Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.2% → 97.3%

Time: 8.4s

Alternatives: 11

Speedup: 1.1×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.2% 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: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(z, b, t\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b))))
   (if (<= t_1 2e+306) t_1 (fma z y (* (fma z b t) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x + (y * z)) + (t * a)) + ((a * z) * b);
	double tmp;
	if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = fma(z, y, (fma(z, b, t) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b))
	tmp = 0.0
	if (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = fma(z, y, Float64(fma(z, b, t) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+306], t$95$1, N[(z * y + N[(N[(z * b + t), $MachinePrecision] * a), $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)) < 2.00000000000000003e306

   1. Initial program 99.5%

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

   if 2.00000000000000003e306 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 91.3

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 91.3

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

   7. Applied rewrites 91.3%

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

Alternative 2: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot z\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+211}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a b) z)))
   (if (<= z -2.8e+211)
     t_1
     (if (<= z -3.1e-55)
       (fma y z x)
       (if (<= z 4.4e+41)
         (fma a t x)
         (if (<= z 1.05e+261) (fma y z x) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) * z;
	double tmp;
	if (z <= -2.8e+211) {
		tmp = t_1;
	} else if (z <= -3.1e-55) {
		tmp = fma(y, z, x);
	} else if (z <= 4.4e+41) {
		tmp = fma(a, t, x);
	} else if (z <= 1.05e+261) {
		tmp = fma(y, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) * z)
	tmp = 0.0
	if (z <= -2.8e+211)
		tmp = t_1;
	elseif (z <= -3.1e-55)
		tmp = fma(y, z, x);
	elseif (z <= 4.4e+41)
		tmp = fma(a, t, x);
	elseif (z <= 1.05e+261)
		tmp = fma(y, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.8e+211], t$95$1, If[LessEqual[z, -3.1e-55], N[(y * z + x), $MachinePrecision], If[LessEqual[z, 4.4e+41], N[(a * t + x), $MachinePrecision], If[LessEqual[z, 1.05e+261], N[(y * z + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8e211 or 1.05e261 < z

   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 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 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.7%

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

   if -2.8e211 < z < -3.09999999999999997e-55 or 4.3999999999999998e41 < z < 1.05e261

   1. Initial program 89.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 16.4

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

   5. Applied rewrites 16.4%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-+r+ N/A

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

      3. div-add-rev N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 65.0

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

   11. Applied rewrites 65.0%

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

   if -3.09999999999999997e-55 < z < 4.3999999999999998e41

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 37.6

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

   5. Applied rewrites 37.6%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-+r+ N/A

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

      3. div-add-rev N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 86.3%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 77.6

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

   11. Applied rewrites 77.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+211}:\\ \;\;\;\;\left(a \cdot b\right) \cdot z\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot a, b, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{elif}\;x \leq 0.0022:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -2.65e+54)
   (fma (* z a) b (fma t a x))
   (if (<= x 0.0022) (fma (fma b z t) a (* z y)) (fma (fma b a y) z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.65e+54) {
		tmp = fma((z * a), b, fma(t, a, x));
	} else if (x <= 0.0022) {
		tmp = fma(fma(b, z, t), a, (z * y));
	} else {
		tmp = fma(fma(b, a, y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.65e+54)
		tmp = fma(Float64(z * a), b, fma(t, a, x));
	elseif (x <= 0.0022)
		tmp = fma(fma(b, z, t), a, Float64(z * y));
	else
		tmp = fma(fma(b, a, y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.65e+54], N[(N[(z * a), $MachinePrecision] * b + N[(t * a + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0022], N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.65000000000000009e54

   1. Initial program 96.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 91.2

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

   5. Applied rewrites 91.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 91.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

   if -2.65000000000000009e54 < x < 0.00220000000000000013

   1. Initial program 92.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 0.00220000000000000013 < x

   1. Initial program 97.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot a, b, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{elif}\;x \leq 0.0022:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x + a \cdot \mathsf{fma}\left(b, z, t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.7e+158) (fma (fma b a y) z x) (fma z y (+ x (* a (fma b z t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.7e+158) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(z, y, (x + (a * fma(b, z, t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.7e+158)
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(z, y, Float64(x + Float64(a * fma(b, z, t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.7e+158], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(z * y + N[(x + N[(a * N[(b * z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.70000000000000011e158

   1. Initial program 93.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -3.70000000000000011e158 < z

   1. Initial program 94.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 97.0

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

   4. Applied rewrites 97.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x + a \cdot \mathsf{fma}\left(b, z, t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-55} \lor \neg \left(z \leq 4.5 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.1e-55) (not (<= z 4.5e+41)))
   (fma (fma b a y) z x)
   (fma (fma b z t) a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.1e-55) || !(z <= 4.5e+41)) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(fma(b, z, t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3.1e-55) || !(z <= 4.5e+41))
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(fma(b, z, t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3.1e-55], N[Not[LessEqual[z, 4.5e+41]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.09999999999999997e-55 or 4.5000000000000001e41 < z

   1. Initial program 89.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if -3.09999999999999997e-55 < z < 4.5000000000000001e41

   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 y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-55} \lor \neg \left(z \leq 4.5 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-58} \lor \neg \left(z \leq 8.5 \cdot 10^{-169}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.8e-58) (not (<= z 8.5e-169)))
   (fma (fma b a y) z x)
   (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.8e-58) || !(z <= 8.5e-169)) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.8e-58) || !(z <= 8.5e-169))
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.8e-58], N[Not[LessEqual[z, 8.5e-169]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-58 or 8.50000000000000054e-169 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -2.8000000000000001e-58 < z < 8.50000000000000054e-169

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 43.3

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-+r+ N/A

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

      3. div-add-rev N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 86.1%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 86.1

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

   11. Applied rewrites 86.1%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-58} \lor \neg \left(z \leq 8.5 \cdot 10^{-169}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-51} \lor \neg \left(z \leq 1.55 \cdot 10^{+106}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.8e-51) (not (<= z 1.55e+106)))
   (* (fma b a y) z)
   (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.8e-51) || !(z <= 1.55e+106)) {
		tmp = fma(b, a, y) * z;
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -4.8e-51) || !(z <= 1.55e+106))
		tmp = Float64(fma(b, a, y) * z);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -4.8e-51], N[Not[LessEqual[z, 1.55e+106]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8e-51 or 1.55e106 < z

   1. Initial program 88.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if -4.8e-51 < z < 1.55e106

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 34.7

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-+r+ N/A

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

      3. div-add-rev N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 84.3%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 75.9

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

   11. Applied rewrites 75.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-51} \lor \neg \left(z \leq 1.55 \cdot 10^{+106}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-55} \lor \neg \left(z \leq 4.4 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.1e-55) (not (<= z 4.4e+41))) (fma y z x) (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.1e-55) || !(z <= 4.4e+41)) {
		tmp = fma(y, z, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3.1e-55) || !(z <= 4.4e+41))
		tmp = fma(y, z, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3.1e-55], N[Not[LessEqual[z, 4.4e+41]], $MachinePrecision]], N[(y * z + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.09999999999999997e-55 or 4.3999999999999998e41 < z

   1. Initial program 89.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 15.9

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

   5. Applied rewrites 15.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-+r+ N/A

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

      3. div-add-rev N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 81.3%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 59.1

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

   11. Applied rewrites 59.1%

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

   if -3.09999999999999997e-55 < z < 4.3999999999999998e41

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 37.6

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

   5. Applied rewrites 37.6%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-+r+ N/A

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

      3. div-add-rev N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 86.3%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 77.6

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

   11. Applied rewrites 77.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-55} \lor \neg \left(z \leq 4.4 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{+63} \lor \neg \left(z \leq 4.2 \cdot 10^{+121}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.62e+63) (not (<= z 4.2e+121))) (* y z) (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.62e+63) || !(z <= 4.2e+121)) {
		tmp = y * z;
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.62e+63) || !(z <= 4.2e+121))
		tmp = Float64(y * z);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.62e+63], N[Not[LessEqual[z, 4.2e+121]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.62e63 or 4.2000000000000003e121 < z

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 89.0

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

   4. Applied rewrites 89.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 51.3

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

   7. Applied rewrites 51.3%

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

   if -1.62e63 < z < 4.2000000000000003e121

   1. Initial program 99.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 32.1

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-+r+ N/A

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

      3. div-add-rev N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 69.0

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

   11. Applied rewrites 69.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{+63} \lor \neg \left(z \leq 4.2 \cdot 10^{+121}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-68} \lor \neg \left(z \leq 1.18 \cdot 10^{+42}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -9e-68) (not (<= z 1.18e+42))) (* y z) (* a t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -9e-68) || !(z <= 1.18e+42)) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-9d-68)) .or. (.not. (z <= 1.18d+42))) then
        tmp = y * z
    else
        tmp = a * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -9e-68) || !(z <= 1.18e+42)) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -9e-68) or not (z <= 1.18e+42):
		tmp = y * z
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -9e-68) || !(z <= 1.18e+42))
		tmp = Float64(y * z);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -9e-68) || ~((z <= 1.18e+42)))
		tmp = y * z;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -9e-68], N[Not[LessEqual[z, 1.18e+42]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(a * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999998e-68 or 1.18e42 < z

   1. Initial program 89.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 90.6

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 43.5

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

   7. Applied rewrites 43.5%

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

   if -8.99999999999999998e-68 < z < 1.18e42

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 38.2

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

   5. Applied rewrites 38.2%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-68} \lor \neg \left(z \leq 1.18 \cdot 10^{+42}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. distribute-lft-out N/A

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

   18. lower-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 95.1

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

4. Applied rewrites 95.1%

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

5. Taylor expanded in y around inf

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

   1. lower-*.f64 28.6

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

7. Applied rewrites 28.6%

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

Developer Target 1: 97.2% 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 2024337 
(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)))