Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.3% → 97.4%

Time: 9.6s

Alternatives: 9

Speedup: 0.5×

• 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: 92.3% 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: 97.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * N[(a * z), $MachinePrecision]), $MachinePrecision] + N[(N[(a * t), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(a * N[(N[(b * z + t), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision] * y), $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)) < +inf.0

   1. Initial program 98.4%

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

   if +inf.0 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 0.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}}}} \]

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 52.9

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

   7. Applied rewrites 52.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 100.0%

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

4. Final simplification 98.5%

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

Alternative 2: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a \cdot z\right) + \left(a \cdot t + \left(z \cdot y + x\right)\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(b, z, t\right)}{y}, z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* b (* a z)) (+ (* a t) (+ (* z y) x)))))
   (if (<= t_1 -4e+217)
     (fma (fma b z t) a (* z y))
     (if (<= t_1 1e+308)
       (fma z y (fma t a x))
       (* (fma a (/ (fma b z t) y) z) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * (a * z)) + ((a * t) + ((z * y) + x));
	double tmp;
	if (t_1 <= -4e+217) {
		tmp = fma(fma(b, z, t), a, (z * y));
	} else if (t_1 <= 1e+308) {
		tmp = fma(z, y, fma(t, a, x));
	} else {
		tmp = fma(a, (fma(b, z, t) / y), z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * Float64(a * z)) + Float64(Float64(a * t) + Float64(Float64(z * y) + x)))
	tmp = 0.0
	if (t_1 <= -4e+217)
		tmp = fma(fma(b, z, t), a, Float64(z * y));
	elseif (t_1 <= 1e+308)
		tmp = fma(z, y, fma(t, a, x));
	else
		tmp = Float64(fma(a, Float64(fma(b, z, t) / y), z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * N[(a * z), $MachinePrecision]), $MachinePrecision] + N[(N[(a * t), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+217], N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(b * z + t), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < -3.99999999999999984e217

   1. Initial program 95.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}}}} \]

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if -3.99999999999999984e217 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < 1e308

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 85.1

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

   5. Applied rewrites 85.1%

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

   if 1e308 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 61.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}}}} \]

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 83.0

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 98.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a \cdot z\right) + \left(a \cdot t + \left(z \cdot y + x\right)\right) \leq -4 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{elif}\;b \cdot \left(a \cdot z\right) + \left(a \cdot t + \left(z \cdot y + x\right)\right) \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(b, z, t\right)}{y}, z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.32e+67)
   (* (fma b a y) z)
   (if (<= b 2.45e+108) (fma z y (fma t a x)) (fma (fma b z t) a (* z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.32e+67) {
		tmp = fma(b, a, y) * z;
	} else if (b <= 2.45e+108) {
		tmp = fma(z, y, fma(t, a, x));
	} else {
		tmp = fma(fma(b, z, t), a, (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.32e+67)
		tmp = Float64(fma(b, a, y) * z);
	elseif (b <= 2.45e+108)
		tmp = fma(z, y, fma(t, a, x));
	else
		tmp = fma(fma(b, z, t), a, Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.32e+67], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 2.45e+108], N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.3200000000000001e67

   1. Initial program 90.6%

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

   3. 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 74.2

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

   5. Applied rewrites 74.2%

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

   if -1.3200000000000001e67 < b < 2.45000000000000007e108

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 89.9

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

   5. Applied rewrites 89.9%

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

   if 2.45000000000000007e108 < b

   1. Initial program 92.1%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(\left(x + y \cdot z\right) + t \cdot a\right)}^{3} + {\left(\left(a \cdot z\right) \cdot b\right)}^{3}}{\left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\left(\left(a \cdot z\right) \cdot b\right) \cdot \left(\left(a \cdot z\right) \cdot b\right) - \left(\left(x + y \cdot z\right) + t \cdot a\right) \cdot \left(\left(a \cdot z\right) \cdot b\right)\right)}}}} \]

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 88.2

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

   7. Applied rewrites 88.2%

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

Alternative 4: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -9.5e+178) t_1 (if (<= z 1.9e+112) (fma z y (fma t a x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, a, y) * z;
	double tmp;
	if (z <= -9.5e+178) {
		tmp = t_1;
	} else if (z <= 1.9e+112) {
		tmp = fma(z, y, fma(t, a, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(b, a, y) * z)
	tmp = 0.0
	if (z <= -9.5e+178)
		tmp = t_1;
	elseif (z <= 1.9e+112)
		tmp = fma(z, y, fma(t, a, x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -9.5e+178], t$95$1, If[LessEqual[z, 1.9e+112], N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.5e178 or 1.90000000000000004e112 < z

   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 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 88.0

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

   5. Applied rewrites 88.0%

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

   if -9.5e178 < z < 1.90000000000000004e112

   1. Initial program 95.8%

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

   3. Taylor expanded in 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 84.2

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

   5. Applied rewrites 84.2%

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

Alternative 5: 74.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -4700000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -4700000.0) t_1 (if (<= z 3.8e-31) (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 <= -4700000.0) {
		tmp = t_1;
	} else if (z <= 3.8e-31) {
		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 <= -4700000.0)
		tmp = t_1;
	elseif (z <= 3.8e-31)
		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, -4700000.0], t$95$1, If[LessEqual[z, 3.8e-31], N[(t * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.7e6 or 3.8e-31 < z

   1. Initial program 84.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if -4.7e6 < z < 3.8e-31

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 76.0

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

   5. Applied rewrites 76.0%

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

Alternative 6: 64.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;y \leq 600000000000:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3e+41)
   (fma z y x)
   (if (<= y 600000000000.0) (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 <= -3e+41) {
		tmp = fma(z, y, x);
	} else if (y <= 600000000000.0) {
		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 <= -3e+41)
		tmp = fma(z, y, x);
	elseif (y <= 600000000000.0)
		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, -3e+41], N[(z * y + x), $MachinePrecision], If[LessEqual[y, 600000000000.0], N[(t * a + x), $MachinePrecision], N[(z * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9999999999999998e41 or 6e11 < y

   1. Initial program 89.7%

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

   3. Taylor expanded in a around 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 68.6

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

   5. Applied rewrites 68.6%

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

   if -2.9999999999999998e41 < y < 6e11

   1. Initial program 93.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 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 7: 58.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+45}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.2e+45) (* z y) (if (<= y 2.2e+160) (fma t a x) (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.2e+45) {
		tmp = z * y;
	} else if (y <= 2.2e+160) {
		tmp = fma(t, a, x);
	} else {
		tmp = z * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.2e+45)
		tmp = Float64(z * y);
	elseif (y <= 2.2e+160)
		tmp = fma(t, a, x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.2e+45], N[(z * y), $MachinePrecision], If[LessEqual[y, 2.2e+160], N[(t * a + x), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999999e45 or 2.19999999999999992e160 < y

   1. Initial program 88.8%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 57.7

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

   5. Applied rewrites 57.7%

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

   if -4.1999999999999999e45 < y < 2.19999999999999992e160

   1. Initial program 93.3%

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

   3. Taylor expanded in 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 61.3

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

   5. Applied rewrites 61.3%

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

Alternative 8: 39.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+41}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3e+41) (* z y) (if (<= y 0.00045) (* a t) (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3e+41) {
		tmp = z * y;
	} else if (y <= 0.00045) {
		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 <= (-3d+41)) then
        tmp = z * y
    else if (y <= 0.00045d0) 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 <= -3e+41) {
		tmp = z * y;
	} else if (y <= 0.00045) {
		tmp = a * t;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3e+41:
		tmp = z * y
	elif y <= 0.00045:
		tmp = a * t
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3e+41)
		tmp = Float64(z * y);
	elseif (y <= 0.00045)
		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 <= -3e+41)
		tmp = z * y;
	elseif (y <= 0.00045)
		tmp = a * t;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3e+41], N[(z * y), $MachinePrecision], If[LessEqual[y, 0.00045], N[(a * t), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9999999999999998e41 or 4.4999999999999999e-4 < y

   1. Initial program 90.0%

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

   3. Taylor expanded in 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 49.9

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

   5. Applied rewrites 49.9%

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

   if -2.9999999999999998e41 < y < 4.4999999999999999e-4

   1. Initial program 93.3%

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

   3. Taylor expanded in 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 34.2

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

   5. Applied rewrites 34.2%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+41}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.0% 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 91.9%

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

3. Taylor expanded in t around 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 27.3

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

5. Applied rewrites 27.3%

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

6. Final simplification 27.3%

   \[\leadsto a \cdot t \]
7. 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 2024248 
(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)))