (y <= 2.2e+82) {
		tmp = fma(t, a, x);
	} else {
		tmp = fma(z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.9e+24)
		tmp = fma(z, y, x);
	elseif (y <= 2.2e+82)
		tmp = fma(t, a, x);
	else
		tmp = fma(z, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.9e+24], N[(z * y + x), $MachinePrecision], If[LessEqual[y, 2.2e+82], N[(t * a + x), $MachinePrecision], N[(z * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.89999999999999979e24 or 2.2000000000000001e82 < y

   1. Initial program 89.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if -2.89999999999999979e24 < y < 2.2000000000000001e82

   1. Initial program 92.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 64.0

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

   5. Applied rewrites 64.0%

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

Alternative 8: 80.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -4e+81) (* (fma b z t) a) (fma t a (fma z y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -4e+81) {
		tmp = fma(b, z, t) * a;
	} else {
		tmp = fma(t, a, fma(z, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -4e+81)
		tmp = Float64(fma(b, z, t) * a);
	else
		tmp = fma(t, a, fma(z, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -4e+81], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], N[(t * a + N[(z * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.99999999999999969e81

   1. Initial program 80.2%

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

   3. Taylor expanded in 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 86.3

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

   5. Applied rewrites 86.3%

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

   if -3.99999999999999969e81 < a

   1. Initial program 94.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 82.9%

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

Alternative 9: 94.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 95.0

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 95.0

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

4. Applied rewrites 95.0%

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

Alternative 10: 38.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+29}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.25e+29) (* z y) (if (<= y 2.3e+82) (* a t) (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.25e+29) {
		tmp = z * y;
	} else if (y <= 2.3e+82) {
		tmp = a * t;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-1.25d+29)) then
        tmp = z * y
    else if (y <= 2.3d+82) then
        tmp = a * t
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.25e+29) {
		tmp = z * y;
	} else if (y <= 2.3e+82) {
		tmp = a * t;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.25e+29:
		tmp = z * y
	elif y <= 2.3e+82:
		tmp = a * t
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.25e+29)
		tmp = Float64(z * y);
	elseif (y <= 2.3e+82)
		tmp = Float64(a * t);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.25e+29)
		tmp = z * y;
	elseif (y <= 2.3e+82)
		tmp = a * t;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.25e+29], N[(z * y), $MachinePrecision], If[LessEqual[y, 2.3e+82], N[(a * t), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e29 or 2.29999999999999988e82 < y

   1. Initial program 89.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.5

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

   5. Applied rewrites 50.5%

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

   if -1.25e29 < y < 2.29999999999999988e82

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 32.7

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

   5. Applied rewrites 32.7%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+29}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+123}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.1e+123) (* z y) (fma t a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.1e+123) {
		tmp = z * y;
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.1e+123)
		tmp = Float64(z * y);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.1e+123], N[(z * y), $MachinePrecision], N[(t * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.09999999999999996e123

   1. Initial program 86.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   if -1.09999999999999996e123 < y

   1. Initial program 92.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 58.8

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

   5. Applied rewrites 58.8%

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

Alternative 12: 28.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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 = a * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 27.1

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

5. Applied rewrites 27.1%

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

6. Final simplification 27.1%

   \[\leadsto a \cdot t \]
7. Add Preprocessing

Developer Target 1: 97.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024249 
(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)))