Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 5.9s

Alternatives: 12

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1 \]
6. Add Preprocessing

Alternative 2: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + \left(b \cdot b\right) \cdot 4 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (+ (pow (+ (* b b) (* a a)) 2.0) (* (* b b) 4.0)) 0.01)
   (fma (* b b) 4.0 -1.0)
   (* (* (fma b b 4.0) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((b * b) + (a * a)), 2.0) + ((b * b) * 4.0)) <= 0.01) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = (fma(b, b, 4.0) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(Float64(b * b) * 4.0)) <= 0.01)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = Float64(Float64(fma(b, b, 4.0) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $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 b b))) < 0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.8%

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

   if 0.0100000000000000002 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (*.f64 b b)))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 64.0%

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

   10. Applied rewrites 63.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + \left(b \cdot b\right) \cdot 4 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot 4\\ \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + t\_0 \leq 0.01:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) 4.0)))
   (if (<= (+ (pow (+ (* b b) (* a a)) 2.0) t_0) 0.01) -1.0 t_0)))

C

double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if ((pow(((b * b) + (a * a)), 2.0) + t_0) <= 0.01) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) * 4.0d0
    if (((((b * b) + (a * a)) ** 2.0d0) + t_0) <= 0.01d0) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if ((Math.pow(((b * b) + (a * a)), 2.0) + t_0) <= 0.01) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b * b) * 4.0
	tmp = 0
	if (math.pow(((b * b) + (a * a)), 2.0) + t_0) <= 0.01:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * 4.0)
	tmp = 0.0
	if (Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + t_0) <= 0.01)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b * b) * 4.0;
	tmp = 0.0;
	if (((((b * b) + (a * a)) ^ 2.0) + t_0) <= 0.01)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$0), $MachinePrecision], 0.01], -1.0, t$95$0]]

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 b b))) < 0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.4%

      \[\leadsto -1 \]

   if 0.0100000000000000002 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (*.f64 b b)))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 64.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 36.6%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + \left(b \cdot b\right) \cdot 4 \leq 0.01:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-8}:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(2, a \cdot a, b \cdot b\right) \cdot b\right) \cdot b - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-8)
   (- (pow a 4.0) 1.0)
   (fma (* b b) 4.0 (- (* (* (fma 2.0 (* a a) (* b b)) b) b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-8) {
		tmp = pow(a, 4.0) - 1.0;
	} else {
		tmp = fma((b * b), 4.0, (((fma(2.0, (a * a), (b * b)) * b) * b) - 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-8)
		tmp = Float64((a ^ 4.0) - 1.0);
	else
		tmp = fma(Float64(b * b), 4.0, Float64(Float64(Float64(fma(2.0, Float64(a * a), Float64(b * b)) * b) * b) - 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-8], N[(N[Power[a, 4.0], $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(2.0 * N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 4.9999999999999998e-8 < (*.f64 b b)

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 96.9

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

   7. Applied rewrites 96.9%

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

Alternative 5: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(2, a \cdot a, b \cdot b\right) \cdot b\right) \cdot b - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-8)
   (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0)
   (fma (* b b) 4.0 (- (* (* (fma 2.0 (* a a) (* b b)) b) b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-8) {
		tmp = ((fma((b * b), 2.0, (a * a)) * a) * a) - 1.0;
	} else {
		tmp = fma((b * b), 4.0, (((fma(2.0, (a * a), (b * b)) * b) * b) - 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-8)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a) - 1.0);
	else
		tmp = fma(Float64(b * b), 4.0, Float64(Float64(Float64(fma(2.0, Float64(a * a), Float64(b * b)) * b) * b) - 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-8], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(2.0 * N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999998e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 100.0%

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

   if 4.9999999999999998e-8 < (*.f64 b b)

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 96.9

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

   7. Applied rewrites 96.9%

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

Alternative 6: 98.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a \cdot a, 2, 4\right)\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-8)
   (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0)
   (fma (* (fma b b (fma (* a a) 2.0 4.0)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-8) {
		tmp = ((fma((b * b), 2.0, (a * a)) * a) * a) - 1.0;
	} else {
		tmp = fma((fma(b, b, fma((a * a), 2.0, 4.0)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-8)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, fma(Float64(a * a), 2.0, 4.0)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-8], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + N[(N[(a * a), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999998e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 100.0%

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

   if 4.9999999999999998e-8 < (*.f64 b b)

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 96.9

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   10. Applied rewrites 96.9%

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

Alternative 7: 86.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \left(a \cdot a\right)\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 4e+89)
   (fma (* (fma b b 4.0) b) b -1.0)
   (fma (* (* 2.0 (* a a)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 4e+89) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma(((2.0 * (a * a)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 4e+89)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(Float64(2.0 * Float64(a * a)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 4e+89], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 3.99999999999999998e89

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if 3.99999999999999998e89 < (*.f64 a a)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   10. Applied rewrites 77.3%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 77.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \left(a \cdot a\right)\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a \cdot a, 2, 4\right)\right) \cdot b, b, -1\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (fma (* (fma b b (fma (* a a) 2.0 4.0)) b) b -1.0))

C

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

Julia

function code(a, b)
	return fma(Float64(fma(b, b, fma(Float64(a * a), 2.0, 4.0)) * b), b, -1.0)
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 86.9

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

5. Applied rewrites 86.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 86.9

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

7. Applied rewrites 86.9%

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

8. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. pow-sqr N/A

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

   8. distribute-lft-in N/A

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

   9. associate-+r+ N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

10. Applied rewrites 86.9%

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

Alternative 9: 70.4% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-rgt-in N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. lower-fma.f64 70.6

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

7. Applied rewrites 70.6%

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

Alternative 10: 70.4% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 70.6

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

5. Applied rewrites 70.6%

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

Alternative 11: 51.7% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 70.6

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

5. Applied rewrites 70.6%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 48.4%

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

Alternative 12: 25.1% accurate, 131.0× 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 99.9%

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

3. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 70.6

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

5. Applied rewrites 70.6%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 20.0%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))