Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.9% → 98.9%

Time: 7.3s

Alternatives: 9

Speedup: 5.2×

• 60.0% 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: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Final simplification 98.4%

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

Alternative 2: 97.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, 1 - a, a \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right) \cdot b, b, \left(4 \cdot a\right) \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+18)
   (fma (* (fma 4.0 (- 1.0 a) (* a a)) a) a -1.0)
   (-
    (fma (* (fma (fma 2.0 a 4.0) a (fma b b 12.0)) b) b (* (* 4.0 a) a))
    1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+18) {
		tmp = fma((fma(4.0, (1.0 - a), (a * a)) * a), a, -1.0);
	} else {
		tmp = fma((fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)) * b), b, ((4.0 * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+18)
		tmp = fma(Float64(fma(4.0, Float64(1.0 - a), Float64(a * a)) * a), a, -1.0);
	else
		tmp = Float64(fma(Float64(fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)) * b), b, Float64(Float64(4.0 * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+18], N[(N[(N[(4.0 * N[(1.0 - a), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e18

   1. Initial program 80.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-lft-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 98.1

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

   8. Applied rewrites 98.1%

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

   10. Applied rewrites 98.2%

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

   if 5e18 < (*.f64 b b)

   1. Initial program 74.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.3%

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

Alternative 3: 93.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, 1 - a, a \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+85)
   (fma (* (fma 4.0 (- 1.0 a) (* a a)) a) a -1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+85) {
		tmp = fma((fma(4.0, (1.0 - a), (a * a)) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+85)
		tmp = fma(Float64(fma(4.0, Float64(1.0 - a), Float64(a * a)) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+85], N[(N[(N[(4.0 * N[(1.0 - a), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e85

   1. Initial program 80.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-lft-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 96.3

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

   8. Applied rewrites 96.3%

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

   10. Applied rewrites 96.4%

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

   if 2e85 < (*.f64 b b)

   1. Initial program 73.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 97.9

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

   8. Applied rewrites 97.9%

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

Alternative 4: 93.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+85)
   (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+85) {
		tmp = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+85)
		tmp = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+85], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e85

   1. Initial program 80.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-lft-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 96.3

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

   8. Applied rewrites 96.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 96.3%

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

   if 2e85 < (*.f64 b b)

   1. Initial program 73.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 97.9

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

   8. Applied rewrites 97.9%

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

Alternative 5: 86.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e+102)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 9.6e+119)
     (fma (* (fma b b 12.0) b) b -1.0)
     (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.3e+102) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 9.6e+119) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.3e+102)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 9.6e+119)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.3e+102], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 9.6e+119], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.29999999999999999e102

   1. Initial program 62.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-lft-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

   if -3.29999999999999999e102 < a < 9.6e119

   1. Initial program 97.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 80.4

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

   8. Applied rewrites 80.4%

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

   if 9.6e119 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-lft-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 84.6%

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

Alternative 6: 72.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+119}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e+102)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 9.6e+119) (- (* 12.0 (* b b)) 1.0) (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.3e+102) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 9.6e+119) {
		tmp = (12.0 * (b * b)) - 1.0;
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.3e+102)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 9.6e+119)
		tmp = Float64(Float64(12.0 * Float64(b * b)) - 1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.3e+102], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 9.6e+119], N[(N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.29999999999999999e102

   1. Initial program 62.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-lft-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

   if -3.29999999999999999e102 < a < 9.6e119

   1. Initial program 97.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 62.0%

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

   if 9.6e119 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-lft-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 84.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+119}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+241}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+241) (fma (* a a) 4.0 -1.0) (- (* 12.0 (* b b)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+241) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = (12.0 * (b * b)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+241)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(Float64(12.0 * Float64(b * b)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+241], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.0000000000000001e241

   1. Initial program 81.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-lft-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 57.8%

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

   if 1.0000000000000001e241 < (*.f64 b b)

   1. Initial program 68.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 90.9%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+241}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.7% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. distribute-lft-out N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower--.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 71.2

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

8. Applied rewrites 71.2%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 49.0%

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

Alternative 9: 25.0% 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 77.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. distribute-lft-out N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower--.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 71.2

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

8. Applied rewrites 71.2%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 25.0%

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

Reproduce

herbie shell --seed 2024331 
(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))