Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.6% → 99.0%

Time: 8.1s

Alternatives: 8

Speedup: 1.2×

• 61.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

   1. sub-neg 74.9%

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

3. Simplified 75.3%

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

5. Taylor expanded in a around 0 99.8%

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

   1. pow2 99.8%

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

7. Applied egg-rr 99.8%

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

   1. fma-define 99.8%

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

   2. unpow2 99.8%

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

   3. distribute-lft-in 84.6%

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

   4. fma-define 84.6%

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

   5. add-sqr-sqrt 84.6%

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

   6. pow2 84.6%

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

   7. fma-define 84.6%

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

   8. hypot-define 84.6%

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

   9. pow2 84.6%

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

   10. fma-define 84.6%

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

   11. add-sqr-sqrt 84.6%

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

   12. pow2 84.6%

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

   13. fma-define 84.6%

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

   14. hypot-define 84.6%

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

   15. pow2 84.6%

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

9. Applied egg-rr 84.6%

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

    1. distribute-lft-out 99.8%

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

    2. rem-square-sqrt 99.8%

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

    3. unpow2 99.8%

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

    4. unpow2 99.8%

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

    5. hypot-undefine 99.8%

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

    6. unpow2 99.8%

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

    7. unpow2 99.8%

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

    8. hypot-undefine 99.8%

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

    9. unpow2 99.8%

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

    10. pow-sqr 99.9%

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

    11. metadata-eval 99.9%

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

11. Simplified 99.9%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 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(-1.0 + t_0);
	else
		tmp = Float64(-1.0 + Float64((a ^ 3.0) * Float64(4.0 + a)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $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[(-1.0 + t$95$0), $MachinePrecision], N[(-1.0 + N[(N[Power[a, 3.0], $MachinePrecision] * N[(4.0 + a), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

      1. sub-neg 0.0%

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

   3. Simplified 1.6%

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

   5. Taylor expanded in a around inf 92.6%

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

      1. associate-*r/ 92.6%

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

      2. metadata-eval 92.6%

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

   7. Simplified 92.6%

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

   8. Taylor expanded in a around 0 92.6%

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

4. Final simplification 98.0%

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

Alternative 3: 83.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 105000:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4} \cdot \left(1 + 2 \cdot \frac{\frac{a}{b}}{\frac{b}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 105000.0)
   (+ -1.0 (* (pow a 4.0) (+ 1.0 (/ 4.0 a))))
   (+ -1.0 (* (pow b 4.0) (+ 1.0 (* 2.0 (/ (/ a b) (/ b a))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 105000.0) {
		tmp = -1.0 + (pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else {
		tmp = -1.0 + (pow(b, 4.0) * (1.0 + (2.0 * ((a / b) / (b / a)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 105000.0d0) then
        tmp = (-1.0d0) + ((a ** 4.0d0) * (1.0d0 + (4.0d0 / a)))
    else
        tmp = (-1.0d0) + ((b ** 4.0d0) * (1.0d0 + (2.0d0 * ((a / b) / (b / a)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 105000.0) {
		tmp = -1.0 + (Math.pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else {
		tmp = -1.0 + (Math.pow(b, 4.0) * (1.0 + (2.0 * ((a / b) / (b / a)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 105000.0:
		tmp = -1.0 + (math.pow(a, 4.0) * (1.0 + (4.0 / a)))
	else:
		tmp = -1.0 + (math.pow(b, 4.0) * (1.0 + (2.0 * ((a / b) / (b / a)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 105000.0)
		tmp = Float64(-1.0 + Float64((a ^ 4.0) * Float64(1.0 + Float64(4.0 / a))));
	else
		tmp = Float64(-1.0 + Float64((b ^ 4.0) * Float64(1.0 + Float64(2.0 * Float64(Float64(a / b) / Float64(b / a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 105000.0)
		tmp = -1.0 + ((a ^ 4.0) * (1.0 + (4.0 / a)));
	else
		tmp = -1.0 + ((b ^ 4.0) * (1.0 + (2.0 * ((a / b) / (b / a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 105000.0], N[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] * N[(1.0 + N[(2.0 * N[(N[(a / b), $MachinePrecision] / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 105000

   1. Initial program 77.5%

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

      1. sub-neg 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in a around inf 73.4%

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

      1. associate-*r/ 73.4%

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

      2. metadata-eval 73.4%

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

   7. Simplified 73.4%

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

   if 105000 < b

   1. Initial program 67.0%

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

   3. Taylor expanded in b around inf 86.6%

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

   4. Taylor expanded in a around inf 86.3%

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

      1. unpow2 86.3%

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

      2. unpow2 86.3%

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

      3. times-frac 97.3%

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

      4. unpow2 97.3%

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

   6. Simplified 97.3%

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

      1. unpow2 97.3%

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

      2. clear-num 97.3%

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

      3. un-div-inv 97.3%

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

   8. Applied egg-rr 97.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 105000:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4} \cdot \left(1 + 2 \cdot \frac{\frac{a}{b}}{\frac{b}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{+76} \lor \neg \left(a \leq 8 \cdot 10^{+38}\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.15e+76) (not (<= a 8e+38)))
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (+ (* 4.0 (* b b)) (pow b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.15e+76) || !(a <= 8e+38)) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + ((4.0 * (b * b)) + 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 ((a <= (-2.15d+76)) .or. (.not. (a <= 8d+38))) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((4.0d0 * (b * b)) + (b ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.15e+76) || !(a <= 8e+38)) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((4.0 * (b * b)) + Math.pow(b, 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.15e+76) or not (a <= 8e+38):
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((4.0 * (b * b)) + math.pow(b, 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.15e+76) || !(a <= 8e+38))
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(4.0 * Float64(b * b)) + (b ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.15e+76) || ~((a <= 8e+38)))
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((4.0 * (b * b)) + (b ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.15e+76], N[Not[LessEqual[a, 8e+38]], $MachinePrecision]], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.14999999999999989e76 or 7.99999999999999982e38 < a

   1. Initial program 38.9%

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

      1. sub-neg 38.9%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in a around inf 96.7%

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

   if -2.14999999999999989e76 < a < 7.99999999999999982e38

   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. Step-by-step derivation

      1. sub-neg 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in a around 0 99.8%

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

      1. pow2 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. fma-define 99.8%

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

      2. unpow2 99.8%

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

      3. distribute-lft-in 86.9%

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

      4. fma-define 86.9%

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

      5. add-sqr-sqrt 86.9%

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

      6. pow2 86.9%

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

      7. fma-define 86.9%

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

      8. hypot-define 86.9%

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

      9. pow2 86.9%

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

      10. fma-define 86.9%

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

      11. add-sqr-sqrt 86.9%

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

      12. pow2 86.9%

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

      13. fma-define 86.9%

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

      14. hypot-define 86.9%

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

      15. pow2 86.9%

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

   9. Applied egg-rr 86.9%

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

       1. distribute-lft-out 99.8%

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

       2. rem-square-sqrt 99.8%

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

       3. unpow2 99.8%

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

       4. unpow2 99.8%

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

       5. hypot-undefine 99.8%

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

       6. unpow2 99.8%

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

       7. unpow2 99.8%

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

       8. hypot-undefine 99.8%

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

       9. unpow2 99.8%

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

       10. pow-sqr 99.9%

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

       11. metadata-eval 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in a around 0 94.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{+76} \lor \neg \left(a \leq 8 \cdot 10^{+38}\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+28}:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8e+28)
   (+ -1.0 (* (pow a 4.0) (+ 1.0 (/ 4.0 a))))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8e+28) {
		tmp = -1.0 + (pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else {
		tmp = -1.0 + 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 <= 8d+28) then
        tmp = (-1.0d0) + ((a ** 4.0d0) * (1.0d0 + (4.0d0 / a)))
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8e+28) {
		tmp = -1.0 + (Math.pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8e+28:
		tmp = -1.0 + (math.pow(a, 4.0) * (1.0 + (4.0 / a)))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8e+28)
		tmp = Float64(-1.0 + Float64((a ^ 4.0) * Float64(1.0 + Float64(4.0 / a))));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8e+28)
		tmp = -1.0 + ((a ^ 4.0) * (1.0 + (4.0 / a)));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8e+28], N[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.99999999999999967e28

   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. Step-by-step derivation

      1. sub-neg 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in a around inf 72.5%

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

      1. associate-*r/ 72.5%

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

      2. metadata-eval 72.5%

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

   7. Simplified 72.5%

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

   if 7.99999999999999967e28 < b

   1. Initial program 67.6%

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

      1. sub-neg 67.6%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in b around inf 88.4%

      \[\leadsto \color{blue}{{b}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+28}:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+28}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e+28) (+ -1.0 (pow a 4.0)) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e+28) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + 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 <= 5d+28) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5e+28) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5e+28:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e+28)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e+28)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e+28], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999957e28

   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. Step-by-step derivation

      1. sub-neg 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in a around inf 72.5%

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

   if 4.99999999999999957e28 < b

   1. Initial program 67.6%

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

      1. sub-neg 67.6%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in b around inf 88.4%

      \[\leadsto \color{blue}{{b}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+28}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ -1 + {a}^{4} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. sub-neg 74.9%

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

3. Simplified 75.3%

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

5. Taylor expanded in a around inf 63.9%

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

6. Final simplification 63.9%

   \[\leadsto -1 + {a}^{4} \]
7. Add Preprocessing

Alternative 8: 24.9% 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 74.9%

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

   1. sub-neg 74.9%

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

3. Simplified 75.3%

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

5. Taylor expanded in a around inf 63.9%

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

6. Taylor expanded in a around 0 23.2%

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

Reproduce

herbie shell --seed 2024139 
(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))