Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 10.8s

Alternatives: 10

Speedup: 3.3×

• 61.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + ((b * b) * 4.0)) + -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[(N[(b * b), $MachinePrecision] * 4.0), $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. Final simplification 99.9%

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

Alternative 2: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (+ (pow (+ (* a a) (* b b)) 2.0) (* (* b b) 4.0)) 5e-7)
   -1.0
   (* a (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + ((b * b) * 4.0)) <= 5e-7) {
		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 * a) + (b * b)) ** 2.0d0) + ((b * b) * 4.0d0)) <= 5d-7) 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(((a * a) + (b * b)), 2.0) + ((b * b) * 4.0)) <= 5e-7) {
		tmp = -1.0;
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((a * a) + (b * b)), 2.0) + ((b * b) * 4.0)) <= 5e-7:
		tmp = -1.0
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(Float64(b * b) * 4.0)) <= 5e-7)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((a * a) + (b * b)) ^ 2.0) + ((b * b) * 4.0)) <= 5e-7)
		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[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], 5e-7], -1.0, N[(a * N[(a * N[(a * a), $MachinePrecision]), $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))) < 4.99999999999999977e-7

   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

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 98.8%

      \[\leadsto -1 \]

   if 4.99999999999999977e-7 < (+.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 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. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \cdot a \leq 5 \cdot 10^{-36}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \cdot a \leq 5 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) (* b b))))
   (if (<= (* a a) 5e-202)
     t_0
     (if (<= (* a a) 5e-36)
       -1.0
       (if (<= (* a a) 5e+41) t_0 (* a (* a (* a a))))))))

C

double code(double a, double b) {
	double t_0 = (b * b) * (b * b);
	double tmp;
	if ((a * a) <= 5e-202) {
		tmp = t_0;
	} else if ((a * a) <= 5e-36) {
		tmp = -1.0;
	} else if ((a * a) <= 5e+41) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (b * b) * (b * b)
    if ((a * a) <= 5d-202) then
        tmp = t_0
    else if ((a * a) <= 5d-36) then
        tmp = -1.0d0
    else if ((a * a) <= 5d+41) then
        tmp = t_0
    else
        tmp = a * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (b * b) * (b * b);
	double tmp;
	if ((a * a) <= 5e-202) {
		tmp = t_0;
	} else if ((a * a) <= 5e-36) {
		tmp = -1.0;
	} else if ((a * a) <= 5e+41) {
		tmp = t_0;
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b * b) * (b * b)
	tmp = 0
	if (a * a) <= 5e-202:
		tmp = t_0
	elif (a * a) <= 5e-36:
		tmp = -1.0
	elif (a * a) <= 5e+41:
		tmp = t_0
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * Float64(b * b))
	tmp = 0.0
	if (Float64(a * a) <= 5e-202)
		tmp = t_0;
	elseif (Float64(a * a) <= 5e-36)
		tmp = -1.0;
	elseif (Float64(a * a) <= 5e+41)
		tmp = t_0;
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b * b) * (b * b);
	tmp = 0.0;
	if ((a * a) <= 5e-202)
		tmp = t_0;
	elseif ((a * a) <= 5e-36)
		tmp = -1.0;
	elseif ((a * a) <= 5e+41)
		tmp = t_0;
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * a), $MachinePrecision], 5e-202], t$95$0, If[LessEqual[N[(a * a), $MachinePrecision], 5e-36], -1.0, If[LessEqual[N[(a * a), $MachinePrecision], 5e+41], t$95$0, N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a a) < 4.99999999999999973e-202 or 5.00000000000000004e-36 < (*.f64 a a) < 5.00000000000000022e41

   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

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   6. 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. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 61.9

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

   7. Applied rewrites 61.9%

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

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   9. 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. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.9

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

   10. Applied rewrites 61.9%

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

   if 4.99999999999999973e-202 < (*.f64 a a) < 5.00000000000000004e-36

   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

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 63.1%

      \[\leadsto -1 \]

   if 5.00000000000000022e41 < (*.f64 a a)

   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 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. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

Alternative 4: 69.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \cdot a \leq 5 \cdot 10^{-36}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \cdot a \leq 5 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b (* b b)))))
   (if (<= (* a a) 5e-202)
     t_0
     (if (<= (* a a) 5e-36)
       -1.0
       (if (<= (* a a) 5e+41) t_0 (* a (* a (* a a))))))))

C

double code(double a, double b) {
	double t_0 = b * (b * (b * b));
	double tmp;
	if ((a * a) <= 5e-202) {
		tmp = t_0;
	} else if ((a * a) <= 5e-36) {
		tmp = -1.0;
	} else if ((a * a) <= 5e+41) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = b * (b * (b * b))
    if ((a * a) <= 5d-202) then
        tmp = t_0
    else if ((a * a) <= 5d-36) then
        tmp = -1.0d0
    else if ((a * a) <= 5d+41) then
        tmp = t_0
    else
        tmp = a * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * (b * b));
	double tmp;
	if ((a * a) <= 5e-202) {
		tmp = t_0;
	} else if ((a * a) <= 5e-36) {
		tmp = -1.0;
	} else if ((a * a) <= 5e+41) {
		tmp = t_0;
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * (b * b))
	tmp = 0
	if (a * a) <= 5e-202:
		tmp = t_0
	elif (a * a) <= 5e-36:
		tmp = -1.0
	elif (a * a) <= 5e+41:
		tmp = t_0
	else:
		tmp = a * (a * (a * a))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * Float64(b * b)))
	tmp = 0.0
	if (Float64(a * a) <= 5e-202)
		tmp = t_0;
	elseif (Float64(a * a) <= 5e-36)
		tmp = -1.0;
	elseif (Float64(a * a) <= 5e+41)
		tmp = t_0;
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * (b * b));
	tmp = 0.0;
	if ((a * a) <= 5e-202)
		tmp = t_0;
	elseif ((a * a) <= 5e-36)
		tmp = -1.0;
	elseif ((a * a) <= 5e+41)
		tmp = t_0;
	else
		tmp = a * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * a), $MachinePrecision], 5e-202], t$95$0, If[LessEqual[N[(a * a), $MachinePrecision], 5e-36], -1.0, If[LessEqual[N[(a * a), $MachinePrecision], 5e+41], t$95$0, N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a a) < 4.99999999999999973e-202 or 5.00000000000000004e-36 < (*.f64 a a) < 5.00000000000000022e41

   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. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   if 4.99999999999999973e-202 < (*.f64 a a) < 5.00000000000000004e-36

   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

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 63.1%

      \[\leadsto -1 \]

   if 5.00000000000000022e41 < (*.f64 a a)

   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 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. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

Alternative 5: 97.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(a \cdot a\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(2, a \cdot a, 4\right)\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-18)
   (fma a (* a (* a a)) -1.0)
   (fma (* b b) (fma b b (fma 2.0 (* a a) 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-18) {
		tmp = fma(a, (a * (a * a)), -1.0);
	} else {
		tmp = fma((b * b), fma(b, b, fma(2.0, (a * a), 4.0)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-18)
		tmp = fma(a, Float64(a * Float64(a * a)), -1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, fma(2.0, Float64(a * a), 4.0)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-18], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(2.0 * N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.0000000000000001e-18

   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. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.0000000000000001e-18 < (*.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

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 96.0%

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

Alternative 6: 99.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 7: 94.3% accurate, 4.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5.0)
   (fma b (* b (fma b b 4.0)) -1.0)
   (fma a (* a (* a a)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5.0) {
		tmp = fma(b, (b * fma(b, b, 4.0)), -1.0);
	} else {
		tmp = fma(a, (a * (a * a)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5.0)
		tmp = fma(b, Float64(b * fma(b, b, 4.0)), -1.0);
	else
		tmp = fma(a, Float64(a * Float64(a * a)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5.0], N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 5

   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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-lft-out N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   if 5 < (*.f64 a a)

   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. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 90.4

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

   5. Applied rewrites 90.4%

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

Alternative 8: 93.8% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5.0) (fma (* b b) (* b b) -1.0) (fma a (* a (* a a)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5.0) {
		tmp = fma((b * b), (b * b), -1.0);
	} else {
		tmp = fma(a, (a * (a * a)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5.0)
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	else
		tmp = fma(a, Float64(a * Float64(a * a)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5.0], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 5

   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

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 98.5%

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

   if 5 < (*.f64 a a)

   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. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 90.4

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

   5. Applied rewrites 90.4%

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

Alternative 9: 93.3% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+78) (fma a (* a (* a a)) -1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+78) {
		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+78)
		tmp = fma(a, Float64(a * Float64(a * a)), -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+78], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000003e78

   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. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 93.6

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

   5. Applied rewrites 93.6%

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

   if 4.00000000000000003e78 < (*.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--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   6. 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. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 93.8

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   9. 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. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 93.8

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

   10. Applied rewrites 93.8%

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

Alternative 10: 24.4% 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

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

   1. associate-+r- N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. +-commutative N/A

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

   7. sub-neg N/A

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

   8. associate-+l+ N/A

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

   9. metadata-eval N/A

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

   10. pow-sqr N/A

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

   11. distribute-lft-in N/A

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

   12. associate-+r+ N/A

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

   13. lower-fma.f64 N/A

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

5. Applied rewrites 81.2%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 22.2%

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

Reproduce

herbie shell --seed 2024232 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))