Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.6% → 96.8%

Time: 11.0s

Alternatives: 14

Speedup: 0.5×

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.6% 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: 96.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. Taylor expanded in b around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-/l* N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 97.3%

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

Alternative 2: 96.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 90.0

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

   5. Simplified 90.0%

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

4. Final simplification 96.9%

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

Alternative 3: 55.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+221}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -2e+221)
   (* z (* a b))
   (if (<= b -3.8e+29)
     (fma a t x)
     (if (<= b 2.65e-6) (fma z y x) (* (* z a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2e+221) {
		tmp = z * (a * b);
	} else if (b <= -3.8e+29) {
		tmp = fma(a, t, x);
	} else if (b <= 2.65e-6) {
		tmp = fma(z, y, x);
	} else {
		tmp = (z * a) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2e+221)
		tmp = Float64(z * Float64(a * b));
	elseif (b <= -3.8e+29)
		tmp = fma(a, t, x);
	elseif (b <= 2.65e-6)
		tmp = fma(z, y, x);
	else
		tmp = Float64(Float64(z * a) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -2e+221], N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.8e+29], N[(a * t + x), $MachinePrecision], If[LessEqual[b, 2.65e-6], N[(z * y + x), $MachinePrecision], N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0000000000000001e221

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 76.6%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 67.4

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

   8. Simplified 67.4%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 67.7

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

   10. Applied egg-rr 67.7%

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

   if -2.0000000000000001e221 < b < -3.79999999999999971e29

   1. Initial program 94.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. accelerator-lowering-fma.f64 80.0

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

   5. Simplified 80.0%

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

   if -3.79999999999999971e29 < b < 2.65e-6

   1. Initial program 92.4%

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

   3. Taylor expanded in a around 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. accelerator-lowering-fma.f64 67.4

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

   5. Simplified 67.4%

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

   if 2.65e-6 < b

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 82.8%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 64.9

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

   8. Simplified 64.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+221}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+220}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -7.5e+220)
   (* a (* z b))
   (if (<= b -1.8e+29)
     (fma a t x)
     (if (<= b 2.65e-6) (fma z y x) (* (* z a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -7.5e+220) {
		tmp = a * (z * b);
	} else if (b <= -1.8e+29) {
		tmp = fma(a, t, x);
	} else if (b <= 2.65e-6) {
		tmp = fma(z, y, x);
	} else {
		tmp = (z * a) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -7.5e+220)
		tmp = Float64(a * Float64(z * b));
	elseif (b <= -1.8e+29)
		tmp = fma(a, t, x);
	elseif (b <= 2.65e-6)
		tmp = fma(z, y, x);
	else
		tmp = Float64(Float64(z * a) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -7.5e+220], N[(a * N[(z * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.8e+29], N[(a * t + x), $MachinePrecision], If[LessEqual[b, 2.65e-6], N[(z * y + x), $MachinePrecision], N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -7.5000000000000003e220

   1. Initial program 95.1%

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

   3. Taylor expanded in b around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 67.5

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

   5. Simplified 67.5%

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

   if -7.5000000000000003e220 < b < -1.79999999999999988e29

   1. Initial program 94.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. accelerator-lowering-fma.f64 80.0

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

   5. Simplified 80.0%

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

   if -1.79999999999999988e29 < b < 2.65e-6

   1. Initial program 92.4%

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

   3. Taylor expanded in a around 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. accelerator-lowering-fma.f64 67.4

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

   5. Simplified 67.4%

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

   if 2.65e-6 < b

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 82.8%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 64.9

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

   8. Simplified 64.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+220}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(z \cdot b\right)\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+221}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* z b))))
   (if (<= b -1.5e+221)
     t_1
     (if (<= b -1.15e+29) (fma a t x) (if (<= b 2.65e-6) (fma z y x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (z * b);
	double tmp;
	if (b <= -1.5e+221) {
		tmp = t_1;
	} else if (b <= -1.15e+29) {
		tmp = fma(a, t, x);
	} else if (b <= 2.65e-6) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(z * b))
	tmp = 0.0
	if (b <= -1.5e+221)
		tmp = t_1;
	elseif (b <= -1.15e+29)
		tmp = fma(a, t, x);
	elseif (b <= 2.65e-6)
		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[(a * N[(z * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.5e+221], t$95$1, If[LessEqual[b, -1.15e+29], N[(a * t + x), $MachinePrecision], If[LessEqual[b, 2.65e-6], N[(z * y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5000000000000001e221 or 2.65e-6 < b

   1. Initial program 94.6%

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

   3. Taylor expanded in b around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 64.5

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

   5. Simplified 64.5%

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

   if -1.5000000000000001e221 < b < -1.1500000000000001e29

   1. Initial program 94.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. accelerator-lowering-fma.f64 80.0

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

   5. Simplified 80.0%

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

   if -1.1500000000000001e29 < b < 2.65e-6

   1. Initial program 92.4%

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

   3. Taylor expanded in a around 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. accelerator-lowering-fma.f64 67.4

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

   5. Simplified 67.4%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+221}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\\ \mathbf{if}\;z \leq -18000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (fma a b y) x)))
   (if (<= z -18000.0) t_1 (if (<= z 2e-51) (fma a t (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(z, fma(a, b, y), x);
	double tmp;
	if (z <= -18000.0) {
		tmp = t_1;
	} else if (z <= 2e-51) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, fma(a, b, y), x)
	tmp = 0.0
	if (z <= -18000.0)
		tmp = t_1;
	elseif (z <= 2e-51)
		tmp = fma(a, t, 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[(z * N[(a * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -18000.0], t$95$1, If[LessEqual[z, 2e-51], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -18000 or 2e-51 < z

   1. Initial program 89.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 89.5

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

   5. Simplified 89.5%

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

   if -18000 < z < 2e-51

   1. Initial program 99.1%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 90.6%

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

Alternative 7: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+221}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(a, b, y\right)\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot a, b, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6.2e+221)
   (* z (fma a b y))
   (if (<= b 5.9e+82) (fma a t (fma z y x)) (fma (* z a) b x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6.2e+221) {
		tmp = z * fma(a, b, y);
	} else if (b <= 5.9e+82) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = fma((z * a), b, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6.2e+221)
		tmp = Float64(z * fma(a, b, y));
	elseif (b <= 5.9e+82)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = fma(Float64(z * a), b, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6.2e+221], N[(z * N[(a * b + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.9e+82], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(z * a), $MachinePrecision] * b + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000013e221

   1. Initial program 95.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 86.2

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

   5. Simplified 86.2%

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

   if -6.20000000000000013e221 < b < 5.8999999999999997e82

   1. Initial program 93.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 87.7%

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

   if 5.8999999999999997e82 < b

   1. Initial program 94.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 85.1%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 85.1

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

   7. Applied egg-rr 85.1%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+221}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(a, b, y\right)\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot a, b, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+148}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-182}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -2.6e+148)
   x
   (if (<= x 2.2e-182) (* t a) (if (<= x 2.2e+99) (* y z) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.6e+148) {
		tmp = x;
	} else if (x <= 2.2e-182) {
		tmp = t * a;
	} else if (x <= 2.2e+99) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-2.6d+148)) then
        tmp = x
    else if (x <= 2.2d-182) then
        tmp = t * a
    else if (x <= 2.2d+99) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.6e+148) {
		tmp = x;
	} else if (x <= 2.2e-182) {
		tmp = t * a;
	} else if (x <= 2.2e+99) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -2.6e+148:
		tmp = x
	elif x <= 2.2e-182:
		tmp = t * a
	elif x <= 2.2e+99:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.6e+148)
		tmp = x;
	elseif (x <= 2.2e-182)
		tmp = Float64(t * a);
	elseif (x <= 2.2e+99)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -2.6e+148)
		tmp = x;
	elseif (x <= 2.2e-182)
		tmp = t * a;
	elseif (x <= 2.2e+99)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.6e+148], x, If[LessEqual[x, 2.2e-182], N[(t * a), $MachinePrecision], If[LessEqual[x, 2.2e+99], N[(y * z), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e148 or 2.19999999999999978e99 < x

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 55.7%

      \[\leadsto \color{blue}{x} \]

   if -2.6e148 < x < 2.2e-182

   1. Initial program 95.0%

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

   3. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 37.6

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

   5. Simplified 37.6%

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

   if 2.2e-182 < x < 2.19999999999999978e99

   1. Initial program 87.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 46.1

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

   5. Simplified 46.1%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+148}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-182}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(a, b, y\right)\\ \mathbf{if}\;z \leq -2050000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (fma a b y))))
   (if (<= z -2050000000000.0) t_1 (if (<= z 1.5e-51) (fma a t x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * fma(a, b, y);
	double tmp;
	if (z <= -2050000000000.0) {
		tmp = t_1;
	} else if (z <= 1.5e-51) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * fma(a, b, y))
	tmp = 0.0
	if (z <= -2050000000000.0)
		tmp = t_1;
	elseif (z <= 1.5e-51)
		tmp = fma(a, t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(a * b + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2050000000000.0], t$95$1, If[LessEqual[z, 1.5e-51], N[(a * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.05e12 or 1.50000000000000001e-51 < z

   1. Initial program 89.0%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 78.9

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

   5. Simplified 78.9%

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

   if -2.05e12 < z < 1.50000000000000001e-51

   1. Initial program 99.1%

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

   3. Taylor expanded in 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. accelerator-lowering-fma.f64 80.8

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

   5. Simplified 80.8%

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

Alternative 10: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(b, z, t\right)\\ \mathbf{if}\;a \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (fma b z t))))
   (if (<= a -3.6e-6) t_1 (if (<= a 6.5e-38) (fma z y x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * fma(b, z, t);
	double tmp;
	if (a <= -3.6e-6) {
		tmp = t_1;
	} else if (a <= 6.5e-38) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * fma(b, z, t))
	tmp = 0.0
	if (a <= -3.6e-6)
		tmp = t_1;
	elseif (a <= 6.5e-38)
		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[(a * N[(b * z + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.6e-6], t$95$1, If[LessEqual[a, 6.5e-38], N[(z * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.59999999999999984e-6 or 6.49999999999999949e-38 < a

   1. Initial program 88.4%

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

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 73.7

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

   5. Simplified 73.7%

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

   if -3.59999999999999984e-6 < a < 6.49999999999999949e-38

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 77.0

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

   5. Simplified 77.0%

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

Alternative 11: 62.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3.7e+180) (fma a t x) (if (<= t 1.9e-52) (fma z y x) (fma a t x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3.7e+180) {
		tmp = fma(a, t, x);
	} else if (t <= 1.9e-52) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -3.7e+180)
		tmp = fma(a, t, x);
	elseif (t <= 1.9e-52)
		tmp = fma(z, y, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3.7e+180], N[(a * t + x), $MachinePrecision], If[LessEqual[t, 1.9e-52], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.7000000000000002e180 or 1.9000000000000002e-52 < t

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 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. accelerator-lowering-fma.f64 68.8

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

   5. Simplified 68.8%

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

   if -3.7000000000000002e180 < t < 1.9000000000000002e-52

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 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. accelerator-lowering-fma.f64 64.7

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

   5. Simplified 64.7%

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

Alternative 12: 57.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+132}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.02e+132) (* y z) (if (<= y 5.6e+208) (fma a t x) (* y z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.02e+132) {
		tmp = y * z;
	} else if (y <= 5.6e+208) {
		tmp = fma(a, t, x);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.02e+132)
		tmp = Float64(y * z);
	elseif (y <= 5.6e+208)
		tmp = fma(a, t, x);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.02e+132], N[(y * z), $MachinePrecision], If[LessEqual[y, 5.6e+208], N[(a * t + x), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0200000000000001e132 or 5.60000000000000045e208 < y

   1. Initial program 90.7%

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

   3. Taylor expanded in 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. *-lowering-*.f64 68.9

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

   5. Simplified 68.9%

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

   if -1.0200000000000001e132 < y < 5.60000000000000045e208

   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 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. accelerator-lowering-fma.f64 56.0

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

   5. Simplified 56.0%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+132}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+70}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.02e+70) (* t a) (if (<= t 3.5e-75) x (* t a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.02e+70) {
		tmp = t * a;
	} else if (t <= 3.5e-75) {
		tmp = x;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-1.02d+70)) then
        tmp = t * a
    else if (t <= 3.5d-75) then
        tmp = x
    else
        tmp = t * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.02e+70) {
		tmp = t * a;
	} else if (t <= 3.5e-75) {
		tmp = x;
	} else {
		tmp = t * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.02e+70:
		tmp = t * a
	elif t <= 3.5e-75:
		tmp = x
	else:
		tmp = t * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.02e+70)
		tmp = Float64(t * a);
	elseif (t <= 3.5e-75)
		tmp = x;
	else
		tmp = Float64(t * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.02e+70)
		tmp = t * a;
	elseif (t <= 3.5e-75)
		tmp = x;
	else
		tmp = t * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.02e+70], N[(t * a), $MachinePrecision], If[LessEqual[t, 3.5e-75], x, N[(t * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.02e70 or 3.49999999999999985e-75 < t

   1. Initial program 89.2%

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

   3. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 48.5

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

   5. Simplified 48.5%

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

   if -1.02e70 < t < 3.49999999999999985e-75

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 35.3%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+70}:\\ \;\;\;\;t \cdot a\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.2% accurate, 30.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

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 x around inf

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

5. Simplified 27.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 97.4% 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 2024196 
(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)))