Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.8% → 97.9%

Time: 8.9s

Alternatives: 13

Speedup: 5.7×

• 61.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} \cdot \left(\frac{\frac{\mathsf{fma}\left(-12, a, 4\right)}{b}}{b} + \mathsf{fma}\left(\frac{a}{b} \cdot 2, \frac{a}{b}, 1\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-30)
   (fma (* a a) (fma a a (fma a 4.0 4.0)) -1.0)
   (-
    (*
     (pow b 4.0)
     (+ (/ (/ (fma -12.0 a 4.0) b) b) (fma (* (/ a b) 2.0) (/ a b) 1.0)))
    1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-30) {
		tmp = fma((a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0);
	} else {
		tmp = (pow(b, 4.0) * (((fma(-12.0, a, 4.0) / b) / b) + fma(((a / b) * 2.0), (a / b), 1.0))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-30)
		tmp = fma(Float64(a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0);
	else
		tmp = Float64(Float64((b ^ 4.0) * Float64(Float64(Float64(fma(-12.0, a, 4.0) / b) / b) + fma(Float64(Float64(a / b) * 2.0), Float64(a / b), 1.0))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-30], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(a * 4.0 + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[Power[b, 4.0], $MachinePrecision] * N[(N[(N[(N[(-12.0 * a + 4.0), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(a / b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(a / b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e-30

   1. Initial program 81.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 1e-30 < (*.f64 b b)

   1. Initial program 63.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4} \cdot \left(1 + \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{1 - 3 \cdot a}{{b}^{2}}\right)\right)} - 1 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{1 - 3 \cdot a}{{b}^{2}}\right)\right) \cdot {b}^{4}} - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{1 - 3 \cdot a}{{b}^{2}}\right)\right) \cdot {b}^{4}} - 1 \]

   5. Applied rewrites 96.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} \cdot \left(\frac{\frac{\mathsf{fma}\left(-12, a, 4\right)}{b}}{b} + \mathsf{fma}\left(\frac{a}{b} \cdot 2, \frac{a}{b}, 1\right)\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 - 1.0), $MachinePrecision], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -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 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 92.4%

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1 \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(2 \cdot \left(a \cdot {b}^{2}\right)\right) \cdot a\right) - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 79.0%

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

   12. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt1-in N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. pow-sqr N/A

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

       10. distribute-lft-in N/A

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

       11. associate-+r+ N/A

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

       12. unpow2 N/A

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

       13. distribute-rgt-in N/A

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

       16. *-commutative N/A

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

   14. Applied rewrites 93.6%

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

4. Final simplification 98.0%

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

Alternative 3: 97.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-17)
   (fma (* a a) (fma a a (fma a 4.0 4.0)) -1.0)
   (- (fma (* (* b b) b) b (* (* (* 2.0 (* b b)) a) a)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-17) {
		tmp = fma((a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0);
	} else {
		tmp = fma(((b * b) * b), b, (((2.0 * (b * b)) * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-17)
		tmp = fma(Float64(a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0);
	else
		tmp = Float64(fma(Float64(Float64(b * b) * b), b, Float64(Float64(Float64(2.0 * Float64(b * b)) * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-17], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(a * 4.0 + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999999e-17

   1. Initial program 81.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 4.9999999999999999e-17 < (*.f64 b b)

   1. Initial program 62.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 95.2%

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1 \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(2 \cdot \left(a \cdot {b}^{2}\right)\right) \cdot a\right) - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 95.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{4}{b} + b\right) \cdot b\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.2)
   (fma (* a a) (fma a a (fma a 4.0 4.0)) -1.0)
   (- (* (* (* (+ (/ 4.0 b) b) b) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.2) {
		tmp = fma((a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0);
	} else {
		tmp = (((((4.0 / b) + b) * b) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.2)
		tmp = fma(Float64(a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(4.0 / b) + b) * b) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.2], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(a * 4.0 + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(4.0 / b), $MachinePrecision] + b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.20000000000000001

   1. Initial program 81.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 0.20000000000000001 < (*.f64 b b)

   1. Initial program 62.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + 4 \cdot {b}^{2}\right) + {b}^{4}\right)} - 1 \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 4 \cdot {b}^{2}\right) + {b}^{4}\right) - 1 \]

      3. distribute-rgt-out N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot \left(-12 \cdot a + 4\right)} + {b}^{4}\right) - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({b}^{2} \cdot \left(\color{blue}{\left(4 \cdot -3\right)} \cdot a + 4\right) + {b}^{4}\right) - 1 \]

      5. associate-*r* N/A

         \[\leadsto \left({b}^{2} \cdot \left(\color{blue}{4 \cdot \left(-3 \cdot a\right)} + 4\right) + {b}^{4}\right) - 1 \]

      6. metadata-eval N/A

         \[\leadsto \left({b}^{2} \cdot \left(4 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) + 4\right) + {b}^{4}\right) - 1 \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left({b}^{2} \cdot \left(4 \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} + 4\right) + {b}^{4}\right) - 1 \]

      8. metadata-eval N/A

         \[\leadsto \left({b}^{2} \cdot \left(4 \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right) + \color{blue}{4 \cdot 1}\right) + {b}^{4}\right) - 1 \]

      9. distribute-lft-in N/A

         \[\leadsto \left({b}^{2} \cdot \color{blue}{\left(4 \cdot \left(\left(\mathsf{neg}\left(3 \cdot a\right)\right) + 1\right)\right)} + {b}^{4}\right) - 1 \]

      10. +-commutative N/A

          \[\leadsto \left({b}^{2} \cdot \left(4 \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)}\right) + {b}^{4}\right) - 1 \]

      11. sub-neg N/A

          \[\leadsto \left({b}^{2} \cdot \left(4 \cdot \color{blue}{\left(1 - 3 \cdot a\right)}\right) + {b}^{4}\right) - 1 \]

      12. metadata-eval N/A

          \[\leadsto \left({b}^{2} \cdot \left(4 \cdot \left(1 - 3 \cdot a\right)\right) + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) - 1 \]

      13. pow-sqr N/A

          \[\leadsto \left({b}^{2} \cdot \left(4 \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) - 1 \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{{b}^{2} \cdot \left(4 \cdot \left(1 - 3 \cdot a\right) + {b}^{2}\right)} - 1 \]

      15. unpow2 N/A

          \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(4 \cdot \left(1 - 3 \cdot a\right) + {b}^{2}\right) - 1 \]

      16. associate-*l* N/A

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(4 \cdot \left(1 - 3 \cdot a\right) + {b}^{2}\right)\right)} - 1 \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(b \cdot \left(4 \cdot \left(1 - 3 \cdot a\right) + {b}^{2}\right)\right) \cdot b} - 1 \]

      18. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(b \cdot \left(4 \cdot \left(1 - 3 \cdot a\right) + {b}^{2}\right)\right) \cdot b} - 1 \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b\right) \cdot b} - 1 \]

   6. Taylor expanded in b around inf

      \[\leadsto \left(\left({b}^{2} \cdot \left(1 + \left(-12 \cdot \frac{a}{{b}^{2}} + 4 \cdot \frac{1}{{b}^{2}}\right)\right)\right) \cdot b\right) \cdot b - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 85.9%

      \[\leadsto \left(\left(b \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{b \cdot b}, -12, \frac{4}{b \cdot b}\right), b, b\right)\right) \cdot b\right) \cdot b - 1 \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(\left(b \cdot \left(b + 4 \cdot \frac{1}{b}\right)\right) \cdot b\right) \cdot b - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 89.0%

       \[\leadsto \left(\left(b \cdot \left(\frac{4}{b} + b\right)\right) \cdot b\right) \cdot b - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{4}{b} + b\right) \cdot b\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.2)
   (fma (* a a) (fma a a (fma a 4.0 4.0)) -1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.2) {
		tmp = fma((a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.2)
		tmp = fma(Float64(a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.2], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(a * 4.0 + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.20000000000000001

   1. Initial program 81.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 0.20000000000000001 < (*.f64 b b)

   1. Initial program 62.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 88.9

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

   8. Applied rewrites 88.9%

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

Alternative 6: 94.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.2)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.2) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.2)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.2], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.20000000000000001

   1. Initial program 81.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 80.7%

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1 \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(2 \cdot \left(a \cdot {b}^{2}\right)\right) \cdot a\right) - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 61.1%

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

   12. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt1-in N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. pow-sqr N/A

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

       10. distribute-lft-in N/A

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

       11. associate-+r+ N/A

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

       12. unpow2 N/A

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

       13. distribute-rgt-in N/A

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

       16. *-commutative N/A

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

   14. Applied rewrites 99.8%

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

   if 0.20000000000000001 < (*.f64 b b)

   1. Initial program 62.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 88.9

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

   8. Applied rewrites 88.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* a a) (* a a) -1.0)))
   (if (<= a -4.1e+70)
     t_0
     (if (<= a 2.35e+67) (fma (* (fma b b 4.0) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((a * a), (a * a), -1.0);
	double tmp;
	if (a <= -4.1e+70) {
		tmp = t_0;
	} else if (a <= 2.35e+67) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a * a), Float64(a * a), -1.0)
	tmp = 0.0
	if (a <= -4.1e+70)
		tmp = t_0;
	elseif (a <= 2.35e+67)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -4.1e+70], t$95$0, If[LessEqual[a, 2.35e+67], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1000000000000002e70 or 2.35000000000000009e67 < a

   1. Initial program 33.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

   if -4.1000000000000002e70 < a < 2.35000000000000009e67

   1. Initial program 96.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 91.9

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

   8. Applied rewrites 91.9%

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

Alternative 8: 92.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* a a) (* a a) -1.0)))
   (if (<= a -4.1e+70)
     t_0
     (if (<= a 2.35e+67) (fma (* (* b b) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((a * a), (a * a), -1.0);
	double tmp;
	if (a <= -4.1e+70) {
		tmp = t_0;
	} else if (a <= 2.35e+67) {
		tmp = fma(((b * b) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a * a), Float64(a * a), -1.0)
	tmp = 0.0
	if (a <= -4.1e+70)
		tmp = t_0;
	elseif (a <= 2.35e+67)
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -4.1e+70], t$95$0, If[LessEqual[a, 2.35e+67], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1000000000000002e70 or 2.35000000000000009e67 < a

   1. Initial program 33.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

   if -4.1000000000000002e70 < a < 2.35000000000000009e67

   1. Initial program 96.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right) + \color{blue}{-1} \]

   8. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, -1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.6%

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

Alternative 9: 92.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* a a) (* a a) -1.0)))
   (if (<= a -4.1e+70)
     t_0
     (if (<= a 2.35e+67) (fma (* b b) (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((a * a), (a * a), -1.0);
	double tmp;
	if (a <= -4.1e+70) {
		tmp = t_0;
	} else if (a <= 2.35e+67) {
		tmp = fma((b * b), (b * b), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a * a), Float64(a * a), -1.0)
	tmp = 0.0
	if (a <= -4.1e+70)
		tmp = t_0;
	elseif (a <= 2.35e+67)
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -4.1e+70], t$95$0, If[LessEqual[a, 2.35e+67], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1000000000000002e70 or 2.35000000000000009e67 < a

   1. Initial program 33.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

   if -4.1000000000000002e70 < a < 2.35000000000000009e67

   1. Initial program 96.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right) + \color{blue}{-1} \]

   8. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, -1\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.6%

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

   13. Applied rewrites 90.6%

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

Alternative 10: 84.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.1 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 3.1e+307) (fma (* a a) (* a a) -1.0) (- (* 4.0 (* b b)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.1e+307) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = (4.0 * (b * b)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 3.1e+307)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(4.0 * Float64(b * b)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 3.1e+307], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.1000000000000002e307

   1. Initial program 76.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.0%

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

   if 3.1000000000000002e307 < (*.f64 b b)

   1. Initial program 56.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, 4, {b}^{4}\right)} - 1 \]

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto 4 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.1 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2.9 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2.9e+307) (fma (* a a) 4.0 -1.0) (- (* 4.0 (* b b)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2.9e+307) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = (4.0 * (b * b)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2.9e+307)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(Float64(4.0 * Float64(b * b)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2.9e+307], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.89999999999999997e307

   1. Initial program 76.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 59.6%

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

   if 2.89999999999999997e307 < (*.f64 b b)

   1. Initial program 56.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, 4, {b}^{4}\right)} - 1 \]

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto 4 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2.9 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a \cdot a, 4, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma (* a a) 4.0 -1.0))

C

double code(double a, double b) {
	return fma((a * a), 4.0, -1.0);
}

Julia

function code(a, b)
	return fma(Float64(a * a), 4.0, -1.0)
end

Wolfram

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

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

5. Applied rewrites 88.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

6. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-fma.f64 N/A

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

   14. +-commutative N/A

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

   15. distribute-rgt-in N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 67.7

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

8. Applied rewrites 67.7%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 50.2%

    \[\leadsto \mathsf{fma}\left(a \cdot a, 4, -1\right) \]
12. Add Preprocessing

Alternative 13: 24.7% accurate, 160.0× 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 71.0%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]

5. Applied rewrites 88.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right) + \color{blue}{-1} \]

8. Applied rewrites 69.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, -1\right)} \]

9. Taylor expanded in b around 0

   \[\leadsto -1 \]
10. Step-by-step derivation

11. Applied rewrites 22.4%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))