Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.2% → 98.3%

Time: 8.8s

Alternatives: 17

Speedup: 5.9×

• 61.1% 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: 73.2% 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: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 - 1.0;
	} else {
		tmp = ((((4.0 + a) * a) * a) * a) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 - 1.0
	else:
		tmp = ((((4.0 + a) * a) * a) * a) - 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 - 1.0;
	else
		tmp = ((((4.0 + a) * a) * a) * a) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 51.1

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

   5. Applied rewrites 51.1%

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

   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. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. unpow2 N/A

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

      4. unpow3 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. distribute-lft-in N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.6%

       \[\leadsto \left(\left(\left(4 + a\right) \cdot a\right) \cdot a\right) \cdot a - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 520000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{4}{a} + 1\right) \cdot {a}^{4} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.7e-6)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 520000000000.0)
     (- (fma (* b b) 4.0 (pow b 4.0)) 1.0)
     (- (* (+ (/ 4.0 a) 1.0) (pow a 4.0)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-6) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 520000000000.0) {
		tmp = fma((b * b), 4.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = (((4.0 / a) + 1.0) * pow(a, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.7e-6)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 520000000000.0)
		tmp = Float64(fma(Float64(b * b), 4.0, (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(4.0 / a) + 1.0) * (a ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.7e-6], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 520000000000.0], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(4.0 / a), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7000000000000002e-6

   1. Initial program 32.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.3

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

   8. Applied rewrites 33.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 33.3

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

   11. Applied rewrites 33.3%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 92.9%

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

   if -3.7000000000000002e-6 < a < 5.2e11

   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

   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 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.2e11 < a

   1. Initial program 59.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 98.2

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

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(\frac{4}{a} + 1\right) \cdot {a}^{4}} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 520000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right) \cdot a, 4, {a}^{4}\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.7e-6)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 520000000000.0)
     (- (fma (* b b) 4.0 (pow b 4.0)) 1.0)
     (- (fma (* (fma a a a) a) 4.0 (pow a 4.0)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-6) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 520000000000.0) {
		tmp = fma((b * b), 4.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma((fma(a, a, a) * a), 4.0, pow(a, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.7e-6)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 520000000000.0)
		tmp = Float64(fma(Float64(b * b), 4.0, (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(fma(a, a, a) * a), 4.0, (a ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.7e-6], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 520000000000.0], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7000000000000002e-6

   1. Initial program 32.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.3

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

   8. Applied rewrites 33.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 33.3

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

   11. Applied rewrites 33.3%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 92.9%

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

   if -3.7000000000000002e-6 < a < 5.2e11

   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

   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 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.2e11 < a

   1. Initial program 59.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-pow.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 4: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.7e-6)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 1.2e+23) (- (fma (* b b) 4.0 (pow b 4.0)) 1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-6) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 1.2e+23) {
		tmp = fma((b * b), 4.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.7e-6)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 1.2e+23)
		tmp = Float64(fma(Float64(b * b), 4.0, (b ^ 4.0)) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.7e-6], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 1.2e+23], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7000000000000002e-6

   1. Initial program 32.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.3

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

   8. Applied rewrites 33.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 33.3

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

   11. Applied rewrites 33.3%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 92.9%

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

   if -3.7000000000000002e-6 < a < 1.2e23

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   if 1.2e23 < a

   1. Initial program 60.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 35.5

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

   5. Applied rewrites 35.5%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 35.5

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 35.5

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

   11. Applied rewrites 35.5%

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

   12. Taylor expanded in a around inf

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

       1. lower-pow.f64 100.0

          \[\leadsto \color{blue}{{a}^{4}} \]

   14. Applied rewrites 100.0%

       \[\leadsto \color{blue}{{a}^{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.7e-6)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 1.2e+23) (fma (fma b b 4.0) (* b b) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-6) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 1.2e+23) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.7e-6)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 1.2e+23)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.7e-6], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 1.2e+23], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7000000000000002e-6

   1. Initial program 32.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.3

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

   8. Applied rewrites 33.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 33.3

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

   11. Applied rewrites 33.3%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 92.9%

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

   if -3.7000000000000002e-6 < a < 1.2e23

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 98.8

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

   8. Applied rewrites 98.8%

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

   if 1.2e23 < a

   1. Initial program 60.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 35.5

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

   5. Applied rewrites 35.5%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 35.5

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 35.5

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

   11. Applied rewrites 35.5%

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

   12. Taylor expanded in a around inf

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

       1. lower-pow.f64 100.0

          \[\leadsto \color{blue}{{a}^{4}} \]

   14. Applied rewrites 100.0%

       \[\leadsto \color{blue}{{a}^{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 520000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(4 + a\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.7e-6)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 520000000000.0)
     (fma (fma b b 4.0) (* b b) -1.0)
     (- (* (* (* (+ 4.0 a) a) a) a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-6) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 520000000000.0) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = ((((4.0 + a) * a) * a) * a) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.7e-6)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 520000000000.0)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(4.0 + a) * a) * a) * a) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.7e-6], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 520000000000.0], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7000000000000002e-6

   1. Initial program 32.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.3

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

   8. Applied rewrites 33.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 33.3

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

   11. Applied rewrites 33.3%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 92.9%

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

   if -3.7000000000000002e-6 < a < 5.2e11

   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

   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 99.6

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

   5. Applied rewrites 99.6%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   if 5.2e11 < a

   1. Initial program 59.9%

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

   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 36.1

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

   5. Applied rewrites 36.1%

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

   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. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. unpow2 N/A

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

      4. unpow3 N/A

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

      5. distribute-lft-in N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. distribute-lft-in N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.2%

       \[\leadsto \left(\left(\left(4 + a\right) \cdot a\right) \cdot a\right) \cdot a - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 520000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(4 + a\right) \cdot a, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.7e-6)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 520000000000.0)
     (fma (fma b b 4.0) (* b b) -1.0)
     (fma (* (+ 4.0 a) a) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-6) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 520000000000.0) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = fma(((4.0 + a) * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.7e-6)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 520000000000.0)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = fma(Float64(Float64(4.0 + a) * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.7e-6], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 520000000000.0], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7000000000000002e-6

   1. Initial program 32.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.3

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

   8. Applied rewrites 33.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 33.3

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

   11. Applied rewrites 33.3%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 92.9%

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

   if -3.7000000000000002e-6 < a < 5.2e11

   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

   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 99.6

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

   5. Applied rewrites 99.6%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   if 5.2e11 < a

   1. Initial program 59.9%

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

   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 36.1

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

   5. Applied rewrites 36.1%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.2%

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

Alternative 8: 94.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+16}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{elif}\;a \leq 520000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(4 + a\right) \cdot a, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+16)
   (- (* (* a a) (* a a)) 1.0)
   (if (<= a 520000000000.0)
     (fma (fma b b 4.0) (* b b) -1.0)
     (fma (* (+ 4.0 a) a) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+16) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else if (a <= 520000000000.0) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = fma(((4.0 + a) * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e+16)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	elseif (a <= 520000000000.0)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = fma(Float64(Float64(4.0 + a) * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e+16], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 520000000000.0], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8e16

   1. Initial program 22.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   4. Step-by-step derivation

      1. lower-pow.f64 98.4

         \[\leadsto \color{blue}{{a}^{4}} - 1 \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 98.4%

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

   if -3.8e16 < a < 5.2e11

   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

   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 96.8

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

   5. Applied rewrites 96.8%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if 5.2e11 < a

   1. Initial program 59.9%

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

   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 36.1

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

   5. Applied rewrites 36.1%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.2%

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

Alternative 9: 94.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+16} \lor \neg \left(a \leq 1.2 \cdot 10^{+23}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.8e+16) (not (<= a 1.2e+23)))
   (- (* (* a a) (* a a)) 1.0)
   (fma (fma b b 4.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.8e+16) || !(a <= 1.2e+23)) {
		tmp = ((a * a) * (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 ((a <= -3.8e+16) || !(a <= 1.2e+23))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.8e+16], N[Not[LessEqual[a, 1.2e+23]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8e16 or 1.2e23 < a

   1. Initial program 40.1%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   4. Step-by-step derivation

      1. lower-pow.f64 99.2

         \[\leadsto \color{blue}{{a}^{4}} - 1 \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

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

   if -3.8e16 < a < 1.2e23

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 96.2

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

   5. Applied rewrites 96.2%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 96.1

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

   8. Applied rewrites 96.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+16} \lor \neg \left(a \leq 1.2 \cdot 10^{+23}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right) \cdot a, a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+154)
   (fma 4.0 (* a a) -1.0)
   (if (<= a 5e+40)
     (fma (fma b b 4.0) (* b b) -1.0)
     (fma (* (fma 4.0 a 4.0) a) a -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+154) {
		tmp = fma(4.0, (a * a), -1.0);
	} else if (a <= 5e+40) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = fma((fma(4.0, a, 4.0) * a), a, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+154)
		tmp = fma(4.0, Float64(a * a), -1.0);
	elseif (a <= 5e+40)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = fma(Float64(fma(4.0, a, 4.0) * a), a, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+154], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 5e+40], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000007e154

   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

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 27.7

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

   5. Applied rewrites 27.7%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

   if -2.00000000000000007e154 < a < 5.00000000000000003e40

   1. Initial program 92.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 86.8

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

   8. Applied rewrites 86.8%

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

   if 5.00000000000000003e40 < a

   1. Initial program 60.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 35.5

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

   5. Applied rewrites 35.5%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 84.4%

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

Alternative 11: 86.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right) \cdot a, a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+154)
   (fma 4.0 (* a a) -1.0)
   (if (<= a 5e+40)
     (fma (* (fma b b 4.0) b) b -1.0)
     (fma (* (fma 4.0 a 4.0) a) a -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+154) {
		tmp = fma(4.0, (a * a), -1.0);
	} else if (a <= 5e+40) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma((fma(4.0, a, 4.0) * a), a, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+154)
		tmp = fma(4.0, Float64(a * a), -1.0);
	elseif (a <= 5e+40)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(fma(4.0, a, 4.0) * a), a, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+154], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 5e+40], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(4.0 * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000007e154

   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

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 27.7

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

   5. Applied rewrites 27.7%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

   if -2.00000000000000007e154 < a < 5.00000000000000003e40

   1. Initial program 92.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 86.8

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

   8. Applied rewrites 86.8%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 86.7

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

   11. Applied rewrites 86.7%

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

   if 5.00000000000000003e40 < a

   1. Initial program 60.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 35.5

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

   5. Applied rewrites 35.5%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 84.4%

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

Alternative 12: 85.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right) \cdot a, a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+154)
   (fma 4.0 (* a a) -1.0)
   (if (<= a 5e+40)
     (fma (* (* b b) b) b -1.0)
     (fma (* (fma 4.0 a 4.0) a) a -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+154) {
		tmp = fma(4.0, (a * a), -1.0);
	} else if (a <= 5e+40) {
		tmp = fma(((b * b) * b), b, -1.0);
	} else {
		tmp = fma((fma(4.0, a, 4.0) * a), a, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+154)
		tmp = fma(4.0, Float64(a * a), -1.0);
	elseif (a <= 5e+40)
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	else
		tmp = fma(Float64(fma(4.0, a, 4.0) * a), a, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+154], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 5e+40], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(4.0 * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000007e154

   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

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 27.7

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

   5. Applied rewrites 27.7%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

   if -2.00000000000000007e154 < a < 5.00000000000000003e40

   1. Initial program 92.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 86.8

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

   8. Applied rewrites 86.8%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 86.7

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

   11. Applied rewrites 86.7%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 86.0%

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

   if 5.00000000000000003e40 < a

   1. Initial program 60.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 35.5

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

   5. Applied rewrites 35.5%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 84.4%

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

Alternative 13: 84.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154} \lor \neg \left(a \leq 6.5 \cdot 10^{+143}\right):\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2e+154) (not (<= a 6.5e+143)))
   (fma 4.0 (* a a) -1.0)
   (fma (* (* b b) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2e+154) || !(a <= 6.5e+143)) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = fma(((b * b) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2e+154) || !(a <= 6.5e+143))
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2e+154], N[Not[LessEqual[a, 6.5e+143]], $MachinePrecision]], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000007e154 or 6.4999999999999997e143 < a

   1. Initial program 29.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 33.4

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

   5. Applied rewrites 33.4%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 96.3%

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

   if -2.00000000000000007e154 < a < 6.4999999999999997e143

   1. Initial program 90.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.0

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   10. 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. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 82.0

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

   11. Applied rewrites 82.0%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 81.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154} \lor \neg \left(a \leq 6.5 \cdot 10^{+143}\right):\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+18) (fma 4.0 (* a a) -1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+18) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+18)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+18], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e18

   1. Initial program 80.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 75.7%

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

   if 5e18 < (*.f64 b b)

   1. Initial program 66.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 87.1

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

   5. Applied rewrites 87.1%

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

   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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.1

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

   8. Applied rewrites 87.1%

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

   9. Taylor expanded in b around inf

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

       1. lower-pow.f64 87.1

          \[\leadsto \color{blue}{{b}^{4}} \]

   11. Applied rewrites 87.1%

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

   13. Applied rewrites 87.1%

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

Alternative 15: 69.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+296) (fma 4.0 (* a a) -1.0) (fma (* b b) 4.0 -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+296) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = fma((b * b), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+296)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = fma(Float64(b * b), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+296], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e296

   1. Initial program 75.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 58.4

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

   5. Applied rewrites 58.4%

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

   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. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. distribute-lft-in N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

   8. Applied rewrites 78.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 60.9%

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

   if 5.0000000000000001e296 < (*.f64 b b)

   1. Initial program 65.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   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 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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 97.1%

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

Alternative 16: 51.4% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
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 68.3

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

5. Applied rewrites 68.3%

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

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. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 68.3

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

8. Applied rewrites 68.3%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 47.8%

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

Alternative 17: 24.6% 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 73.4%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
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 68.3

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

5. Applied rewrites 68.3%

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

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. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 68.3

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

8. Applied rewrites 68.3%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 23.0%

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

Reproduce

herbie shell --seed 2024340 
(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))