Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.1% → 99.9%

Time: 8.1s

Alternatives: 10

Speedup: 5.5×

• 61.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

4. Applied rewrites 70.2%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

4. Applied rewrites 70.2%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

4. Applied rewrites 70.2%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 99.7%

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

Alternative 4: 99.8% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

4. Applied rewrites 70.2%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 99.9%

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

11. Taylor expanded in a around 0

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

13. Applied rewrites 99.7%

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

Alternative 5: 99.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

4. Applied rewrites 70.2%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 99.9%

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

11. Taylor expanded in b around inf

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

13. Applied rewrites 99.2%

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

Alternative 6: 86.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+143}:\\ \;\;\;\;\left(4 \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(4, a, 4\right) \cdot a\right) \cdot a - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.2e+143)
   (- (* (* 4.0 a) a) 1.0)
   (if (<= a 3.5e+102)
     (fma (fma b b 4.0) (* b b) -1.0)
     (- (* (* (fma 4.0 a 4.0) a) a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.2e+143) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else if (a <= 3.5e+102) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = ((fma(4.0, a, 4.0) * a) * a) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.2e+143)
		tmp = Float64(Float64(Float64(4.0 * a) * a) - 1.0);
	elseif (a <= 3.5e+102)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = Float64(Float64(Float64(fma(4.0, a, 4.0) * a) * a) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.2e+143], N[(N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 3.5e+102], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(4.0 * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.1999999999999998e143

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt1-in N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. pow-sqr N/A

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

      17. distribute-lft-out N/A

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

      18. associate-+r+ N/A

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

      19. unpow2 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

      \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 97.3%

       \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

   if -5.1999999999999998e143 < a < 3.50000000000000011e102

   1. Initial program 86.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

   if 3.50000000000000011e102 < a

   1. Initial program 55.3%

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

   4. Applied rewrites 59.6%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt1-in N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. pow-sqr N/A

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

      17. distribute-lft-out N/A

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

      18. associate-+r+ N/A

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

      19. unpow2 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 100.0%

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

Alternative 7: 94.4% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-10)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (fma (fma b b 4.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-10) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-10)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-10], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000007e-10

   1. Initial program 77.5%

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r* N/A

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

       5. unpow2 N/A

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

       6. distribute-rgt-in N/A

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. distribute-lft-in N/A

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

       10. associate-+r+ N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-in N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

       17. metadata-eval N/A

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

   13. Applied rewrites 99.9%

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

   if 2.00000000000000007e-10 < (*.f64 b b)

   1. Initial program 63.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 94.7

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

   8. Applied rewrites 94.7%

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

Alternative 8: 82.3% accurate, 5.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-10)
   (- (* (* 4.0 a) a) 1.0)
   (fma (fma b b 4.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-10) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-10)
		tmp = Float64(Float64(Float64(4.0 * a) * a) - 1.0);
	else
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-10], N[(N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000007e-10

   1. Initial program 77.5%

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt1-in N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. pow-sqr N/A

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

      17. distribute-lft-out N/A

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

      18. associate-+r+ N/A

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

      19. unpow2 N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

      \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 71.8%

       \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

   if 2.00000000000000007e-10 < (*.f64 b b)

   1. Initial program 63.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 94.7

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

   8. Applied rewrites 94.7%

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

Alternative 9: 69.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\left(4 \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+292) (- (* (* 4.0 a) a) 1.0) (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+292) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+292) then
        tmp = ((4.0d0 * a) * a) - 1.0d0
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+292) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+292:
		tmp = ((4.0 * a) * a) - 1.0
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+292)
		tmp = Float64(Float64(Float64(4.0 * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+292)
		tmp = ((4.0 * a) * a) - 1.0;
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+292], N[(N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000001e292

   1. Initial program 73.3%

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt1-in N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. pow-sqr N/A

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

      17. distribute-lft-out N/A

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

      18. associate-+r+ N/A

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

      19. unpow2 N/A

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 57.2%

       \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

   if 4.0000000000000001e292 < (*.f64 b b)

   1. Initial program 60.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 92.3%

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

Alternative 10: 51.0% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return ((b * b) * 4.0) - 1.0

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((b * b) * 4.0) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 69.4%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 49.0%

   \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{4} - 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))