Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.4% → 99.9%

Time: 10.3s

Alternatives: 14

Speedup: 5.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, -a, 4\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0)
   (+ (* (* a a) (fma a (+ a -4.0) 4.0)) -1.0)
   (+
    (fma
     (* b b)
     (fma b b (fma a (fma 2.0 a 4.0) 12.0))
     (* (* a a) (fma a a (fma 4.0 (- a) 4.0))))
    -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0) {
		tmp = ((a * a) * fma(a, (a + -4.0), 4.0)) + -1.0;
	} else {
		tmp = fma((b * b), fma(b, b, fma(a, fma(2.0, a, 4.0), 12.0)), ((a * a) * fma(a, a, fma(4.0, -a, 4.0)))) + -1.0;
	}
	return tmp;
}

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0], N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(a * N[(2.0 * a + 4.0), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(4.0 * (-a) + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0

   1. Initial program 74.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 68.1%

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

   6. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 100.0

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

   8. Simplified 100.0%

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

   if 0.0 < (*.f64 b b)

   1. Initial program 72.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, -a, 4\right)\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* b b) (* a a)) 2.0)
       (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
      4e-5)
   -1.0
   (* b (* b 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 4e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((b * b) + (a * a)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (a + 3.0d0))))) <= 4d-5) then
        tmp = -1.0d0
    else
        tmp = b * (b * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 4e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 4e-5:
		tmp = -1.0
	else:
		tmp = b * (b * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0))))) <= 4e-5)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((b * b) + (a * a)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 4e-5)
		tmp = -1.0;
	else
		tmp = b * (b * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], -1.0, N[(b * N[(b * 12.0), $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 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 4.00000000000000033e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 98.2%

      \[\leadsto \color{blue}{-1} \]

   if 4.00000000000000033e-5 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 61.2%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 56.6

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

   5. Simplified 56.6%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 36.5

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

   8. Simplified 36.5%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

          \[\leadsto 12 \cdot \color{blue}{\left(b \cdot b\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(12 \cdot b\right) \cdot b} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \left(12 \cdot b\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(12 \cdot b\right)} \]

       5. *-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(b \cdot 12\right)} \]

       6. lower-*.f64 37.0

          \[\leadsto b \cdot \color{blue}{\left(b \cdot 12\right)} \]

   11. Simplified 37.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-72)
   (+ (* (* a a) (fma a (+ a -4.0) 4.0)) -1.0)
   (+
    (fma (* b b) (fma b b (fma a (fma 2.0 a 4.0) 12.0)) (* (* a a) (* a a)))
    -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-72) {
		tmp = ((a * a) * fma(a, (a + -4.0), 4.0)) + -1.0;
	} else {
		tmp = fma((b * b), fma(b, b, fma(a, fma(2.0, a, 4.0), 12.0)), ((a * a) * (a * a))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-72)
		tmp = Float64(Float64(Float64(a * a) * fma(a, Float64(a + -4.0), 4.0)) + -1.0);
	else
		tmp = Float64(fma(Float64(b * b), fma(b, b, fma(a, fma(2.0, a, 4.0), 12.0)), Float64(Float64(a * a) * Float64(a * a))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-72], N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(a * N[(2.0 * a + 4.0), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999996e-72

   1. Initial program 81.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 83.2%

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

   6. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.9

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

   8. Simplified 99.9%

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

   if 4.9999999999999996e-72 < (*.f64 b b)

   1. Initial program 63.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \left(a \cdot a\right) \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0004)
   (+ (* (* a a) (fma a (+ a -4.0) 4.0)) -1.0)
   (+
    (fma (* b b) (fma b b (fma a (fma 2.0 a 4.0) 12.0)) (* (* a a) 4.0))
    -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0004) {
		tmp = ((a * a) * fma(a, (a + -4.0), 4.0)) + -1.0;
	} else {
		tmp = fma((b * b), fma(b, b, fma(a, fma(2.0, a, 4.0), 12.0)), ((a * a) * 4.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0004)
		tmp = Float64(Float64(Float64(a * a) * fma(a, Float64(a + -4.0), 4.0)) + -1.0);
	else
		tmp = Float64(fma(Float64(b * b), fma(b, b, fma(a, fma(2.0, a, 4.0), 12.0)), Float64(Float64(a * a) * 4.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0004], N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(a * N[(2.0 * a + 4.0), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000019e-4

   1. Initial program 82.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 85.0%

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

   6. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.7

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

   8. Simplified 99.7%

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

   if 4.00000000000000019e-4 < (*.f64 b b)

   1. Initial program 60.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 96.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \left(a \cdot a\right) \cdot 4\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, 4\right), \mathsf{fma}\left(b, b, 12\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0004)
   (+ (* (* a a) (fma a (+ a -4.0) 4.0)) -1.0)
   (+ (* (* b b) (fma a (fma a 2.0 4.0) (fma b b 12.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0004) {
		tmp = ((a * a) * fma(a, (a + -4.0), 4.0)) + -1.0;
	} else {
		tmp = ((b * b) * fma(a, fma(a, 2.0, 4.0), fma(b, b, 12.0))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0004)
		tmp = Float64(Float64(Float64(a * a) * fma(a, Float64(a + -4.0), 4.0)) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * fma(a, fma(a, 2.0, 4.0), fma(b, b, 12.0))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0004], N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(a * N[(a * 2.0 + 4.0), $MachinePrecision] + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000019e-4

   1. Initial program 82.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 85.0%

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

   6. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.7

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

   8. Simplified 99.7%

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

   if 4.00000000000000019e-4 < (*.f64 b b)

   1. Initial program 60.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

   9. Taylor expanded in b around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. distribute-lft-in N/A

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

       5. *-rgt-identity N/A

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

       6. +-commutative N/A

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

       7. distribute-lft-in N/A

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

   11. Simplified 96.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0004:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, 4\right), \mathsf{fma}\left(b, b, 12\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot 2\right) + -1\\ \mathbf{if}\;a \leq -0.0022:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 16000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* (* a a) (fma a a (* (* b b) 2.0))) -1.0)))
   (if (<= a -0.0022)
     t_0
     (if (<= a 16000000.0) (fma (fma b b 12.0) (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = ((a * a) * fma(a, a, ((b * b) * 2.0))) + -1.0;
	double tmp;
	if (a <= -0.0022) {
		tmp = t_0;
	} else if (a <= 16000000.0) {
		tmp = fma(fma(b, b, 12.0), (b * b), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * fma(a, a, Float64(Float64(b * b) * 2.0))) + -1.0)
	tmp = 0.0
	if (a <= -0.0022)
		tmp = t_0;
	elseif (a <= 16000000.0)
		tmp = fma(fma(b, b, 12.0), Float64(b * b), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -0.0022], t$95$0, If[LessEqual[a, 16000000.0], N[(N[(b * b + 12.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.00220000000000000013 or 1.6e7 < a

   1. Initial program 43.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 83.1%

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

   6. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.1

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

   8. Simplified 81.1%

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

   9. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

   11. Simplified 97.2%

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

   if -0.00220000000000000013 < a < 1.6e7

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 98.7

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

   5. Simplified 98.7%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 12\right)} - 1 \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 98.7

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

   7. Applied egg-rr 98.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0022:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot 2\right) + -1\\ \mathbf{elif}\;a \leq 16000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot 2\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+87)
   (+ (* (* a a) (fma a (+ a -4.0) 4.0)) -1.0)
   (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+87) {
		tmp = ((a * a) * fma(a, (a + -4.0), 4.0)) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+87)
		tmp = Float64(Float64(Float64(a * a) * fma(a, Float64(a + -4.0), 4.0)) + -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+87], N[(N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999998e87

   1. Initial program 80.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 86.6%

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

   6. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 95.2

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

   8. Simplified 95.2%

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

   if 4.9999999999999998e87 < (*.f64 b b)

   1. Initial program 60.1%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   7. 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 96.6

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

   8. Simplified 96.6%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a + -4, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+87)
   (+ (* (+ a -4.0) (* a (* a a))) -1.0)
   (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+87) {
		tmp = ((a + -4.0) * (a * (a * a))) + -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) <= 5d+87) then
        tmp = ((a + (-4.0d0)) * (a * (a * a))) + (-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) <= 5e+87) {
		tmp = ((a + -4.0) * (a * (a * a))) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+87:
		tmp = ((a + -4.0) * (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) <= 5e+87)
		tmp = Float64(Float64(Float64(a + -4.0) * Float64(a * Float64(a * a))) + -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+87)
		tmp = ((a + -4.0) * (a * (a * a))) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+87], N[(N[(N[(a + -4.0), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999998e87

   1. Initial program 80.2%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 93.8

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

   5. Simplified 93.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{3} \cdot \left(a - 4\right)} - 1 \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 93.8

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

   8. Simplified 93.8%

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

   if 4.9999999999999998e87 < (*.f64 b b)

   1. Initial program 60.1%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   7. 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 96.6

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

   8. Simplified 96.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+87}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+87) (+ (* a (* a (* a a))) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+87) {
		tmp = (a * (a * (a * a))) + -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) <= 5d+87) then
        tmp = (a * (a * (a * a))) + (-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) <= 5e+87) {
		tmp = (a * (a * (a * a))) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+87:
		tmp = (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) <= 5e+87)
		tmp = Float64(Float64(a * Float64(a * Float64(a * a))) + -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+87)
		tmp = (a * (a * (a * a))) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+87], N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999998e87

   1. Initial program 80.2%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.9

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

   5. Simplified 92.9%

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

   if 4.9999999999999998e87 < (*.f64 b b)

   1. Initial program 60.1%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   7. 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 96.6

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

   8. Simplified 96.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+87}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -1.52 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 145:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 12, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -1.52e+29) t_0 (if (<= a 145.0) (fma b (* b 12.0) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -1.52e+29) {
		tmp = t_0;
	} else if (a <= 145.0) {
		tmp = fma(b, (b * 12.0), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (a <= -1.52e+29)
		tmp = t_0;
	elseif (a <= 145.0)
		tmp = fma(b, Float64(b * 12.0), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.52e+29], t$95$0, If[LessEqual[a, 145.0], N[(b * N[(b * 12.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.52e29 or 145 < a

   1. Initial program 44.6%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

   6. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

   if -1.52e29 < a < 145

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.1

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

   8. Simplified 81.1%

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

Alternative 11: 92.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+87) (fma (* a a) (* a a) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+87) {
		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) <= 5e+87)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+87], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999998e87

   1. Initial program 80.2%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.9

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

   5. Simplified 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 92.8

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

   7. Applied egg-rr 92.8%

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

   if 4.9999999999999998e87 < (*.f64 b b)

   1. Initial program 60.1%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   7. 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 96.6

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

   8. Simplified 96.6%

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

Alternative 12: 70.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+302}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+302) (+ (* a (* a 4.0)) -1.0) (* b (* b 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+302) {
		tmp = (a * (a * 4.0)) + -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+302) then
        tmp = (a * (a * 4.0d0)) + (-1.0d0)
    else
        tmp = b * (b * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+302) {
		tmp = (a * (a * 4.0)) + -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+302:
		tmp = (a * (a * 4.0)) + -1.0
	else:
		tmp = b * (b * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+302)
		tmp = Float64(Float64(a * Float64(a * 4.0)) + -1.0);
	else
		tmp = Float64(b * Float64(b * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+302)
		tmp = (a * (a * 4.0)) + -1.0;
	else
		tmp = b * (b * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+302], N[(N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.0000000000000001e302

   1. Initial program 75.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 88.9%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 70.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{4 \cdot {a}^{2}} - 1 \]
   10. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 66.3

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

   11. Simplified 66.3%

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

   if 1.0000000000000001e302 < (*.f64 b b)

   1. Initial program 62.5%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.4

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

   8. Simplified 97.4%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

          \[\leadsto 12 \cdot \color{blue}{\left(b \cdot b\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(12 \cdot b\right) \cdot b} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \left(12 \cdot b\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(12 \cdot b\right)} \]

       5. *-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(b \cdot 12\right)} \]

       6. lower-*.f64 97.4

          \[\leadsto b \cdot \color{blue}{\left(b \cdot 12\right)} \]

   11. Simplified 97.4%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+302}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.4% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-rgt-out N/A

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

   4. +-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 68.9

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

5. Simplified 68.9%

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

6. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 54.7

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

8. Simplified 54.7%

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

Alternative 14: 25.1% accurate, 155.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in a around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

      \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 72.9

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

5. Simplified 72.9%

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

6. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 29.3%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))