Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 9.5s

Alternatives: 10

Speedup: N/A×

• 61.2% 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, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \left(\left(b \cdot b\right) \cdot 4 + \frac{t\_0}{\frac{1}{t\_0}}\right) - 1 \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (- (+ (* (* b b) 4.0) (/ t_0 (/ 1.0 t_0))) 1.0)))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	return (((b * b) * 4.0) + (t_0 / (1.0 / t_0))) - 1.0;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	return Float64(Float64(Float64(Float64(b * b) * 4.0) + Float64(t_0 / Float64(1.0 / t_0))) - 1.0)
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[(t$95$0 / N[(1.0 / t$95$0), $MachinePrecision]), $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-pow.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. unpow2 N/A

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

   3. lift-+.f64 N/A

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

   4. flip-+ N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. clear-num N/A

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

   13. flip-+ N/A

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

   14. lift-+.f64 N/A

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

   15. lower-/.f64 99.9

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 68.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + \left(b \cdot b\right) \cdot 4 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (+ (pow (+ (* b b) (* a a)) 2.0) (* (* b b) 4.0)) 2e-19)
   -1.0
   (* (* (* a a) a) a)))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((b * b) + (a * a)), 2.0) + ((b * b) * 4.0)) <= 2e-19) {
		tmp = -1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((b * b) + (a * a)) ** 2.0d0) + ((b * b) * 4.0d0)) <= 2d-19) then
        tmp = -1.0d0
    else
        tmp = ((a * a) * a) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((b * b) + (a * a)), 2.0) + ((b * b) * 4.0)) <= 2e-19) {
		tmp = -1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((b * b) + (a * a)), 2.0) + ((b * b) * 4.0)) <= 2e-19:
		tmp = -1.0
	else:
		tmp = ((a * a) * a) * a
	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)) <= 2e-19)
		tmp = -1.0;
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((b * b) + (a * a)) ^ 2.0) + ((b * b) * 4.0)) <= 2e-19)
		tmp = -1.0;
	else
		tmp = ((a * a) * a) * a;
	end
	tmp_2 = 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], 2e-19], -1.0, N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $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))) < 2e-19

   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 b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto -1 \]

   if 2e-19 < (+.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 99.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + \left(b \cdot b\right) \cdot 4 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + \left(b \cdot b\right) \cdot 4 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (+ (pow (+ (* b b) (* a a)) 2.0) (* (* b b) 4.0)) 2e-19)
   -1.0
   (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((b * b) + (a * a)), 2.0) + ((b * b) * 4.0)) <= 2e-19) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((b * b) + (a * a)) ** 2.0d0) + ((b * b) * 4.0d0)) <= 2d-19) then
        tmp = -1.0d0
    else
        tmp = (a * a) * (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((b * b) + (a * a)), 2.0) + ((b * b) * 4.0)) <= 2e-19) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((b * b) + (a * a)), 2.0) + ((b * b) * 4.0)) <= 2e-19:
		tmp = -1.0
	else:
		tmp = (a * a) * (a * a)
	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)) <= 2e-19)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((b * b) + (a * a)) ^ 2.0) + ((b * b) * 4.0)) <= 2e-19)
		tmp = -1.0;
	else
		tmp = (a * a) * (a * a);
	end
	tmp_2 = 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], 2e-19], -1.0, N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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))) < 2e-19

   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 b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto -1 \]

   if 2e-19 < (+.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 99.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   7. Applied rewrites 64.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + \left(b \cdot b\right) \cdot 4 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(a - b, \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \frac{1}{a}, \mathsf{fma}\left(b \cdot b, 4, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(b, b, a \cdot a\right)}{\frac{1}{b \cdot b}} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+33)
   (fma (- a b) (* (* (* (* a a) a) a) (/ 1.0 a)) (fma (* b b) 4.0 -1.0))
   (- (+ (/ (fma b b (* a a)) (/ 1.0 (* b b))) (* (* b b) 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+33) {
		tmp = fma((a - b), ((((a * a) * a) * a) * (1.0 / a)), fma((b * b), 4.0, -1.0));
	} else {
		tmp = ((fma(b, b, (a * a)) / (1.0 / (b * b))) + ((b * b) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+33)
		tmp = fma(Float64(a - b), Float64(Float64(Float64(Float64(a * a) * a) * a) * Float64(1.0 / a)), fma(Float64(b * b), 4.0, -1.0));
	else
		tmp = Float64(Float64(Float64(fma(b, b, Float64(a * a)) / Float64(1.0 / Float64(b * b))) + Float64(Float64(b * b) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+33], N[(N[(a - b), $MachinePrecision] * N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999999e33

   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-pow.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. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. clear-num N/A

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

      13. flip-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.9

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

   9. Applied rewrites 97.9%

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

   10. Taylor expanded in b around 0

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

       1. lower-/.f64 97.9

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

   12. Applied rewrites 97.9%

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

   if 1.9999999999999999e33 < (*.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. Step-by-step derivation

      1. lift-pow.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. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. clear-num N/A

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

      13. flip-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.9

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

   7. Applied rewrites 96.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(a - b, \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \frac{1}{a}, \mathsf{fma}\left(b \cdot b, 4, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(b, b, a \cdot a\right)}{\frac{1}{b \cdot b}} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(b, b, a \cdot a\right)}{\frac{1}{a \cdot a}} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+125)
   (- (+ (/ (fma b b (* a a)) (/ 1.0 (* a a))) (* (* b b) 4.0)) 1.0)
   (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+125) {
		tmp = ((fma(b, b, (a * a)) / (1.0 / (a * a))) + ((b * b) * 4.0)) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+125)
		tmp = Float64(Float64(Float64(fma(b, b, Float64(a * a)) / Float64(1.0 / Float64(a * a))) + Float64(Float64(b * b) * 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+125], N[(N[(N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.9999999999999997e125

   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-pow.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. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. clear-num N/A

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

      13. flip-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.3

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

   7. Applied rewrites 92.3%

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

   if 3.9999999999999997e125 < (*.f64 b b)

   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 b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

      10. lower-*.f64 100.0

          \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

   5. Applied rewrites 100.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(b, b, a \cdot a\right)}{\frac{1}{a \cdot a}} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{a \cdot a}{\frac{1}{a \cdot a}} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+125)
   (- (+ (/ (* a a) (/ 1.0 (* a a))) (* (* b b) 4.0)) 1.0)
   (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+125) {
		tmp = (((a * a) / (1.0 / (a * a))) + ((b * b) * 4.0)) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+125) then
        tmp = (((a * a) / (1.0d0 / (a * a))) + ((b * b) * 4.0d0)) - 1.0d0
    else
        tmp = ((b * b) * b) * b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+125) {
		tmp = (((a * a) / (1.0 / (a * a))) + ((b * b) * 4.0)) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+125:
		tmp = (((a * a) / (1.0 / (a * a))) + ((b * b) * 4.0)) - 1.0
	else:
		tmp = ((b * b) * b) * b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+125)
		tmp = Float64(Float64(Float64(Float64(a * a) / Float64(1.0 / Float64(a * a))) + Float64(Float64(b * b) * 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+125)
		tmp = (((a * a) / (1.0 / (a * a))) + ((b * b) * 4.0)) - 1.0;
	else
		tmp = ((b * b) * b) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+125], N[(N[(N[(N[(a * a), $MachinePrecision] / N[(1.0 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.9999999999999997e125

   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-pow.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. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. clear-num N/A

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

      13. flip-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.3

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.3

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

   10. Applied rewrites 92.3%

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

   if 3.9999999999999997e125 < (*.f64 b b)

   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 b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

      10. lower-*.f64 100.0

          \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

   5. Applied rewrites 100.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{a \cdot a}{\frac{1}{a \cdot a}} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8.5 \cdot 10^{-246}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 8.5e-246)
   (* (* (* b b) b) b)
   (if (<= a 1.0) -1.0 (* (* (* a a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 8.5e-246) {
		tmp = ((b * b) * b) * b;
	} else if (a <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 8.5d-246) then
        tmp = ((b * b) * b) * b
    else if (a <= 1.0d0) then
        tmp = -1.0d0
    else
        tmp = ((a * a) * a) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 8.5e-246) {
		tmp = ((b * b) * b) * b;
	} else if (a <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 8.5e-246:
		tmp = ((b * b) * b) * b
	elif a <= 1.0:
		tmp = -1.0
	else:
		tmp = ((a * a) * a) * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 8.5e-246)
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	elseif (a <= 1.0)
		tmp = -1.0;
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 8.5e-246)
		tmp = ((b * b) * b) * b;
	elseif (a <= 1.0)
		tmp = -1.0;
	else
		tmp = ((a * a) * a) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 8.5e-246], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[a, 1.0], -1.0, N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 8.4999999999999998e-246

   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 b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

      10. lower-*.f64 41.7

          \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \]

   if 8.4999999999999998e-246 < a < 1

   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 b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 51.4%

      \[\leadsto -1 \]

   if 1 < a

   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 inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.7

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

   5. Applied rewrites 88.7%

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

Alternative 8: 93.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+125) (fma (* (* a a) a) a -1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+125) {
		tmp = fma(((a * a) * a), a, -1.0);
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+125)
		tmp = fma(Float64(Float64(a * a) * a), a, -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+125], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.9999999999999997e125

   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 b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 91.9%

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

   if 3.9999999999999997e125 < (*.f64 b b)

   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 b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

      10. lower-*.f64 100.0

          \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

   5. Applied rewrites 100.0%

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

Alternative 9: 93.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+125) (fma (* a a) (* a a) -1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+125) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+125)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+125], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.9999999999999997e125

   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 b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 91.8

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

   5. Applied rewrites 91.8%

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

   if 3.9999999999999997e125 < (*.f64 b b)

   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 b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

      10. lower-*.f64 100.0

          \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) \cdot b \]

   5. Applied rewrites 100.0%

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

Alternative 10: 25.2% 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 b around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 72.9

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

5. Applied rewrites 72.9%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 25.5%

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

Reproduce

herbie shell --seed 2024237 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))