Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.8% → 98.2%

Time: 8.7s

Alternatives: 7

Speedup: 1.2×

• 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: 73.8% 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: 98.2% accurate, 0.5× speedup

Math

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

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)
     (+ (* a (* a (+ 4.0 (* a (+ a 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 = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0;
	}
	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 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0;
	}
	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 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = (a * (a * (4.0 + (a * (a + 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 = Float64(Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a + 4.0))))) + -1.0);
	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 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0;
	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, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(a * N[(a * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 99.8%

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

   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 33.9%

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

   4. Taylor expanded in a around 0 93.0%

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

      1. *-commutative 93.0%

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

      2. unpow2 93.0%

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

      3. associate-*r* 93.0%

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

      4. +-commutative 93.0%

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

      5. fma-define 93.0%

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

      6. +-commutative 93.0%

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

   6. Applied egg-rr 93.0%

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

   7. Taylor expanded in a around 0 93.0%

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

4. Final simplification 98.1%

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

Alternative 2: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot \left(a + 4\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.45e+15) (+ (* a (* a (+ 4.0 (* a (+ a 4.0))))) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.45e+15) {
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0;
	} else {
		tmp = pow(b, 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 (b <= 1.45d+15) then
        tmp = (a * (a * (4.0d0 + (a * (a + 4.0d0))))) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.45e+15) {
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.45e+15:
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.45e+15)
		tmp = Float64(Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a + 4.0))))) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.45e+15)
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.45e+15], N[(N[(a * N[(a * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45e15

   1. Initial program 76.9%

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

   3. Taylor expanded in b around 0 62.1%

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

   4. Taylor expanded in a around 0 77.8%

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

      1. *-commutative 77.8%

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

      2. unpow2 77.8%

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

      3. associate-*r* 77.8%

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

      4. +-commutative 77.8%

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

      5. fma-define 77.8%

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

      6. +-commutative 77.8%

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

   6. Applied egg-rr 77.8%

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

   7. Taylor expanded in a around 0 77.8%

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

   if 1.45e15 < b

   1. Initial program 64.9%

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

      1. associate--l+ 64.9%

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

      2. +-commutative 64.9%

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

      3. fma-define 64.9%

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

      4. fma-neg 64.9%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in b around inf 93.8%

      \[\leadsto \color{blue}{{b}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot \left(a + 4\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.95 \lor \neg \left(a \leq 1.95\right):\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot 4\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.95) (not (<= a 1.95)))
   (* a (* a (* a (+ a 4.0))))
   (+ (* a (* a (+ 4.0 (* a 4.0)))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.95) || !(a <= 1.95)) {
		tmp = a * (a * (a * (a + 4.0)));
	} else {
		tmp = (a * (a * (4.0 + (a * 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 ((a <= (-1.95d0)) .or. (.not. (a <= 1.95d0))) then
        tmp = a * (a * (a * (a + 4.0d0)))
    else
        tmp = (a * (a * (4.0d0 + (a * 4.0d0)))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.95) || !(a <= 1.95)) {
		tmp = a * (a * (a * (a + 4.0)));
	} else {
		tmp = (a * (a * (4.0 + (a * 4.0)))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.95) or not (a <= 1.95):
		tmp = a * (a * (a * (a + 4.0)))
	else:
		tmp = (a * (a * (4.0 + (a * 4.0)))) + -1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.95) || ~((a <= 1.95)))
		tmp = a * (a * (a * (a + 4.0)));
	else
		tmp = (a * (a * (4.0 + (a * 4.0)))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.95], N[Not[LessEqual[a, 1.95]], $MachinePrecision]], N[(a * N[(a * N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(a * N[(4.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.94999999999999996 or 1.94999999999999996 < a

   1. Initial program 48.3%

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

      1. associate--l+ 48.3%

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

      2. +-commutative 48.3%

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

      3. fma-define 48.3%

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

      4. fma-neg 48.3%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in a around inf 87.1%

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

      1. associate-*r/ 87.1%

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

      2. metadata-eval 87.1%

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

   7. Simplified 87.1%

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

   8. Taylor expanded in a around 0 87.0%

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

      1. unpow3 87.0%

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

      2. unpow2 87.0%

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

      3. associate-*r* 86.9%

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

      4. *-commutative 86.9%

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

      5. unpow2 86.9%

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

      6. associate-*r* 87.0%

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

      7. +-commutative 87.0%

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

   10. Applied egg-rr 87.0%

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

   if -1.94999999999999996 < a < 1.94999999999999996

   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

   3. Taylor expanded in b around 0 47.2%

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

   4. Taylor expanded in a around 0 47.2%

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

      1. *-commutative 47.2%

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

      2. unpow2 47.2%

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

      3. associate-*r* 47.2%

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

      4. +-commutative 47.2%

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

      5. fma-define 47.2%

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

      6. +-commutative 47.2%

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

   6. Applied egg-rr 47.2%

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

   7. Taylor expanded in a around 0 45.9%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.95 \lor \neg \left(a \leq 1.95\right):\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot 4\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.8) (not (<= a 0.41)))
   (* a (* a (* a (+ a 4.0))))
   (+ (* a (* a 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -4.8) || !(a <= 0.41)) {
		tmp = a * (a * (a * (a + 4.0)));
	} else {
		tmp = (a * (a * 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 ((a <= (-4.8d0)) .or. (.not. (a <= 0.41d0))) then
        tmp = a * (a * (a * (a + 4.0d0)))
    else
        tmp = (a * (a * 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -4.8) || !(a <= 0.41)) {
		tmp = a * (a * (a * (a + 4.0)));
	} else {
		tmp = (a * (a * 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -4.8) or not (a <= 0.41):
		tmp = a * (a * (a * (a + 4.0)))
	else:
		tmp = (a * (a * 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -4.8) || !(a <= 0.41))
		tmp = Float64(a * Float64(a * Float64(a * Float64(a + 4.0))));
	else
		tmp = Float64(Float64(a * Float64(a * 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -4.8) || ~((a <= 0.41)))
		tmp = a * (a * (a * (a + 4.0)));
	else
		tmp = (a * (a * 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -4.8], N[Not[LessEqual[a, 0.41]], $MachinePrecision]], N[(a * N[(a * N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.79999999999999982 or 0.409999999999999976 < a

   1. Initial program 47.9%

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

      1. associate--l+ 47.9%

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

      2. +-commutative 47.9%

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

      3. fma-define 47.9%

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

      4. fma-neg 47.9%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in a around inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. metadata-eval 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in a around 0 87.7%

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

      1. unpow3 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. *-commutative 87.6%

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

      5. unpow2 87.6%

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

      6. associate-*r* 87.6%

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

      7. +-commutative 87.6%

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

   10. Applied egg-rr 87.6%

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

   if -4.79999999999999982 < a < 0.409999999999999976

   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

   3. Taylor expanded in b around 0 46.9%

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

   4. Taylor expanded in a around 0 46.9%

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

   5. Taylor expanded in a around 0 45.4%

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

      1. *-commutative 45.4%

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

      2. unpow2 45.4%

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

      3. associate-*r* 45.4%

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

      4. *-commutative 45.4%

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

   7. Applied egg-rr 45.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in b around 0 52.4%

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

4. Taylor expanded in a around 0 67.6%

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

   1. *-commutative 67.6%

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

   2. unpow2 67.6%

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

   3. associate-*r* 67.6%

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

   4. +-commutative 67.6%

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

   5. fma-define 67.6%

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

   6. +-commutative 67.6%

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

6. Applied egg-rr 67.6%

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

7. Taylor expanded in a around 0 67.6%

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

8. Final simplification 67.6%

   \[\leadsto a \cdot \left(a \cdot \left(4 + a \cdot \left(a + 4\right)\right)\right) + -1 \]
9. Add Preprocessing

Alternative 6: 51.1% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in b around 0 52.4%

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

4. Taylor expanded in a around 0 67.6%

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

5. Taylor expanded in a around 0 47.3%

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

   1. *-commutative 47.3%

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

   2. unpow2 47.3%

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

   3. associate-*r* 47.3%

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

   4. *-commutative 47.3%

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

7. Applied egg-rr 47.3%

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

8. Final simplification 47.3%

   \[\leadsto a \cdot \left(a \cdot 4\right) + -1 \]
9. Add Preprocessing

Alternative 7: 24.4% accurate, 130.0× 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 74.1%

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

3. Taylor expanded in b around 0 52.4%

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

4. Taylor expanded in a around 0 23.1%

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

Reproduce

herbie shell --seed 2024096 
(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))