Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.1% → 97.9%

Time: 11.3s

Alternatives: 11

Speedup: 5.7×

• 60.1% 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: 74.1% 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: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))))
   (if (<= t_0 INFINITY)
     (+ t_0 -1.0)
     (fma (* a a) (fma a (+ a 4.0) 4.0) -1.0))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = fma((a * a), fma(a, (a + 4.0), 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = 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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.9%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 93.5

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

   8. Simplified 93.5%

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

4. Final simplification 98.4%

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

Alternative 2: 58.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))))
   (if (<= t_0 1.0)
     -1.0
     (if (<= t_0 INFINITY) (* (* b b) 4.0) (* (* a a) 4.0)))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= 1.0) {
		tmp = -1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (b * b) * 4.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= 1.0) {
		tmp = -1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (b * b) * 4.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= 1.0:
		tmp = -1.0
	elif t_0 <= math.inf:
		tmp = (b * b) * 4.0
	else:
		tmp = (a * a) * 4.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= 1.0)
		tmp = -1.0;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(b * b) * 4.0);
	else
		tmp = Float64(Float64(a * a) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= 1.0)
		tmp = -1.0;
	elseif (t_0 <= Inf)
		tmp = (b * b) * 4.0;
	else
		tmp = (a * a) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = 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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.0], -1.0, If[LessEqual[t$95$0, Infinity], N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 98.4%

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

   if 1 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.8%

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 61.7

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

   5. Simplified 61.7%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 31.4

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

   8. Simplified 31.4%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 31.8

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

   11. Simplified 31.8%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 93.5

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

   5. Simplified 93.5%

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

   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. lower-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. lower-*.f64 78.9

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

   8. Simplified 78.9%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.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. lower-*.f64 78.9

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

   11. Simplified 78.9%

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

4. Final simplification 61.2%

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

Alternative 3: 51.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 98.4%

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

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

   1. Initial program 68.3%

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

   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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 56.7

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

   5. Simplified 56.7%

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

   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. lower-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. lower-*.f64 35.2

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

   8. Simplified 35.2%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.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. lower-*.f64 35.7

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

   11. Simplified 35.7%

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

4. Final simplification 53.4%

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

Alternative 4: 82.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot 4, -1\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-17)
   (fma a (* a 4.0) -1.0)
   (if (<= (* b b) 5e+15) (* a (* a (* a a))) (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-17) {
		tmp = fma(a, (a * 4.0), -1.0);
	} else if ((b * b) <= 5e+15) {
		tmp = a * (a * (a * a));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-17)
		tmp = fma(a, Float64(a * 4.0), -1.0);
	elseif (Float64(b * b) <= 5e+15)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-17], N[(a * N[(a * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 5e+15], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 2.00000000000000014e-17

   1. Initial program 84.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 99.9

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

   5. Simplified 99.9%

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

   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. lower-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. lower-*.f64 81.3

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

   8. Simplified 81.3%

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

   if 2.00000000000000014e-17 < (*.f64 b b) < 5e15

   1. Initial program 79.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.6

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

   5. Simplified 91.6%

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

   6. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 90.0

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

   8. Simplified 90.0%

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

   if 5e15 < (*.f64 b b)

   1. Initial program 69.3%

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

   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. lower-*.f64 N/A

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

      6. lower-*.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. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

   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. lower-*.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. lower-*.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. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

Alternative 5: 94.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+15)
   (fma (* a a) (fma a (+ a 4.0) 4.0) -1.0)
   (+ -1.0 (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+15) {
		tmp = fma((a * a), fma(a, (a + 4.0), 4.0), -1.0);
	} else {
		tmp = -1.0 + (b * (b * (b * b)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+15)
		tmp = fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), -1.0);
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+15], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e15

   1. Initial program 84.4%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.3

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

   8. Simplified 99.3%

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

   if 5e15 < (*.f64 b b)

   1. Initial program 69.3%

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

   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. lower-*.f64 N/A

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

      6. lower-*.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. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+15)
   (fma (* a a) (* a a) -1.0)
   (+ -1.0 (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+15) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = -1.0 + (b * (b * (b * b)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+15)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+15], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e15

   1. Initial program 84.4%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   7. 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. lower-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. lower-*.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. lower-*.f64 98.3

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

   8. Simplified 98.3%

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

   if 5e15 < (*.f64 b b)

   1. Initial program 69.3%

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

   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. lower-*.f64 N/A

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

      6. lower-*.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. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -34000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -34000000000000.0)
     t_0
     (if (<= a 2.7e+51) (fma 4.0 (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -34000000000000.0) {
		tmp = t_0;
	} else if (a <= 2.7e+51) {
		tmp = fma(4.0, (b * b), -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 <= -34000000000000.0)
		tmp = t_0;
	elseif (a <= 2.7e+51)
		tmp = fma(4.0, Float64(b * b), -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, -34000000000000.0], t$95$0, If[LessEqual[a, 2.7e+51], N[(4.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4e13 or 2.69999999999999992e51 < a

   1. Initial program 49.4%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.1

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

   8. Simplified 92.1%

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

   if -3.4e13 < a < 2.69999999999999992e51

   1. Initial program 97.9%

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 73.8

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

   8. Simplified 73.8%

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

Alternative 8: 93.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+15) (fma (* a a) (* a a) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+15) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+15)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+15], 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) < 5e15

   1. Initial program 84.4%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   7. 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. lower-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. lower-*.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. lower-*.f64 98.3

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

   8. Simplified 98.3%

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

   if 5e15 < (*.f64 b b)

   1. Initial program 69.3%

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

   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. lower-*.f64 N/A

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

      6. lower-*.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. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

   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. lower-*.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. lower-*.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. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

Alternative 9: 69.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot 4\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) 4.0)))
   (if (<= a -2.4e+129) t_0 (if (<= a 7e+143) (fma 4.0 (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * 4.0;
	double tmp;
	if (a <= -2.4e+129) {
		tmp = t_0;
	} else if (a <= 7e+143) {
		tmp = fma(4.0, (b * b), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * 4.0)
	tmp = 0.0
	if (a <= -2.4e+129)
		tmp = t_0;
	elseif (a <= 7e+143)
		tmp = fma(4.0, Float64(b * b), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[a, -2.4e+129], t$95$0, If[LessEqual[a, 7e+143], N[(4.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3999999999999999e129 or 7.00000000000000017e143 < a

   1. Initial program 26.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   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. lower-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. lower-*.f64 95.7

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

   8. Simplified 95.7%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.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. lower-*.f64 95.7

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

   11. Simplified 95.7%

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

   if -2.3999999999999999e129 < a < 7.00000000000000017e143

   1. Initial program 94.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 82.9

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

   5. Simplified 82.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 61.3

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

   8. Simplified 61.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+129}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.95 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1.95e+294) (fma a (* a 4.0) -1.0) (* (* b b) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1.95e+294) {
		tmp = fma(a, (a * 4.0), -1.0);
	} else {
		tmp = (b * b) * 4.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1.95e+294)
		tmp = fma(a, Float64(a * 4.0), -1.0);
	else
		tmp = Float64(Float64(b * b) * 4.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1.95e+294], N[(a * N[(a * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.95000000000000007e294

   1. Initial program 80.1%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right)} - 1 \]
   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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 80.6

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

   5. Simplified 80.6%

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

   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. lower-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. lower-*.f64 63.0

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

   8. Simplified 63.0%

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

   if 1.95000000000000007e294 < (*.f64 b b)

   1. Initial program 67.8%

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 92.5

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

   8. Simplified 92.5%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 92.5

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

   11. Simplified 92.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.95 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 25.1% accurate, 160.0× 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 77.2%

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

3. Taylor expanded in a around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 68.0

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

5. Simplified 68.0%

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

6. Taylor expanded in a around 0

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

8. Simplified 28.2%

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

Reproduce

herbie shell --seed 2024215 
(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))