Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.2% → 99.9%

Time: 13.7s

Alternatives: 16

Speedup: 5.7×

• 60.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 74.2% 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, 2.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, 4\right), \mathsf{fma}\left(b, b, 12\right)\right), b, \mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(1 - a, 4, \mathsf{fma}\left(a, a, 0\right)\right), -1\right)\right) \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return fma((b * fma(a, fma(a, 2.0, 4.0), fma(b, b, 12.0))), b, fma(fma(a, a, 0.0), fma((1.0 - a), 4.0, fma(a, a, 0.0)), -1.0));
}

Julia

function code(a, b)
	return fma(Float64(b * fma(a, fma(a, 2.0, 4.0), fma(b, b, 12.0))), b, fma(fma(a, a, 0.0), fma(Float64(1.0 - a), 4.0, fma(a, a, 0.0)), -1.0))
end

Wolfram

code[a_, b_] := N[(N[(b * N[(a * N[(a * 2.0 + 4.0), $MachinePrecision] + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a * a + 0.0), $MachinePrecision] * N[(N[(1.0 - a), $MachinePrecision] * 4.0 + N[(a * a + 0.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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 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. accelerator-lowering-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 94.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right) + \left(\left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right)} \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right)\right)} + \left(\left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right)\right) \cdot b} + \left(\left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right)}, b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \color{blue}{\left(\left(a \cdot \left(2 \cdot a + 4\right) + 12\right) + b \cdot b\right)}, b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   7. associate-+l+ N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \color{blue}{\left(a \cdot \left(2 \cdot a + 4\right) + \left(12 + b \cdot b\right)\right)}, b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \left(a \cdot \left(2 \cdot a + 4\right) + \color{blue}{\left(b \cdot b + 12\right)}\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \color{blue}{\mathsf{fma}\left(a, 2 \cdot a + 4, b \cdot b + 12\right)}, b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot 2} + 4, b \cdot b + 12\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 2, 4\right)}, b \cdot b + 12\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, 4\right), \color{blue}{\mathsf{fma}\left(b, b, 12\right)}\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, 4\right), \mathsf{fma}\left(b, b, 12\right)\right), b, \mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(1 - a, 4, \mathsf{fma}\left(a, a, 0\right)\right), -1\right)\right)} \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right)\\ \mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(b, b \cdot 12, -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (fma a a 0.0)))) (fma t_0 t_0 (fma b (* b 12.0) -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, fma(a, a, 0.0));
	return fma(t_0, t_0, fma(b, (b * 12.0), -1.0));
}

Julia

function code(a, b)
	t_0 = fma(b, b, fma(a, a, 0.0))
	return fma(t_0, t_0, fma(b, Float64(b * 12.0), -1.0))
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a + 0.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(b * N[(b * 12.0), $MachinePrecision] + -1.0), $MachinePrecision]), $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. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. +-commutative N/A

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

   12. +-lowering-+.f64 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. *-lowering-*.f64 N/A

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

   17. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 78.4%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 99.6

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

7. Simplified 99.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(b \cdot b\right) \cdot 3}, 4, \mathsf{fma}\left(a, a, b \cdot b\right) \cdot \mathsf{fma}\left(a, a, b \cdot b\right)\right) - 1 \]
8. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(\left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + \left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

   3. metadata-eval N/A

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

   4. associate-+l+ N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \color{blue}{a \cdot a + 0}\right), a \cdot a + b \cdot b, \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \color{blue}{\mathsf{fma}\left(a, a, 0\right)}\right), a \cdot a + b \cdot b, \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{b \cdot b + a \cdot a}, \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{\mathsf{fma}\left(b, b, a \cdot a\right)}, \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   12. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \color{blue}{a \cdot a + 0}\right), \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \color{blue}{\mathsf{fma}\left(a, a, 0\right)}\right), \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{\left(b \cdot b\right) \cdot \left(3 \cdot 4\right)} + -1\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \left(b \cdot b\right) \cdot \color{blue}{12} + -1\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{b \cdot \left(b \cdot 12\right)} + -1\right) \]

   17. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{\mathsf{fma}\left(b, b \cdot 12, -1\right)}\right) \]

   18. *-lowering-*.f64 99.6

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, \color{blue}{b \cdot 12}, -1\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b \cdot 12, -1\right)\right)} \]
10. Add Preprocessing

Alternative 3: 98.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, 4\right), \mathsf{fma}\left(b, b, 12\right)\right), b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-9)
   (fma a (* a (fma a (+ a -4.0) 4.0)) -1.0)
   (fma (* b (fma a (fma a 2.0 4.0) (fma b b 12.0))) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-9) {
		tmp = fma(a, (a * fma(a, (a + -4.0), 4.0)), -1.0);
	} else {
		tmp = fma((b * fma(a, fma(a, 2.0, 4.0), fma(b, b, 12.0))), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-9)
		tmp = fma(a, Float64(a * fma(a, Float64(a + -4.0), 4.0)), -1.0);
	else
		tmp = fma(Float64(b * fma(a, fma(a, 2.0, 4.0), fma(b, b, 12.0))), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-9], N[(a * N[(a * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(a * N[(a * 2.0 + 4.0), $MachinePrecision] + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.00000000000000006e-9

   1. Initial program 85.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. accelerator-lowering-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 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   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. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 99.7%

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

   if 1.00000000000000006e-9 < (*.f64 b b)

   1. Initial program 69.1%

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

   3. Taylor expanded in 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. accelerator-lowering-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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right) + \left(\left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right)} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right)\right)} + \left(\left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right)\right) \cdot b} + \left(\left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot \left(b \cdot b + \left(a \cdot \left(2 \cdot a + 4\right) + 12\right)\right)}, b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot \color{blue}{\left(\left(a \cdot \left(2 \cdot a + 4\right) + 12\right) + b \cdot b\right)}, b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(b \cdot \color{blue}{\left(a \cdot \left(2 \cdot a + 4\right) + \left(12 + b \cdot b\right)\right)}, b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b \cdot \left(a \cdot \left(2 \cdot a + 4\right) + \color{blue}{\left(b \cdot b + 12\right)}\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot \color{blue}{\mathsf{fma}\left(a, 2 \cdot a + 4, b \cdot b + 12\right)}, b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot 2} + 4, b \cdot b + 12\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 2, 4\right)}, b \cdot b + 12\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, 4\right), \color{blue}{\mathsf{fma}\left(b, b, 12\right)}\right), b, \left(a \cdot a + 0\right) \cdot \left(a \cdot a + \left(4 \cdot \left(0 - a\right) + 4\right)\right) - 1\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, 4\right), \mathsf{fma}\left(b, b, 12\right)\right), b, \mathsf{fma}\left(\mathsf{fma}\left(a, a, 0\right), \mathsf{fma}\left(1 - a, 4, \mathsf{fma}\left(a, a, 0\right)\right), -1\right)\right)} \]

   8. Taylor expanded in a around 0

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

   10. Simplified 95.6%

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

Alternative 4: 98.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, a \cdot 2, \mathsf{fma}\left(b, b, 12\right)\right), b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-9)
   (fma a (* a (fma a (+ a -4.0) 4.0)) -1.0)
   (fma (* b (fma a (* a 2.0) (fma b b 12.0))) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-9) {
		tmp = fma(a, (a * fma(a, (a + -4.0), 4.0)), -1.0);
	} else {
		tmp = fma((b * fma(a, (a * 2.0), fma(b, b, 12.0))), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-9)
		tmp = fma(a, Float64(a * fma(a, Float64(a + -4.0), 4.0)), -1.0);
	else
		tmp = fma(Float64(b * fma(a, Float64(a * 2.0), fma(b, b, 12.0))), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-9], N[(a * N[(a * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(a * N[(a * 2.0), $MachinePrecision] + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.00000000000000006e-9

   1. Initial program 85.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. accelerator-lowering-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 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   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. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 99.7%

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

   if 1.00000000000000006e-9 < (*.f64 b b)

   1. Initial program 69.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 70.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 99.7

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

   7. Simplified 99.7%

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

   8. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+r+ N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 95.5%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. accelerator-lowering-fma.f64 95.5

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

   12. Applied egg-rr 95.5%

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

Alternative 5: 98.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, \mathsf{fma}\left(b, b, 12\right)\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-9)
   (fma a (* a (fma a (+ a -4.0) 4.0)) -1.0)
   (fma (* b b) (fma 2.0 (* a a) (fma b b 12.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-9) {
		tmp = fma(a, (a * fma(a, (a + -4.0), 4.0)), -1.0);
	} else {
		tmp = fma((b * b), fma(2.0, (a * a), fma(b, b, 12.0)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-9)
		tmp = fma(a, Float64(a * fma(a, Float64(a + -4.0), 4.0)), -1.0);
	else
		tmp = fma(Float64(b * b), fma(2.0, Float64(a * a), fma(b, b, 12.0)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-9], N[(a * N[(a * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(2.0 * N[(a * a), $MachinePrecision] + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.00000000000000006e-9

   1. Initial program 85.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. accelerator-lowering-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 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   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. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 99.7%

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

   if 1.00000000000000006e-9 < (*.f64 b b)

   1. Initial program 69.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 70.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 99.7

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

   7. Simplified 99.7%

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

   8. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+r+ N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 95.5%

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

Alternative 6: 97.7% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+19)
   (fma a (* a (fma a (+ a -4.0) 4.0)) -1.0)
   (fma (* b b) (fma 2.0 (* a a) (* b b)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+19) {
		tmp = fma(a, (a * fma(a, (a + -4.0), 4.0)), -1.0);
	} else {
		tmp = fma((b * b), fma(2.0, (a * a), (b * b)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+19)
		tmp = fma(a, Float64(a * fma(a, Float64(a + -4.0), 4.0)), -1.0);
	else
		tmp = fma(Float64(b * b), fma(2.0, Float64(a * a), Float64(b * b)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+19], N[(a * N[(a * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(2.0 * N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e19

   1. Initial program 84.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. accelerator-lowering-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 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   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. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 98.8%

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

   if 1e19 < (*.f64 b b)

   1. Initial program 68.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 70.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+r+ N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 96.4%

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

   11. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 96.4

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

   13. Simplified 96.4%

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

Alternative 7: 94.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+62)
   (fma 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) <= 2e+62) {
		tmp = fma(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) <= 2e+62)
		tmp = fma(a, Float64(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], 2e+62], N[(a * N[(a * 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) < 2.00000000000000007e62

   1. Initial program 85.5%

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

   3. Taylor expanded in 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. accelerator-lowering-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 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   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. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 97.5%

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

   if 2.00000000000000007e62 < (*.f64 b b)

   1. Initial program 67.2%

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 92.3

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

   8. Simplified 92.3%

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

Alternative 8: 93.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+62}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \mathsf{fma}\left(a, a, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+62)
   (+ -1.0 (* a (* a (fma a a 0.0))))
   (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+62) {
		tmp = -1.0 + (a * (a * fma(a, a, 0.0)));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+62)
		tmp = Float64(-1.0 + Float64(a * Float64(a * fma(a, a, 0.0))));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+62], N[(-1.0 + N[(a * N[(a * N[(a * a + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000007e62

   1. Initial program 85.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      8. +-rgt-identity N/A

         \[\leadsto a \cdot \left(a \cdot \color{blue}{\left({a}^{2} + 0\right)}\right) - 1 \]

      9. unpow2 N/A

         \[\leadsto a \cdot \left(a \cdot \left(\color{blue}{a \cdot a} + 0\right)\right) - 1 \]

      10. accelerator-lowering-fma.f64 97.1

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, a, 0\right)}\right) - 1 \]

   5. Simplified 97.1%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(a, a, 0\right)\right)} - 1 \]

   if 2.00000000000000007e62 < (*.f64 b b)

   1. Initial program 67.2%

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 92.3

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

   8. Simplified 92.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+62}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \mathsf{fma}\left(a, a, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right)\\ \mathsf{fma}\left(t\_0, t\_0, -1\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (fma a a 0.0)))) (fma t_0 t_0 -1.0)))

C

double code(double a, double b) {
	double t_0 = fma(b, b, fma(a, a, 0.0));
	return fma(t_0, t_0, -1.0);
}

Julia

function code(a, b)
	t_0 = fma(b, b, fma(a, a, 0.0))
	return fma(t_0, t_0, -1.0)
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a + 0.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$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. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. +-commutative N/A

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

   12. +-lowering-+.f64 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. *-lowering-*.f64 N/A

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

   17. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 78.4%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 99.6

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

7. Simplified 99.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(b \cdot b\right) \cdot 3}, 4, \mathsf{fma}\left(a, a, b \cdot b\right) \cdot \mathsf{fma}\left(a, a, b \cdot b\right)\right) - 1 \]
8. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(\left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + \left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

   3. metadata-eval N/A

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

   4. associate-+l+ N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \color{blue}{a \cdot a + 0}\right), a \cdot a + b \cdot b, \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \color{blue}{\mathsf{fma}\left(a, a, 0\right)}\right), a \cdot a + b \cdot b, \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{b \cdot b + a \cdot a}, \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{\mathsf{fma}\left(b, b, a \cdot a\right)}, \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   12. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \color{blue}{a \cdot a + 0}\right), \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \color{blue}{\mathsf{fma}\left(a, a, 0\right)}\right), \left(\left(b \cdot b\right) \cdot 3\right) \cdot 4 + -1\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{\left(b \cdot b\right) \cdot \left(3 \cdot 4\right)} + -1\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \left(b \cdot b\right) \cdot \color{blue}{12} + -1\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{b \cdot \left(b \cdot 12\right)} + -1\right) \]

   17. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{\mathsf{fma}\left(b, b \cdot 12, -1\right)}\right) \]

   18. *-lowering-*.f64 99.6

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, \color{blue}{b \cdot 12}, -1\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b \cdot 12, -1\right)\right)} \]

10. Taylor expanded in b around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{-1}\right) \]
11. Step-by-step derivation

12. Simplified 99.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, 0\right)\right), \color{blue}{-1}\right) \]
13. Add Preprocessing

Alternative 10: 82.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -380000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 12, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -380000.0) t_0 (if (<= a 8.5e+27) (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 <= -380000.0) {
		tmp = t_0;
	} else if (a <= 8.5e+27) {
		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 <= -380000.0)
		tmp = t_0;
	elseif (a <= 8.5e+27)
		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, -380000.0], t$95$0, If[LessEqual[a, 8.5e+27], N[(b * N[(b * 12.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8e5 or 8.5e27 < a

   1. Initial program 47.1%

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

   3. Taylor expanded in 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. accelerator-lowering-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 86.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 90.9

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

   8. Simplified 90.9%

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

   if -3.8e5 < a < 8.5e27

   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 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(3 + 1\right)}} + 12 \cdot {b}^{2}\right) - 1 \]

      3. pow-plus N/A

         \[\leadsto \left(\color{blue}{{b}^{3} \cdot b} + 12 \cdot {b}^{2}\right) - 1 \]

      4. cube-unmult N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. accelerator-lowering-fma.f64 99.2

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

   5. Simplified 99.2%

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

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 82.5

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

   8. Simplified 82.5%

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

Alternative 11: 93.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+62) (fma (* a a) (* a a) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+62) {
		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) <= 2e+62)
		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], 2e+62], 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) < 2.00000000000000007e62

   1. Initial program 85.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

   8. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 97.1

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

   10. Simplified 97.1%

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

   if 2.00000000000000007e62 < (*.f64 b b)

   1. Initial program 67.2%

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 92.3

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

   8. Simplified 92.3%

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

Alternative 12: 82.4% accurate, 5.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+19) (fma a (* a 4.0) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+19) {
		tmp = fma(a, (a * 4.0), -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+19)
		tmp = fma(a, Float64(a * 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], 1e+19], N[(a * N[(a * 4.0), $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) < 1e19

   1. Initial program 84.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left({a}^{2} + 0\right)} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{a \cdot a} + 0\right) \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right)} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \left(\color{blue}{a \cdot a} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot \left(1 - a\right)\right)} - 1 \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)}\right) - 1 \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(1 + \color{blue}{-1 \cdot a}\right)\right) - 1 \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \color{blue}{\left(-1 \cdot a + 1\right)}\right) - 1 \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \color{blue}{4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1}\right) - 1 \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(-1 \cdot a\right) + \color{blue}{4}\right) - 1 \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \color{blue}{\mathsf{fma}\left(4, -1 \cdot a, 4\right)}\right) - 1 \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{\mathsf{neg}\left(a\right)}, 4\right)\right) - 1 \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{0 - a}, 4\right)\right) - 1 \]

      21. --lowering--.f64 98.8

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{0 - a}, 4\right)\right) - 1 \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 81.5

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

   8. Simplified 81.5%

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

   if 1e19 < (*.f64 b b)

   1. Initial program 68.8%

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

   3. Taylor expanded in 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. accelerator-lowering-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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right)\right), \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)\right)} - 1 \]

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 89.4

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

   8. Simplified 89.4%

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

Alternative 13: 51.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -1.16 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.4:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* a a))))
   (if (<= a -1.16e-7) t_0 (if (<= a 2.4) -1.0 t_0))))

C

double code(double a, double b) {
	double t_0 = 4.0 * (a * a);
	double tmp;
	if (a <= -1.16e-7) {
		tmp = t_0;
	} else if (a <= 2.4) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (a * a)
    if (a <= (-1.16d-7)) then
        tmp = t_0
    else if (a <= 2.4d0) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 4.0 * (a * a);
	double tmp;
	if (a <= -1.16e-7) {
		tmp = t_0;
	} else if (a <= 2.4) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 4.0 * (a * a)
	tmp = 0
	if a <= -1.16e-7:
		tmp = t_0
	elif a <= 2.4:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(4.0 * Float64(a * a))
	tmp = 0.0
	if (a <= -1.16e-7)
		tmp = t_0;
	elseif (a <= 2.4)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 4.0 * (a * a);
	tmp = 0.0;
	if (a <= -1.16e-7)
		tmp = t_0;
	elseif (a <= 2.4)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.16e-7], t$95$0, If[LessEqual[a, 2.4], -1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1600000000000001e-7 or 2.39999999999999991 < a

   1. Initial program 49.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left({a}^{2} + 0\right)} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{a \cdot a} + 0\right) \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right)} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \left(\color{blue}{a \cdot a} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot \left(1 - a\right)\right)} - 1 \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)}\right) - 1 \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(1 + \color{blue}{-1 \cdot a}\right)\right) - 1 \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \color{blue}{\left(-1 \cdot a + 1\right)}\right) - 1 \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \color{blue}{4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1}\right) - 1 \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(-1 \cdot a\right) + \color{blue}{4}\right) - 1 \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \color{blue}{\mathsf{fma}\left(4, -1 \cdot a, 4\right)}\right) - 1 \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{\mathsf{neg}\left(a\right)}, 4\right)\right) - 1 \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{0 - a}, 4\right)\right) - 1 \]

      21. --lowering--.f64 89.0

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{0 - a}, 4\right)\right) - 1 \]

   5. Simplified 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 57.1

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

   8. Simplified 57.1%

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

   9. Taylor expanded in a around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{4 \cdot {a}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 57.2

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

   11. Simplified 57.2%

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

   if -1.1600000000000001e-7 < a < 2.39999999999999991

   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 b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 56.2%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 12, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e+140) (fma a (* a 4.0) -1.0) (fma b (* b 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e+140) {
		tmp = fma(a, (a * 4.0), -1.0);
	} else {
		tmp = fma(b, (b * 12.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e+140)
		tmp = fma(a, Float64(a * 4.0), -1.0);
	else
		tmp = fma(b, Float64(b * 12.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e+140], N[(a * N[(a * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0999999999999999e140

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left({a}^{2} + 0\right)} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{a \cdot a} + 0\right) \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right)} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \left(\color{blue}{a \cdot a} + 4 \cdot \left(1 - a\right)\right) - 1 \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot \left(1 - a\right)\right)} - 1 \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)}\right) - 1 \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(1 + \color{blue}{-1 \cdot a}\right)\right) - 1 \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \color{blue}{\left(-1 \cdot a + 1\right)}\right) - 1 \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \color{blue}{4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1}\right) - 1 \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(-1 \cdot a\right) + \color{blue}{4}\right) - 1 \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \color{blue}{\mathsf{fma}\left(4, -1 \cdot a, 4\right)}\right) - 1 \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{\mathsf{neg}\left(a\right)}, 4\right)\right) - 1 \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{0 - a}, 4\right)\right) - 1 \]

      21. --lowering--.f64 76.5

          \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{0 - a}, 4\right)\right) - 1 \]

   5. Simplified 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 61.0

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

   8. Simplified 61.0%

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

   if 1.0999999999999999e140 < b

   1. Initial program 71.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. +-commutative N/A

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

      2. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(3 + 1\right)}} + 12 \cdot {b}^{2}\right) - 1 \]

      3. pow-plus N/A

         \[\leadsto \left(\color{blue}{{b}^{3} \cdot b} + 12 \cdot {b}^{2}\right) - 1 \]

      4. cube-unmult N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 97.2

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

   8. Simplified 97.2%

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

Alternative 15: 52.1% accurate, 12.9× speedup

Math

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

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(a, Float64(a * 4.0), -1.0)
end

Wolfram

code[a_, b_] := N[(a * N[(a * 4.0), $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 b around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left({a}^{2} + 0\right)} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

   9. unpow2 N/A

      \[\leadsto \left(\color{blue}{a \cdot a} + 0\right) \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right)} \cdot \left({a}^{2} + 4 \cdot \left(1 - a\right)\right) - 1 \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \left(\color{blue}{a \cdot a} + 4 \cdot \left(1 - a\right)\right) - 1 \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot \left(1 - a\right)\right)} - 1 \]

   13. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(a\right)\right)\right)}\right) - 1 \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(1 + \color{blue}{-1 \cdot a}\right)\right) - 1 \]

   15. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \color{blue}{\left(-1 \cdot a + 1\right)}\right) - 1 \]

   16. distribute-lft-in N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \color{blue}{4 \cdot \left(-1 \cdot a\right) + 4 \cdot 1}\right) - 1 \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(-1 \cdot a\right) + \color{blue}{4}\right) - 1 \]

   18. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \color{blue}{\mathsf{fma}\left(4, -1 \cdot a, 4\right)}\right) - 1 \]

   19. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{\mathsf{neg}\left(a\right)}, 4\right)\right) - 1 \]

   20. neg-sub0 N/A

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{0 - a}, 4\right)\right) - 1 \]

   21. --lowering--.f64 70.5

       \[\leadsto \mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, \color{blue}{0 - a}, 4\right)\right) - 1 \]

5. Simplified 70.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, 0\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, 0 - a, 4\right)\right)} - 1 \]

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 56.6

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

8. Simplified 56.6%

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

Alternative 16: 25.5% 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 b around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 72.1

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

5. Simplified 72.1%

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

6. Taylor expanded in b around 0

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

8. Simplified 31.9%

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

Reproduce

herbie shell --seed 2024196 
(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))