Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.7% → 100.0%

Time: 15.8s

Alternatives: 10

Speedup: 1.2×

• 60.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(a + 3\right)\\ t_1 := {\left(a \cdot a + b \cdot b\right)}^{2}\\ \mathbf{if}\;t_1 + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + t_0\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, t_0\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(t_1 + \left(a \cdot a\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) (+ a 3.0))) (t_1 (pow (+ (* a a) (* b b)) 2.0)))
   (if (<= (+ t_1 (* 4.0 (+ (* (* a a) (- 1.0 a)) t_0))) 5e+305)
     (fma 4.0 (fma a (- a (* a a)) t_0) (+ (pow (hypot a b) 4.0) -1.0))
     (+ -1.0 (+ t_1 (* (* a a) 4.0))))))

C

double code(double a, double b) {
	double t_0 = (b * b) * (a + 3.0);
	double t_1 = pow(((a * a) + (b * b)), 2.0);
	double tmp;
	if ((t_1 + (4.0 * (((a * a) * (1.0 - a)) + t_0))) <= 5e+305) {
		tmp = fma(4.0, fma(a, (a - (a * a)), t_0), (pow(hypot(a, b), 4.0) + -1.0));
	} else {
		tmp = -1.0 + (t_1 + ((a * a) * 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * Float64(a + 3.0))
	t_1 = Float64(Float64(a * a) + Float64(b * b)) ^ 2.0
	tmp = 0.0
	if (Float64(t_1 + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + t_0))) <= 5e+305)
		tmp = fma(4.0, fma(a, Float64(a - Float64(a * a)), t_0), Float64((hypot(a, b) ^ 4.0) + -1.0));
	else
		tmp = Float64(-1.0 + Float64(t_1 + Float64(Float64(a * a) * 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+305], N[(4.0 * N[(a * N[(a - N[(a * a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(t$95$1 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a))))) < 5.00000000000000009e305

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. fma-def 99.7%

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

      5. associate-*l* 99.7%

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

      6. fma-def 99.7%

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

      7. distribute-lft-out-- 99.7%

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

      8. *-rgt-identity 99.7%

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

      9. +-commutative 99.7%

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

   3. Simplified 99.9%

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

   if 5.00000000000000009e305 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a)))))

   1. Initial program 62.0%

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

      1. +-commutative 62.0%

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

      2. +-commutative 62.0%

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

      3. *-commutative 62.0%

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

      4. fma-def 66.9%

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

      5. associate-*l* 66.9%

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

   3. Applied egg-rr 66.9%

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

   4. Taylor expanded in a around 0 84.9%

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

      1. unpow2 84.9%

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

   6. Simplified 84.9%

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

   7. Taylor expanded in a around inf 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(a \cdot a\right) \cdot 4\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2}\\ \mathbf{if}\;t_0 + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 10^{+64}:\\ \;\;\;\;-1 + \left(t_0 + 4 \cdot \mathsf{fma}\left(a + 3, b \cdot b, a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(t_0 + \left(a \cdot a\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (pow (+ (* a a) (* b b)) 2.0)))
   (if (<=
        (+ t_0 (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
        1e+64)
     (+ -1.0 (+ t_0 (* 4.0 (fma (+ a 3.0) (* b b) (* a (* a (- 1.0 a)))))))
     (+ -1.0 (+ t_0 (* (* a a) 4.0))))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0);
	double tmp;
	if ((t_0 + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 1e+64) {
		tmp = -1.0 + (t_0 + (4.0 * fma((a + 3.0), (b * b), (a * (a * (1.0 - a))))));
	} else {
		tmp = -1.0 + (t_0 + ((a * a) * 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b)) ^ 2.0
	tmp = 0.0
	if (Float64(t_0 + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0))))) <= 1e+64)
		tmp = Float64(-1.0 + Float64(t_0 + Float64(4.0 * fma(Float64(a + 3.0), Float64(b * b), Float64(a * Float64(a * Float64(1.0 - a)))))));
	else
		tmp = Float64(-1.0 + Float64(t_0 + Float64(Float64(a * a) * 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+64], N[(-1.0 + N[(t$95$0 + N[(4.0 * N[(N[(a + 3.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(t$95$0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a))))) < 1.00000000000000002e64

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. associate-*l* 99.9%

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

   3. Applied egg-rr 99.9%

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

   if 1.00000000000000002e64 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a)))))

   1. Initial program 67.1%

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

      1. +-commutative 67.1%

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

      2. +-commutative 67.1%

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

      3. *-commutative 67.1%

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

      4. fma-def 71.2%

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

      5. associate-*l* 71.2%

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

   3. Applied egg-rr 71.2%

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

   4. Taylor expanded in a around 0 86.9%

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

      1. unpow2 86.9%

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

   6. Simplified 86.9%

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

   7. Taylor expanded in a around inf 99.9%

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

      1. unpow2 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 10^{+64}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \mathsf{fma}\left(a + 3, b \cdot b, a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(a \cdot a\right) \cdot 4\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2}\\ t_1 := t_0 + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ \mathbf{if}\;t_1 \leq 10^{+64}:\\ \;\;\;\;t_1 + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(t_0 + \left(a \cdot a\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (pow (+ (* a a) (* b b)) 2.0))
        (t_1 (+ t_0 (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))))
   (if (<= t_1 1e+64) (+ t_1 -1.0) (+ -1.0 (+ t_0 (* (* a a) 4.0))))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0);
	double t_1 = t_0 + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_1 <= 1e+64) {
		tmp = t_1 + -1.0;
	} else {
		tmp = -1.0 + (t_0 + ((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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((a * a) + (b * b)) ** 2.0d0
    t_1 = t_0 + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (a + 3.0d0))))
    if (t_1 <= 1d+64) then
        tmp = t_1 + (-1.0d0)
    else
        tmp = (-1.0d0) + (t_0 + ((a * a) * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0);
	double t_1 = t_0 + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_1 <= 1e+64) {
		tmp = t_1 + -1.0;
	} else {
		tmp = -1.0 + (t_0 + ((a * a) * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0)
	t_1 = t_0 + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))
	tmp = 0
	if t_1 <= 1e+64:
		tmp = t_1 + -1.0
	else:
		tmp = -1.0 + (t_0 + ((a * a) * 4.0))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b)) ^ 2.0
	t_1 = Float64(t_0 + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_1 <= 1e+64)
		tmp = Float64(t_1 + -1.0);
	else
		tmp = Float64(-1.0 + Float64(t_0 + 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;
	t_1 = t_0 + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	tmp = 0.0;
	if (t_1 <= 1e+64)
		tmp = t_1 + -1.0;
	else
		tmp = -1.0 + (t_0 + ((a * a) * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+64], N[(t$95$1 + -1.0), $MachinePrecision], N[(-1.0 + N[(t$95$0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a))))) < 1.00000000000000002e64

   1. Initial program 99.9%

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

   if 1.00000000000000002e64 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a)))))

   1. Initial program 67.1%

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

      1. +-commutative 67.1%

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

      2. +-commutative 67.1%

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

      3. *-commutative 67.1%

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

      4. fma-def 71.2%

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

      5. associate-*l* 71.2%

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

   3. Applied egg-rr 71.2%

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

   4. Taylor expanded in a around 0 86.9%

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

      1. unpow2 86.9%

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

   6. Simplified 86.9%

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

   7. Taylor expanded in a around inf 99.9%

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

      1. unpow2 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 10^{+64}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(a \cdot a\right) \cdot 4\right)\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(a \cdot a\right) \cdot 4 + {\left(b \cdot b\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-9)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (+ (* (* a a) 4.0) (pow (* b b) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-9) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (((a * a) * 4.0) + pow((b * b), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d-9) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (((a * a) * 4.0d0) + ((b * b) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-9) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (((a * a) * 4.0) + Math.pow((b * b), 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-9:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + (((a * a) * 4.0) + math.pow((b * b), 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-9)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(a * a) * 4.0) + (Float64(b * b) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-9)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (((a * a) * 4.0) + ((b * b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-9], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(b * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e-9

   1. Initial program 83.9%

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

      1. sub-neg 83.9%

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

      2. sqr-pow 83.9%

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

      3. sqr-pow 83.9%

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

      4. sqr-neg 83.9%

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

      5. distribute-rgt-in 83.9%

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

      6. sqr-neg 83.9%

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

      7. distribute-rgt-in 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in a around inf 98.2%

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

   if 5.0000000000000001e-9 < (*.f64 b b)

   1. Initial program 67.0%

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

      1. +-commutative 67.0%

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

      2. +-commutative 67.0%

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

      3. *-commutative 67.0%

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

      4. fma-def 73.2%

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

      5. associate-*l* 73.2%

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

   3. Applied egg-rr 73.2%

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

   4. Taylor expanded in a around 0 80.8%

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

      1. unpow2 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in a around inf 98.2%

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

      1. unpow2 98.2%

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

   9. Simplified 98.2%

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

   10. Taylor expanded in a around 0 94.0%

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

       1. unpow2 94.0%

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

   12. Simplified 94.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(a \cdot a\right) \cdot 4 + {\left(b \cdot b\right)}^{2}\right)\\ \end{array} \]

Alternative 5: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

   2. +-commutative 75.3%

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

   3. *-commutative 75.3%

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

   4. fma-def 78.4%

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

   5. associate-*l* 78.4%

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

3. Applied egg-rr 78.4%

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

4. Taylor expanded in a around 0 89.4%

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

   1. unpow2 89.4%

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

6. Simplified 89.4%

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

7. Taylor expanded in a around inf 98.3%

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

   1. unpow2 98.3%

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

9. Simplified 98.3%

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

10. Final simplification 98.3%

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

Alternative 6: 93.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-9)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (+ (pow b 4.0) (* (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-9) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d-9) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-9) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-9:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-9)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-9)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-9], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e-9

   1. Initial program 83.9%

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

      1. sub-neg 83.9%

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

      2. sqr-pow 83.9%

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

      3. sqr-pow 83.9%

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

      4. sqr-neg 83.9%

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

      5. distribute-rgt-in 83.9%

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

      6. sqr-neg 83.9%

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

      7. distribute-rgt-in 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in a around inf 98.2%

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

   if 5.0000000000000001e-9 < (*.f64 b b)

   1. Initial program 67.0%

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

      1. sub-neg 67.0%

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

      2. sqr-pow 67.0%

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

      3. sqr-pow 67.0%

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

      4. sqr-neg 67.0%

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

      5. distribute-rgt-in 67.0%

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

      6. sqr-neg 67.0%

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

      7. distribute-rgt-in 67.0%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in a around 0 58.2%

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

      1. +-commutative 58.2%

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

      2. +-commutative 58.2%

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

      3. associate-+l+ 58.2%

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

      4. unpow2 58.2%

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

      5. unpow2 58.2%

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

      6. associate-*r* 58.2%

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

      7. distribute-rgt-in 72.0%

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

      8. metadata-eval 72.0%

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

      9. distribute-lft-in 72.0%

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

      10. *-commutative 72.0%

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

      11. distribute-lft-in 72.0%

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

      12. metadata-eval 72.0%

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

   6. Simplified 72.0%

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

   7. Taylor expanded in a around 0 91.0%

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

      1. unpow2 91.0%

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

   9. Simplified 91.0%

      \[\leadsto \left({b}^{4} + \color{blue}{12 \cdot \left(b \cdot b\right)}\right) + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 7: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-9)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-9) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d-9) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-9) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-9:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-9)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-9)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-9], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e-9

   1. Initial program 83.9%

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

      1. sub-neg 83.9%

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

      2. sqr-pow 83.9%

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

      3. sqr-pow 83.9%

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

      4. sqr-neg 83.9%

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

      5. distribute-rgt-in 83.9%

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

      6. sqr-neg 83.9%

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

      7. distribute-rgt-in 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in a around inf 98.2%

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

   if 5.0000000000000001e-9 < (*.f64 b b)

   1. Initial program 67.0%

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

      1. sub-neg 67.0%

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

      2. sqr-pow 67.0%

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

      3. sqr-pow 67.0%

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

      4. sqr-neg 67.0%

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

      5. distribute-rgt-in 67.0%

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

      6. sqr-neg 67.0%

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

      7. distribute-rgt-in 67.0%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in a around 0 58.2%

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

      1. +-commutative 58.2%

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

      2. +-commutative 58.2%

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

      3. associate-+l+ 58.2%

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

      4. unpow2 58.2%

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

      5. unpow2 58.2%

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

      6. associate-*r* 58.2%

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

      7. distribute-rgt-in 72.0%

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

      8. metadata-eval 72.0%

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

      9. distribute-lft-in 72.0%

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

      10. *-commutative 72.0%

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

      11. distribute-lft-in 72.0%

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

      12. metadata-eval 72.0%

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

   6. Simplified 72.0%

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

   7. Taylor expanded in a around 0 91.0%

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

      1. unpow2 91.0%

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

   9. Simplified 91.0%

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

       1. metadata-eval 91.0%

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

       2. pow-prod-up 90.9%

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

       3. pow2 90.9%

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

       4. pow2 90.9%

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

       5. distribute-rgt-out 90.9%

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

   11. Applied egg-rr 90.9%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 8: 54.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6.9 \cdot 10^{-6}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 6.9e-6) (+ -1.0 (* (* b b) 12.0)) (+ -1.0 (* b (* b (* a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 6.9e-6) {
		tmp = -1.0 + ((b * b) * 12.0);
	} else {
		tmp = -1.0 + (b * (b * (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 <= 6.9d-6) then
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    else
        tmp = (-1.0d0) + (b * (b * (a * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 6.9e-6) {
		tmp = -1.0 + ((b * b) * 12.0);
	} else {
		tmp = -1.0 + (b * (b * (a * 4.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 6.9e-6:
		tmp = -1.0 + ((b * b) * 12.0)
	else:
		tmp = -1.0 + (b * (b * (a * 4.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 6.9e-6)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(a * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 6.9e-6)
		tmp = -1.0 + ((b * b) * 12.0);
	else
		tmp = -1.0 + (b * (b * (a * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 6.9e-6], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.9e-6

   1. Initial program 85.4%

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

      1. sub-neg 85.4%

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

      2. sqr-pow 85.4%

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

      3. sqr-pow 85.4%

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

      4. sqr-neg 85.4%

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

      5. distribute-rgt-in 85.4%

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

      6. sqr-neg 85.4%

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

      7. distribute-rgt-in 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in a around 0 53.1%

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

      1. +-commutative 53.1%

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

      2. +-commutative 53.1%

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

      3. associate-+l+ 53.1%

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

      4. unpow2 53.1%

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

      5. unpow2 53.1%

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

      6. associate-*r* 53.1%

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

      7. distribute-rgt-in 62.1%

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

      8. metadata-eval 62.1%

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

      9. distribute-lft-in 62.1%

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

      10. *-commutative 62.1%

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

      11. distribute-lft-in 62.1%

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

      12. metadata-eval 62.1%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in b around 0 42.7%

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

      1. +-commutative 42.7%

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

      2. *-commutative 42.7%

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

      3. fma-udef 42.7%

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

      4. unpow2 42.7%

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

      5. associate-*l* 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in a around 0 53.8%

       \[\leadsto \color{blue}{12 \cdot {b}^{2}} + -1 \]
   11. Step-by-step derivation

       1. unpow2 53.8%

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

   12. Simplified 53.8%

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

   if 6.9e-6 < a

   1. Initial program 39.2%

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

      1. sub-neg 39.2%

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

      2. sqr-pow 39.2%

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

      3. sqr-pow 39.2%

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

      4. sqr-neg 39.2%

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

      5. distribute-rgt-in 39.2%

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

      6. sqr-neg 39.2%

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

      7. distribute-rgt-in 39.2%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in a around 0 41.8%

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

      1. +-commutative 41.8%

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

      2. +-commutative 41.8%

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

      3. associate-+l+ 41.8%

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

      4. unpow2 41.8%

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

      5. unpow2 41.8%

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

      6. associate-*r* 41.8%

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

      7. distribute-rgt-in 41.8%

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

      8. metadata-eval 41.8%

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

      9. distribute-lft-in 41.8%

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

      10. *-commutative 41.8%

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

      11. distribute-lft-in 41.8%

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

      12. metadata-eval 41.8%

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

   6. Simplified 41.8%

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

   7. Taylor expanded in b around 0 35.2%

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

      1. +-commutative 35.2%

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

      2. *-commutative 35.2%

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

      3. fma-udef 35.2%

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

      4. unpow2 35.2%

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

      5. associate-*l* 35.2%

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

   9. Simplified 35.2%

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

   10. Taylor expanded in a around inf 35.2%

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

       1. associate-*r* 35.2%

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

       2. *-commutative 35.2%

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

       3. *-commutative 35.2%

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

       4. *-commutative 35.2%

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

   12. Simplified 35.2%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.9 \cdot 10^{-6}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot 4\right)\right)\\ \end{array} \]

Alternative 9: 69.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ -1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
end function

Java

public static double code(double a, double b) {
	return -1.0 + ((b * b) * ((b * b) + 12.0));
}

Python

def code(a, b):
	return -1.0 + ((b * b) * ((b * b) + 12.0))

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
end

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. sub-neg 75.3%

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

   2. sqr-pow 75.3%

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

   3. sqr-pow 75.3%

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

   4. sqr-neg 75.3%

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

   5. distribute-rgt-in 75.3%

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

   6. sqr-neg 75.3%

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

   7. distribute-rgt-in 75.3%

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

3. Simplified 76.4%

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

4. Taylor expanded in a around 0 50.7%

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

   1. +-commutative 50.7%

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

   2. +-commutative 50.7%

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

   3. associate-+l+ 50.7%

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

   4. unpow2 50.7%

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

   5. unpow2 50.7%

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

   6. associate-*r* 50.7%

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

   7. distribute-rgt-in 57.7%

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

   8. metadata-eval 57.7%

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

   9. distribute-lft-in 57.7%

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

   10. *-commutative 57.7%

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

   11. distribute-lft-in 57.7%

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

   12. metadata-eval 57.7%

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

6. Simplified 57.7%

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

7. Taylor expanded in a around 0 67.4%

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

   1. unpow2 67.4%

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

9. Simplified 67.4%

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

    1. metadata-eval 67.4%

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

    2. pow-prod-up 67.3%

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

    3. pow2 67.3%

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

    4. pow2 67.3%

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

    5. distribute-rgt-out 67.3%

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

11. Applied egg-rr 67.3%

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

12. Final simplification 67.3%

    \[\leadsto -1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) \]

Alternative 10: 51.8% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ -1 + \left(b \cdot b\right) \cdot 12 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + ((b * b) * 12.0d0)
end function

Java

public static double code(double a, double b) {
	return -1.0 + ((b * b) * 12.0);
}

Python

def code(a, b):
	return -1.0 + ((b * b) * 12.0)

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + ((b * b) * 12.0);
end

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. sub-neg 75.3%

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

   2. sqr-pow 75.3%

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

   3. sqr-pow 75.3%

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

   4. sqr-neg 75.3%

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

   5. distribute-rgt-in 75.3%

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

   6. sqr-neg 75.3%

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

   7. distribute-rgt-in 75.3%

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

3. Simplified 76.4%

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

4. Taylor expanded in a around 0 50.7%

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

   1. +-commutative 50.7%

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

   2. +-commutative 50.7%

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

   3. associate-+l+ 50.7%

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

   4. unpow2 50.7%

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

   5. unpow2 50.7%

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

   6. associate-*r* 50.7%

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

   7. distribute-rgt-in 57.7%

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

   8. metadata-eval 57.7%

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

   9. distribute-lft-in 57.7%

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

   10. *-commutative 57.7%

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

   11. distribute-lft-in 57.7%

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

   12. metadata-eval 57.7%

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

6. Simplified 57.7%

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

7. Taylor expanded in b around 0 41.1%

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

   1. +-commutative 41.1%

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

   2. *-commutative 41.1%

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

   3. fma-udef 41.1%

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

   4. unpow2 41.1%

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

   5. associate-*l* 41.1%

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

9. Simplified 41.1%

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

10. Taylor expanded in a around 0 47.3%

    \[\leadsto \color{blue}{12 \cdot {b}^{2}} + -1 \]
11. Step-by-step derivation

    1. unpow2 47.3%

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

12. Simplified 47.3%

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

13. Final simplification 47.3%

    \[\leadsto -1 + \left(b \cdot b\right) \cdot 12 \]

Reproduce

herbie shell --seed 2023299 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))