Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.4% → 98.3%

Time: 8.1s

Alternatives: 14

Speedup: 5.3×

• 61.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.3%

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

Alternative 2: 51.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) \leq 0.02:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
      0.02)
   -1.0
   (* (* b b) 12.0)))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) <= 0.02) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) <= 0.02d0) then
        tmp = -1.0d0
    else
        tmp = (b * b) * 12.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) <= 0.02) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) <= 0.02:
		tmp = -1.0
	else:
		tmp = (b * b) * 12.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) <= 0.02)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * 12.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) <= 0.02)
		tmp = -1.0;
	else
		tmp = (b * b) * 12.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], -1.0, N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.6%

      \[\leadsto -1 \]

   if 0.0200000000000000004 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 65.7%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 59.4

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in b around 0

      \[\leadsto 12 \cdot {b}^{2} - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 33.2%

       \[\leadsto \left(b \cdot b\right) \cdot 12 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, a, a\right) \cdot a, 4, {a}^{4}\right) - 1\\ \mathbf{elif}\;a \leq 520000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.7e-6)
   (- (fma (* (fma (- a) a a) a) 4.0 (pow a 4.0)) 1.0)
   (if (<= a 520000000000.0)
     (- (fma (* b b) 12.0 (pow b 4.0)) 1.0)
     (fma (* (fma (- a 4.0) a 4.0) a) a -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-6) {
		tmp = fma((fma(-a, a, a) * a), 4.0, pow(a, 4.0)) - 1.0;
	} else if (a <= 520000000000.0) {
		tmp = fma((b * b), 12.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.7e-6)
		tmp = Float64(fma(Float64(fma(Float64(-a), a, a) * a), 4.0, (a ^ 4.0)) - 1.0);
	elseif (a <= 520000000000.0)
		tmp = Float64(fma(Float64(b * b), 12.0, (b ^ 4.0)) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.7e-6], N[(N[(N[(N[((-a) * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 520000000000.0], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7000000000000002e-6

   1. Initial program 66.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-pow.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if -3.7000000000000002e-6 < a < 5.2e11

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.2e11 < a

   1. Initial program 19.9%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.2%

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

   10. Applied rewrites 98.2%

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

Alternative 4: 94.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6} \lor \neg \left(a \leq 520000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.7e-6) (not (<= a 520000000000.0)))
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (- (fma (* b b) 12.0 (pow b 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.7e-6) || !(a <= 520000000000.0)) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma((b * b), 12.0, pow(b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.7e-6) || !(a <= 520000000000.0))
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = Float64(fma(Float64(b * b), 12.0, (b ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.7e-6], N[Not[LessEqual[a, 520000000000.0]], $MachinePrecision]], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7000000000000002e-6 or 5.2e11 < a

   1. Initial program 45.4%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.3%

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

   10. Applied rewrites 95.3%

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

   if -3.7000000000000002e-6 < a < 5.2e11

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6} \lor \neg \left(a \leq 520000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-4, a, 4\right), a \cdot a, {a}^{4}\right) - 1\\ \mathbf{elif}\;a \leq 520000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.7e-6)
   (- (fma (fma -4.0 a 4.0) (* a a) (pow a 4.0)) 1.0)
   (if (<= a 520000000000.0)
     (- (fma (* b b) 12.0 (pow b 4.0)) 1.0)
     (fma (* (fma (- a 4.0) a 4.0) a) a -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-6) {
		tmp = fma(fma(-4.0, a, 4.0), (a * a), pow(a, 4.0)) - 1.0;
	} else if (a <= 520000000000.0) {
		tmp = fma((b * b), 12.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.7e-6)
		tmp = Float64(fma(fma(-4.0, a, 4.0), Float64(a * a), (a ^ 4.0)) - 1.0);
	elseif (a <= 520000000000.0)
		tmp = Float64(fma(Float64(b * b), 12.0, (b ^ 4.0)) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.7e-6], N[(N[(N[(-4.0 * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 520000000000.0], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7000000000000002e-6

   1. Initial program 66.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 32.8%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. neg-mul-1 N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 92.9

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

   8. Applied rewrites 92.9%

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

   if -3.7000000000000002e-6 < a < 5.2e11

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.2e11 < a

   1. Initial program 19.9%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.2%

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

   10. Applied rewrites 98.2%

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

Alternative 6: 94.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6} \lor \neg \left(a \leq 520000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.7e-6) (not (<= a 520000000000.0)))
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (fma (* b b) (fma b b 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.7e-6) || !(a <= 520000000000.0)) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.7e-6) || !(a <= 520000000000.0))
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.7e-6], N[Not[LessEqual[a, 520000000000.0]], $MachinePrecision]], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7000000000000002e-6 or 5.2e11 < a

   1. Initial program 45.4%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.3%

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

   10. Applied rewrites 95.3%

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

   if -3.7000000000000002e-6 < a < 5.2e11

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6} \lor \neg \left(a \leq 520000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6} \lor \neg \left(a \leq 520000000000\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.7e-6) (not (<= a 520000000000.0)))
   (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)
   (fma (* b b) (fma b b 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.7e-6) || !(a <= 520000000000.0)) {
		tmp = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	} else {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.7e-6) || !(a <= 520000000000.0))
		tmp = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.7e-6], N[Not[LessEqual[a, 520000000000.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7000000000000002e-6 or 5.2e11 < a

   1. Initial program 45.4%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.3%

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

   if -3.7000000000000002e-6 < a < 5.2e11

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-6} \lor \neg \left(a \leq 520000000000\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+20} \lor \neg \left(a \leq 2.45 \cdot 10^{+17}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2e+20) (not (<= a 2.45e+17)))
   (- (* (* a a) (* a a)) 1.0)
   (fma (* b b) (fma b b 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2e+20) || !(a <= 2.45e+17)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2e+20) || !(a <= 2.45e+17))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2e+20], N[Not[LessEqual[a, 2.45e+17]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e20 or 2.45e17 < a

   1. Initial program 42.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 -2e20 < a < 2.45e17

   1. Initial program 97.8%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 96.1

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

   5. Applied rewrites 96.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+20} \lor \neg \left(a \leq 2.45 \cdot 10^{+17}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+78)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 6.5e+143)
     (fma (* b b) (fma b b 12.0) -1.0)
     (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+78) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 6.5e+143) {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+78)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 6.5e+143)
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+78], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 6.5e+143], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000002e78

   1. Initial program 65.4%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.3%

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

   if -2.00000000000000002e78 < a < 6.4999999999999997e143

   1. Initial program 93.3%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 6.4999999999999997e143 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 93.0%

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

Alternative 10: 84.4% accurate, 5.3× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2e+154) (not (<= a 6.5e+143)))
   (fma (* a a) 4.0 -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((a * a), 4.0, -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(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(Float64(b * b), Float64(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[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 29.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.3%

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

   if -2.00000000000000007e154 < a < 6.4999999999999997e143

   1. Initial program 91.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 81.3%

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

4. Final simplification 85.5%

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

Alternative 11: 86.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+78)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 6.5e+143) (fma (* b b) (* b b) -1.0) (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+78) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 6.5e+143) {
		tmp = fma((b * b), (b * b), -1.0);
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+78)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 6.5e+143)
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+78], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 6.5e+143], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000002e78

   1. Initial program 65.4%

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

   3. Taylor expanded in 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. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.3%

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

   if -2.00000000000000002e78 < a < 6.4999999999999997e143

   1. Initial program 93.3%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 86.4%

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

   if 6.4999999999999997e143 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 93.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4e+148) (fma (* a a) 4.0 -1.0) (* (* b b) 12.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4e+148) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4e+148)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(Float64(b * b) * 12.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4e+148], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.0000000000000002e148

   1. Initial program 75.5%

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

   3. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 55.6%

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

   if 4.0000000000000002e148 < b

   1. Initial program 59.3%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto 12 \cdot {b}^{2} - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 96.7%

       \[\leadsto \left(b \cdot b\right) \cdot 12 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(b \cdot b, 12, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma (* b b) 12.0 -1.0))

C

double code(double a, double b) {
	return fma((b * b), 12.0, -1.0);
}

Julia

function code(a, b)
	return fma(Float64(b * b), 12.0, -1.0)
end

Wolfram

code[a_, b_] := N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]

Derivation

1. Initial program 73.8%

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

3. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-fma.f64 68.3

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

5. Applied rewrites 68.3%

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

6. Taylor expanded in b around 0

   \[\leadsto 12 \cdot {b}^{2} - \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 47.8%

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

Alternative 14: 24.6% accurate, 155.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -1.0d0
end function

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0;
end

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 73.8%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
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. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 68.2

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

5. Applied rewrites 68.2%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 23.0%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))