Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.7% → 98.2%

Time: 12.2s

Alternatives: 13

Speedup: 5.5×

• 60.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))))
   (if (<= t_0 INFINITY)
     (+ t_0 -1.0)
     (fma (* a a) (fma a (+ a -4.0) 4.0) -1.0))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = fma((a * a), fma(a, (a + -4.0), 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = fma(Float64(a * a), fma(a, Float64(a + -4.0), 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < +inf.0

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2} + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a + 1\right)\right)\right)\right), 1\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1\right)\right)\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(-1 \cdot a\right), 4\right)\right)\right), 1\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(\mathsf{neg}\left(a\right)\right), 4\right)\right)\right), 1\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(0 - a\right), 4\right)\right)\right), 1\right) \]

      21. --lowering--.f64 91.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \mathsf{\_.f64}\left(0, a\right), 4\right)\right)\right), 1\right) \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + -1 \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(a - 4\right)\right)}, -1\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2} + 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot a + 0\right), \left(4 + a \cdot \left(a - 4\right)\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot \left(a - 4\right) + \color{blue}{4}\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \color{blue}{\left(a - 4\right)}, 4\right), -1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right), 4\right), -1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + -4\right), 4\right), -1\right) \]

      11. +-lowering-+.f64 91.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \mathsf{+.f64}\left(a, \color{blue}{-4}\right), 4\right), -1\right) \]

   8. Simplified 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, a + -4, 4\right), -1\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot a\right), \mathsf{fma.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(a, -4\right), 4\right), -1\right) \]

      2. *-lowering-*.f64 91.8%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{fma.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(a, -4\right), 4\right), -1\right) \]

   10. Applied egg-rr 91.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, \mathsf{fma}\left(a, a + -4, 4\right), -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq \infty:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(a, a + -4, 4\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+14)
   (fma (* a (fma a (+ a -4.0) 4.0)) a -1.0)
   (+ -1.0 (* b (* b (fma b b 12.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+14) {
		tmp = fma((a * fma(a, (a + -4.0), 4.0)), a, -1.0);
	} else {
		tmp = -1.0 + (b * (b * fma(b, b, 12.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+14)
		tmp = fma(Float64(a * fma(a, Float64(a + -4.0), 4.0)), a, -1.0);
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 12.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+14], N[(N[(a * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e14

   1. Initial program 86.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2} + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a + 1\right)\right)\right)\right), 1\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1\right)\right)\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(-1 \cdot a\right), 4\right)\right)\right), 1\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(\mathsf{neg}\left(a\right)\right), 4\right)\right)\right), 1\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(0 - a\right), 4\right)\right)\right), 1\right) \]

      21. --lowering--.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \mathsf{\_.f64}\left(0, a\right), 4\right)\right)\right), 1\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + -1 \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(a - 4\right)\right)}, -1\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2} + 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot a + 0\right), \left(4 + a \cdot \left(a - 4\right)\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot \left(a - 4\right) + \color{blue}{4}\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \color{blue}{\left(a - 4\right)}, 4\right), -1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right), 4\right), -1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + -4\right), 4\right), -1\right) \]

      11. +-lowering-+.f64 99.5%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \mathsf{+.f64}\left(a, \color{blue}{-4}\right), 4\right), -1\right) \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, a + -4, 4\right), -1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot \left(a + -4\right) + 4\right) \cdot \left(a \cdot a + 0\right) + -1 \]

      2. +-rgt-identity N/A

         \[\leadsto \left(a \cdot \left(a + -4\right) + 4\right) \cdot \left(a \cdot a\right) + -1 \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot \left(a + -4\right) + 4\right) \cdot a\right) \cdot a + -1 \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(a \cdot \left(a + -4\right) + 4\right) \cdot a\right), \color{blue}{a}, -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \left(a + -4\right) + 4\right), a\right), a, -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, \left(a + -4\right), 4\right), a\right), a, -1\right) \]

      7. +-lowering-+.f64 99.5%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, \mathsf{+.f64}\left(a, -4\right), 4\right), a\right), a, -1\right) \]

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a + -4, 4\right) \cdot a, a, -1\right)} \]

   if 1e14 < (*.f64 b b)

   1. Initial program 59.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 93.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(a, a + -4, 4\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+14)
   (fma (* a a) (fma a (+ a -4.0) 4.0) -1.0)
   (+ -1.0 (* b (* b (fma b b 12.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+14) {
		tmp = fma((a * a), fma(a, (a + -4.0), 4.0), -1.0);
	} else {
		tmp = -1.0 + (b * (b * fma(b, b, 12.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+14)
		tmp = fma(Float64(a * a), fma(a, Float64(a + -4.0), 4.0), -1.0);
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 12.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+14], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e14

   1. Initial program 86.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2} + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a + 1\right)\right)\right)\right), 1\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1\right)\right)\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(-1 \cdot a\right), 4\right)\right)\right), 1\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(\mathsf{neg}\left(a\right)\right), 4\right)\right)\right), 1\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(0 - a\right), 4\right)\right)\right), 1\right) \]

      21. --lowering--.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \mathsf{\_.f64}\left(0, a\right), 4\right)\right)\right), 1\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + -1 \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(a - 4\right)\right)}, -1\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2} + 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot a + 0\right), \left(4 + a \cdot \left(a - 4\right)\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot \left(a - 4\right) + \color{blue}{4}\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \color{blue}{\left(a - 4\right)}, 4\right), -1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right), 4\right), -1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + -4\right), 4\right), -1\right) \]

      11. +-lowering-+.f64 99.5%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \mathsf{+.f64}\left(a, \color{blue}{-4}\right), 4\right), -1\right) \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, a + -4, 4\right), -1\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot a\right), \mathsf{fma.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(a, -4\right), 4\right), -1\right) \]

      2. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{fma.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(a, -4\right), 4\right), -1\right) \]

   10. Applied egg-rr 99.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, \mathsf{fma}\left(a, a + -4, 4\right), -1\right) \]

   if 1e14 < (*.f64 b b)

   1. Initial program 59.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 93.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot \left(a + -4\right)\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+14)
   (fma (* a (* a (+ a -4.0))) a -1.0)
   (+ -1.0 (* b (* b (fma b b 12.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+14) {
		tmp = fma((a * (a * (a + -4.0))), a, -1.0);
	} else {
		tmp = -1.0 + (b * (b * fma(b, b, 12.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+14)
		tmp = fma(Float64(a * Float64(a * Float64(a + -4.0))), a, -1.0);
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 12.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+14], N[(N[(a * N[(a * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e14

   1. Initial program 86.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2} + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a + 1\right)\right)\right)\right), 1\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1\right)\right)\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(-1 \cdot a\right), 4\right)\right)\right), 1\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(\mathsf{neg}\left(a\right)\right), 4\right)\right)\right), 1\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(0 - a\right), 4\right)\right)\right), 1\right) \]

      21. --lowering--.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \mathsf{\_.f64}\left(0, a\right), 4\right)\right)\right), 1\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) \cdot \left(a \cdot a\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) \cdot a\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) \cdot a\right), \color{blue}{a}, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(4 \cdot \left(0 - a\right) + 4\right) + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. sub0-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(4 \cdot \left(\mathsf{neg}\left(a\right)\right) + 4\right) + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\mathsf{neg}\left(a\right)\right) \cdot 4 + 4\right) + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\mathsf{neg}\left(a\right)\right) + 1\right) \cdot 4 + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \left(\mathsf{neg}\left(a\right)\right)\right) \cdot 4 + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(1 - a\right) \cdot 4 + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\left(1 - a\right), 4, \left(a \cdot a\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \left(a \cdot a\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      15. +-rgt-identity N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \left(a \cdot a + 0\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \mathsf{fma.f64}\left(a, a, 0\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      17. metadata-eval 99.5%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \mathsf{fma.f64}\left(a, a, 0\right)\right), a\right), a, -1\right) \]

   7. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1 - a, 4, \mathsf{fma}\left(a, a, 0\right)\right) \cdot a, a, -1\right)} \]

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left({a}^{2} \cdot \left(1 - 4 \cdot \frac{1}{a}\right)\right)}, a\right), a, -1\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(a \cdot a\right) \cdot \left(1 - 4 \cdot \frac{1}{a}\right)\right), a\right), a, -1\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \left(a \cdot \left(1 - 4 \cdot \frac{1}{a}\right)\right)\right), a\right), a, -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(1 - 4 \cdot \frac{1}{a}\right)\right)\right), a\right), a, -1\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(1 + \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right)\right)\right)\right), a\right), a, -1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(\left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right) + 1\right)\right)\right), a\right), a, -1\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{1}{a}\right)\right) + a \cdot 1\right)\right), a\right), a, -1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(\mathsf{neg}\left(\frac{1}{a} \cdot 4\right)\right) + a \cdot 1\right)\right), a\right), a, -1\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{1}{a} \cdot \left(\mathsf{neg}\left(4\right)\right)\right) + a \cdot 1\right)\right), a\right), a, -1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{1}{a} \cdot -4\right) + a \cdot 1\right)\right), a\right), a, -1\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\left(a \cdot \frac{1}{a}\right) \cdot -4 + a \cdot 1\right)\right), a\right), a, -1\right) \]

      11. rgt-mult-inverse N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(1 \cdot -4 + a \cdot 1\right)\right), a\right), a, -1\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(-4 + a \cdot 1\right)\right), a\right), a, -1\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(-4 + a\right)\right), a\right), a, -1\right) \]

      14. +-lowering-+.f64 97.3%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-4, a\right)\right), a\right), a, -1\right) \]

   10. Simplified 97.3%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(a \cdot \left(-4 + a\right)\right)} \cdot a, a, -1\right) \]

   if 1e14 < (*.f64 b b)

   1. Initial program 59.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 93.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot \left(a + -4\right)\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+14)
   (fma (fma a a 0.0) (fma a a 4.0) -1.0)
   (+ -1.0 (* b (* b (fma b b 12.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+14) {
		tmp = fma(fma(a, a, 0.0), fma(a, a, 4.0), -1.0);
	} else {
		tmp = -1.0 + (b * (b * fma(b, b, 12.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+14)
		tmp = fma(fma(a, a, 0.0), fma(a, a, 4.0), -1.0);
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 12.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+14], N[(N[(a * a + 0.0), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e14

   1. Initial program 86.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2} + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a + 1\right)\right)\right)\right), 1\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1\right)\right)\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(-1 \cdot a\right), 4\right)\right)\right), 1\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(\mathsf{neg}\left(a\right)\right), 4\right)\right)\right), 1\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(0 - a\right), 4\right)\right)\right), 1\right) \]

      21. --lowering--.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \mathsf{\_.f64}\left(0, a\right), 4\right)\right)\right), 1\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + -1 \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(a - 4\right)\right)}, -1\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2} + 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot a + 0\right), \left(4 + a \cdot \left(a - 4\right)\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot \left(a - 4\right) + \color{blue}{4}\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \color{blue}{\left(a - 4\right)}, 4\right), -1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right), 4\right), -1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + -4\right), 4\right), -1\right) \]

      11. +-lowering-+.f64 99.5%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \mathsf{+.f64}\left(a, \color{blue}{-4}\right), 4\right), -1\right) \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, a + -4, 4\right), -1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \color{blue}{a}, 4\right), -1\right) \]
   10. Step-by-step derivation

   11. Simplified 97.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, \color{blue}{a}, 4\right), -1\right) \]

   if 1e14 < (*.f64 b b)

   1. Initial program 59.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 93.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+14)
   (fma (fma a a 0.0) (fma a a 4.0) -1.0)
   (fma (fma b b 12.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+14) {
		tmp = fma(fma(a, a, 0.0), fma(a, a, 4.0), -1.0);
	} else {
		tmp = fma(fma(b, b, 12.0), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+14)
		tmp = fma(fma(a, a, 0.0), fma(a, a, 4.0), -1.0);
	else
		tmp = fma(fma(b, b, 12.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+14], N[(N[(a * a + 0.0), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b + 12.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e14

   1. Initial program 86.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2} + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a + 1\right)\right)\right)\right), 1\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1\right)\right)\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(-1 \cdot a\right), 4\right)\right)\right), 1\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(\mathsf{neg}\left(a\right)\right), 4\right)\right)\right), 1\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(0 - a\right), 4\right)\right)\right), 1\right) \]

      21. --lowering--.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \mathsf{\_.f64}\left(0, a\right), 4\right)\right)\right), 1\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + -1 \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(a - 4\right)\right)}, -1\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{2} + 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot a + 0\right), \left(4 + a \cdot \left(a - 4\right)\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(\color{blue}{4} + a \cdot \left(a - 4\right)\right), -1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot \left(a - 4\right) + \color{blue}{4}\right), -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \color{blue}{\left(a - 4\right)}, 4\right), -1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right), 4\right), -1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \left(a + -4\right), 4\right), -1\right) \]

      11. +-lowering-+.f64 99.5%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \mathsf{+.f64}\left(a, \color{blue}{-4}\right), 4\right), -1\right) \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, a + -4, 4\right), -1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, \color{blue}{a}, 4\right), -1\right) \]
   10. Step-by-step derivation

   11. Simplified 97.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(a, \color{blue}{a}, 4\right), -1\right) \]

   if 1e14 < (*.f64 b b)

   1. Initial program 59.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 93.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot b + 12\right) \cdot \left(b \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left(b \cdot b + 12\right), \color{blue}{\left(b \cdot b\right)}, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(b, b, 12\right), \left(\color{blue}{b} \cdot b\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(b, b, 12\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. metadata-eval 93.3%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(b, b, 12\right), \mathsf{*.f64}\left(b, b\right), -1\right) \]

   7. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right), b \cdot b, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot a\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+14)
   (fma (* a (* a a)) a -1.0)
   (fma (fma b b 12.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+14) {
		tmp = fma((a * (a * a)), a, -1.0);
	} else {
		tmp = fma(fma(b, b, 12.0), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+14)
		tmp = fma(Float64(a * Float64(a * a)), a, -1.0);
	else
		tmp = fma(fma(b, b, 12.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+14], N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b + 12.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e14

   1. Initial program 86.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(3 + 1\right)}\right), 1\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{3} \cdot a\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2} + 0\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a + 0\right)\right)\right), 1\right) \]

      10. accelerator-lowering-fma.f64 96.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{fma.f64}\left(a, a, 0\right)\right)\right), 1\right) \]

   5. Simplified 96.6%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(a, a, 0\right)\right)} - 1 \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a \cdot \left(a \cdot \left(a \cdot a + 0\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto a \cdot \left(a \cdot \left(a \cdot a\right)\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      5. pow3 N/A

         \[\leadsto {a}^{3} \cdot a + \left(\mathsf{neg}\left(1\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{3}\right), \color{blue}{a}, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. cube-unmult N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a + 0\right)\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a\right) + a \cdot 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. mul0-rgt N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a\right) + 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, \left(a \cdot a\right), 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. +-rgt-identity N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, \left(a \cdot a + 0\right), 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, \mathsf{fma.f64}\left(a, a, 0\right), 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      14. metadata-eval 96.6%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, \mathsf{fma.f64}\left(a, a, 0\right), 0\right), a, -1\right) \]

   7. Applied egg-rr 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(a, a, 0\right), 0\right), a, -1\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left({a}^{3}\right)}, a, -1\right) \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), a, -1\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot {a}^{2}\right), a, -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{2}\right)\right), a, -1\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot a\right)\right), a, -1\right) \]

      5. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), a, -1\right) \]

   10. Simplified 96.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot \left(a \cdot a\right)}, a, -1\right) \]

   if 1e14 < (*.f64 b b)

   1. Initial program 59.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 93.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(b \cdot b + 12\right) \cdot \left(b \cdot b\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left(b \cdot b + 12\right), \color{blue}{\left(b \cdot b\right)}, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(b, b, 12\right), \left(\color{blue}{b} \cdot b\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(b, b, 12\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right), \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. metadata-eval 93.3%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(b, b, 12\right), \mathsf{*.f64}\left(b, b\right), -1\right) \]

   7. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right), b \cdot b, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot a\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+14) (fma (* a (* a a)) a -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+14) {
		tmp = fma((a * (a * a)), a, -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+14)
		tmp = fma(Float64(a * Float64(a * a)), a, -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+14], N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e14

   1. Initial program 86.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(3 + 1\right)}\right), 1\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{3} \cdot a\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2} + 0\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a + 0\right)\right)\right), 1\right) \]

      10. accelerator-lowering-fma.f64 96.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{fma.f64}\left(a, a, 0\right)\right)\right), 1\right) \]

   5. Simplified 96.6%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(a, a, 0\right)\right)} - 1 \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a \cdot \left(a \cdot \left(a \cdot a + 0\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto a \cdot \left(a \cdot \left(a \cdot a\right)\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      5. pow3 N/A

         \[\leadsto {a}^{3} \cdot a + \left(\mathsf{neg}\left(1\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left({a}^{3}\right), \color{blue}{a}, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. cube-unmult N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a + 0\right)\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a\right) + a \cdot 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. mul0-rgt N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a\right) + 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, \left(a \cdot a\right), 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. +-rgt-identity N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, \left(a \cdot a + 0\right), 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, \mathsf{fma.f64}\left(a, a, 0\right), 0\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      14. metadata-eval 96.6%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(a, \mathsf{fma.f64}\left(a, a, 0\right), 0\right), a, -1\right) \]

   7. Applied egg-rr 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(a, a, 0\right), 0\right), a, -1\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left({a}^{3}\right)}, a, -1\right) \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), a, -1\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(a \cdot {a}^{2}\right), a, -1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{2}\right)\right), a, -1\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot a\right)\right), a, -1\right) \]

      5. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), a, -1\right) \]

   10. Simplified 96.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot \left(a \cdot a\right)}, a, -1\right) \]

   if 1e14 < (*.f64 b b)

   1. Initial program 59.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 93.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {b}^{\left(3 + \color{blue}{1}\right)} \]

      2. pow-plus N/A

         \[\leadsto {b}^{3} \cdot \color{blue}{b} \]

      3. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{{b}^{3}} \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{3}\right)}\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      9. *-lowering-*.f64 93.3%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 93.3%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 4, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9000000.0) (fma (* a 4.0) a -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9000000.0) {
		tmp = fma((a * 4.0), a, -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9000000.0)
		tmp = fma(Float64(a * 4.0), a, -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9000000.0], N[(N[(a * 4.0), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9e6

   1. Initial program 76.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2} + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a + 1\right)\right)\right)\right), 1\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1\right)\right)\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(-1 \cdot a\right), 4\right)\right)\right), 1\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(\mathsf{neg}\left(a\right)\right), 4\right)\right)\right), 1\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(0 - a\right), 4\right)\right)\right), 1\right) \]

      21. --lowering--.f64 80.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \mathsf{\_.f64}\left(0, a\right), 4\right)\right)\right), 1\right) \]

   5. Simplified 80.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) \cdot \left(a \cdot a\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) \cdot a\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) \cdot a\right), \color{blue}{a}, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(4 \cdot \left(0 - a\right) + 4\right) + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. sub0-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(4 \cdot \left(\mathsf{neg}\left(a\right)\right) + 4\right) + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\mathsf{neg}\left(a\right)\right) \cdot 4 + 4\right) + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\mathsf{neg}\left(a\right)\right) + 1\right) \cdot 4 + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \left(\mathsf{neg}\left(a\right)\right)\right) \cdot 4 + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(1 - a\right) \cdot 4 + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\left(1 - a\right), 4, \left(a \cdot a\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \left(a \cdot a\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      15. +-rgt-identity N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \left(a \cdot a + 0\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \mathsf{fma.f64}\left(a, a, 0\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      17. metadata-eval 80.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \mathsf{fma.f64}\left(a, a, 0\right)\right), a\right), a, -1\right) \]

   7. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1 - a, 4, \mathsf{fma}\left(a, a, 0\right)\right) \cdot a, a, -1\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, a\right), a, -1\right) \]
   9. Step-by-step derivation

   10. Simplified 61.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{4} \cdot a, a, -1\right) \]

   if 9e6 < b

   1. Initial program 62.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 89.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 89.6%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {b}^{\left(3 + \color{blue}{1}\right)} \]

      2. pow-plus N/A

         \[\leadsto {b}^{3} \cdot \color{blue}{b} \]

      3. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{{b}^{3}} \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{3}\right)}\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      9. *-lowering-*.f64 89.6%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   8. Simplified 89.6%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 4, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 5e-7) -1.0 (* (* b b) 12.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-7) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d-7) then
        tmp = -1.0d0
    else
        tmp = (b * b) * 12.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-7) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-7:
		tmp = -1.0
	else:
		tmp = (b * b) * 12.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-7)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * 12.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-7)
		tmp = -1.0;
	else
		tmp = (b * b) * 12.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-7], -1.0, N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999977e-7

   1. Initial program 86.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 55.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 55.0%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 54.6%

      \[\leadsto \color{blue}{-1} \]

   if 4.99999999999999977e-7 < (*.f64 b b)

   1. Initial program 60.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 90.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 90.7%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{12}\right)\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 49.9%

      \[\leadsto b \cdot \left(b \cdot \color{blue}{12}\right) - 1 \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \left(b \cdot 12\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(b \cdot 12\right) \cdot b + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(b \cdot 12\right) \cdot b + -1 \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left(b \cdot 12\right), \color{blue}{b}, -1\right) \]

      5. *-lowering-*.f64 49.9%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(b, 12\right), b, -1\right) \]

   10. Applied egg-rr 49.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot 12, b, -1\right)} \]

   11. Taylor expanded in b around inf

       \[\leadsto \color{blue}{12 \cdot {b}^{2}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(12, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(12, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 49.9%

          \[\leadsto \mathsf{*.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   13. Simplified 49.9%

       \[\leadsto \color{blue}{12 \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.38 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 4, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.38e+128) (fma (* a 4.0) a -1.0) (* (* b b) 12.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.38e+128) {
		tmp = fma((a * 4.0), a, -1.0);
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.38e+128)
		tmp = fma(Float64(a * 4.0), a, -1.0);
	else
		tmp = Float64(Float64(b * b) * 12.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.38e+128], N[(N[(a * 4.0), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3799999999999999e128

   1. Initial program 75.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot {a}^{2}\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2} + \left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}\right), 1\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2} + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left({a}^{2} + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 - a\right)\right)\right)\right), 1\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(1 + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a + 1\right)\right)\right)\right), 1\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1\right)\right)\right), 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \left(4 \cdot \left(-1 \cdot a\right) + 4\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(-1 \cdot a\right), 4\right)\right)\right), 1\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(\mathsf{neg}\left(a\right)\right), 4\right)\right)\right), 1\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \left(0 - a\right), 4\right)\right)\right), 1\right) \]

      21. --lowering--.f64 75.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(a, a, 0\right), \mathsf{fma.f64}\left(a, a, \mathsf{fma.f64}\left(4, \mathsf{\_.f64}\left(0, a\right), 4\right)\right)\right), 1\right) \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) \cdot \left(a \cdot a\right) + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) \cdot a\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) \cdot a\right), \color{blue}{a}, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(4 \cdot \left(0 - a\right) + 4\right) + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      8. sub0-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(4 \cdot \left(\mathsf{neg}\left(a\right)\right) + 4\right) + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\mathsf{neg}\left(a\right)\right) \cdot 4 + 4\right) + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\mathsf{neg}\left(a\right)\right) + 1\right) \cdot 4 + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \left(\mathsf{neg}\left(a\right)\right)\right) \cdot 4 + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\left(1 - a\right) \cdot 4 + a \cdot a\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\left(1 - a\right), 4, \left(a \cdot a\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \left(a \cdot a\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      15. +-rgt-identity N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \left(a \cdot a + 0\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \mathsf{fma.f64}\left(a, a, 0\right)\right), a\right), a, \left(\mathsf{neg}\left(1\right)\right)\right) \]

      17. metadata-eval 75.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(1, a\right), 4, \mathsf{fma.f64}\left(a, a, 0\right)\right), a\right), a, -1\right) \]

   7. Applied egg-rr 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1 - a, 4, \mathsf{fma}\left(a, a, 0\right)\right) \cdot a, a, -1\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, a\right), a, -1\right) \]
   9. Step-by-step derivation

   10. Simplified 57.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{4} \cdot a, a, -1\right) \]

   if 1.3799999999999999e128 < b

   1. Initial program 57.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

      14. accelerator-lowering-fma.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{12}\right)\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 85.9%

      \[\leadsto b \cdot \left(b \cdot \color{blue}{12}\right) - 1 \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \left(b \cdot 12\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(b \cdot 12\right) \cdot b + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(b \cdot 12\right) \cdot b + -1 \]

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

         \[\leadsto \mathsf{fma.f64}\left(\left(b \cdot 12\right), \color{blue}{b}, -1\right) \]

      5. *-lowering-*.f64 85.9%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(b, 12\right), b, -1\right) \]

   10. Applied egg-rr 85.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot 12, b, -1\right)} \]

   11. Taylor expanded in b around inf

       \[\leadsto \color{blue}{12 \cdot {b}^{2}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(12, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(12, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 85.9%

          \[\leadsto \mathsf{*.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   13. Simplified 85.9%

       \[\leadsto \color{blue}{12 \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.38 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 4, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(12, b \cdot b, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma 12.0 (* b b) -1.0))

C

double code(double a, double b) {
	return fma(12.0, (b * b), -1.0);
}

Julia

function code(a, b)
	return fma(12.0, Float64(b * b), -1.0)
end

Wolfram

code[a_, b_] := N[(12.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 73.0%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

   3. pow-plus N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

   4. cube-unmult N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

   8. distribute-rgt-out N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

   14. accelerator-lowering-fma.f64 73.5%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

5. Simplified 73.5%

   \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{12 \cdot {b}^{2} - 1} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto 12 \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto 12 \cdot {b}^{2} + -1 \]

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

      \[\leadsto \mathsf{fma.f64}\left(12, \color{blue}{\left({b}^{2}\right)}, -1\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{fma.f64}\left(12, \left(b \cdot \color{blue}{b}\right), -1\right) \]

   5. *-lowering-*.f64 52.3%

      \[\leadsto \mathsf{fma.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right), -1\right) \]

8. Simplified 52.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(12, b \cdot b, -1\right)} \]
9. Add Preprocessing

Alternative 13: 25.2% accurate, 155.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

double code(double a, double b) {
	return -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -1.0d0
end function

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0;
end

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 73.0%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 12 \cdot {b}^{2}\right), 1\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(3 + 1\right)} + 12 \cdot {b}^{2}\right), 1\right) \]

   3. pow-plus N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{3} \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

   4. cube-unmult N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot {b}^{2}\right), 1\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + 12 \cdot \left(b \cdot b\right)\right), 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(b \cdot {b}^{2}\right) \cdot b + \left(12 \cdot b\right) \cdot b\right), 1\right) \]

   8. distribute-rgt-out N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(b \cdot \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + 12 \cdot b\right)\right), 1\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot {b}^{2} + b \cdot 12\right)\right), 1\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left({b}^{2} + 12\right)\right)\right), 1\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left({b}^{2} + 12\right)\right)\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot b + 12\right)\right)\right), 1\right) \]

   14. accelerator-lowering-fma.f64 73.5%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{fma.f64}\left(b, b, 12\right)\right)\right), 1\right) \]

5. Simplified 73.5%

   \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)} - 1 \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 26.6%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024193 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))