Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.4% → 99.4%

Time: 7.1s

Alternatives: 9

Speedup: 9.8×

• 61.4% 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: 73.4% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+76}:\\ \;\;\;\;\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 4e+76)
   (+ (+ (pow (hypot a b) 4.0) (* 4.0 (* a (* a (- 1.0 a))))) -1.0)
   (+ -1.0 (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 4e+76) {
		tmp = (pow(hypot(a, b), 4.0) + (4.0 * (a * (a * (1.0 - a))))) + -1.0;
	} else {
		tmp = -1.0 + ((a * a) * (a * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 4e+76) {
		tmp = (Math.pow(Math.hypot(a, b), 4.0) + (4.0 * (a * (a * (1.0 - a))))) + -1.0;
	} else {
		tmp = -1.0 + ((a * a) * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 4e+76:
		tmp = (math.pow(math.hypot(a, b), 4.0) + (4.0 * (a * (a * (1.0 - a))))) + -1.0
	else:
		tmp = -1.0 + ((a * a) * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 4e+76)
		tmp = Float64(Float64((hypot(a, b) ^ 4.0) + Float64(4.0 * Float64(a * Float64(a * Float64(1.0 - a))))) + -1.0);
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 4e+76)
		tmp = ((hypot(a, b) ^ 4.0) + (4.0 * (a * (a * (1.0 - a))))) + -1.0;
	else
		tmp = -1.0 + ((a * a) * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 4e+76], N[(N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.0000000000000002e76

   1. Initial program 87.7%

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

      1. sub-neg 87.7%

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

      2. fma-def 87.7%

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

      3. fma-def 87.7%

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

      4. +-commutative 87.7%

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

      5. metadata-eval 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in b around 0 99.1%

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

      1. unpow2 99.1%

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

      2. associate-*r* 99.1%

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

   6. Simplified 99.1%

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

      1. fma-def 99.1%

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

      2. metadata-eval 99.1%

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

      3. sqrt-pow2 99.2%

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

      4. hypot-udef 99.2%

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

      5. expm1-log1p-u 97.8%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      6. expm1-udef 97.8%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

   8. Applied egg-rr 97.8%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]
   9. Step-by-step derivation

      1. expm1-def 97.8%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      2. expm1-log1p 99.2%

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

   10. Simplified 99.2%

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

   if 4.0000000000000002e76 < a

   1. Initial program 10.0%

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

      1. sub-neg 10.0%

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

      2. fma-def 10.0%

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

      3. fma-def 14.0%

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

      4. +-commutative 14.0%

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

      5. metadata-eval 14.0%

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

   3. Simplified 14.0%

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

   4. Taylor expanded in a around inf 100.0%

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

      1. metadata-eval 100.0%

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

      2. pow-prod-up 100.0%

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

      3. pow-prod-down 100.0%

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

      4. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+76}:\\ \;\;\;\;\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]

Alternative 2: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 99.8%

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

   if +inf.0 < (+.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 0.0%

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

      1. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 2.9%

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

      4. +-commutative 2.9%

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

      5. metadata-eval 2.9%

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

   3. Simplified 2.9%

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

   4. Taylor expanded in a around inf 94.7%

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

      1. metadata-eval 94.7%

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

      2. pow-prod-up 94.7%

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

      3. pow-prod-down 94.7%

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

      4. pow2 94.7%

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

   6. Applied egg-rr 94.7%

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

4. Final simplification 98.4%

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

Alternative 3: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+76}:\\ \;\;\;\;-1 + \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} - 4 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 4e+76)
   (+ -1.0 (- (pow (hypot a b) 4.0) (* 4.0 (* a (* a a)))))
   (+ -1.0 (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 4e+76) {
		tmp = -1.0 + (pow(hypot(a, b), 4.0) - (4.0 * (a * (a * a))));
	} else {
		tmp = -1.0 + ((a * a) * (a * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 4e+76) {
		tmp = -1.0 + (Math.pow(Math.hypot(a, b), 4.0) - (4.0 * (a * (a * a))));
	} else {
		tmp = -1.0 + ((a * a) * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 4e+76:
		tmp = -1.0 + (math.pow(math.hypot(a, b), 4.0) - (4.0 * (a * (a * a))))
	else:
		tmp = -1.0 + ((a * a) * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 4e+76)
		tmp = Float64(-1.0 + Float64((hypot(a, b) ^ 4.0) - Float64(4.0 * Float64(a * Float64(a * a)))));
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 4e+76)
		tmp = -1.0 + ((hypot(a, b) ^ 4.0) - (4.0 * (a * (a * a))));
	else
		tmp = -1.0 + ((a * a) * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 4e+76], N[(-1.0 + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] - N[(4.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.0000000000000002e76

   1. Initial program 87.7%

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

      1. sub-neg 87.7%

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

      2. fma-def 87.7%

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

      3. fma-def 87.7%

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

      4. +-commutative 87.7%

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

      5. metadata-eval 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in b around 0 99.1%

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

      1. unpow2 99.1%

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

      2. associate-*r* 99.1%

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

   6. Simplified 99.1%

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

      1. fma-def 99.1%

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

      2. metadata-eval 99.1%

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

      3. sqrt-pow2 99.2%

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

      4. hypot-udef 99.2%

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

      5. expm1-log1p-u 97.8%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      6. expm1-udef 97.8%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

   8. Applied egg-rr 97.8%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]
   9. Step-by-step derivation

      1. expm1-def 97.8%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      2. expm1-log1p 99.2%

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

   10. Simplified 99.2%

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

   11. Taylor expanded in a around inf 98.1%

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

       1. unpow2 98.1%

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

       2. mul-1-neg 98.1%

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

       3. distribute-rgt-neg-out 98.1%

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

   13. Simplified 98.1%

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

   if 4.0000000000000002e76 < a

   1. Initial program 10.0%

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

      1. sub-neg 10.0%

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

      2. fma-def 10.0%

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

      3. fma-def 14.0%

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

      4. +-commutative 14.0%

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

      5. metadata-eval 14.0%

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

   3. Simplified 14.0%

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

   4. Taylor expanded in a around inf 100.0%

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

      1. metadata-eval 100.0%

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

      2. pow-prod-up 100.0%

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

      3. pow-prod-down 100.0%

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

      4. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+76}:\\ \;\;\;\;-1 + \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} - 4 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]

Alternative 4: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+20}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot 12\right) + {b}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+20)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (+ (* b (* b 12.0)) (pow b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+20) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * (b * 12.0)) + 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 * b) <= 2d+20) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b * (b * 12.0d0)) + (b ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+20) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * (b * 12.0)) + Math.pow(b, 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+20:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * (b * 12.0)) + math.pow(b, 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+20)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * Float64(b * 12.0)) + (b ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+20)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b * (b * 12.0)) + (b ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+20], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e20

   1. Initial program 84.5%

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

      1. sub-neg 84.5%

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

      2. fma-def 84.6%

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

      3. fma-def 84.6%

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

      4. +-commutative 84.6%

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

      5. metadata-eval 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in a around inf 96.1%

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

   if 2e20 < (*.f64 b b)

   1. Initial program 59.9%

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

      1. sub-neg 59.9%

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

      2. fma-def 59.9%

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

      3. fma-def 61.5%

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

      4. +-commutative 61.5%

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

      5. metadata-eval 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in a around 0 61.1%

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

      1. associate-+r+ 61.1%

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

      2. associate-*r* 61.1%

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

      3. distribute-rgt-out 74.7%

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

      4. metadata-eval 74.7%

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

      5. distribute-lft-in 74.7%

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

      6. unpow2 74.7%

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

      7. distribute-rgt-in 74.7%

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

      8. metadata-eval 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in a around 0 91.8%

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

      1. unpow2 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*r* 91.8%

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

   9. Simplified 91.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+20}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot 12\right) + {b}^{4}\right)\\ \end{array} \]

Alternative 5: 83.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+304}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+304) (+ -1.0 (pow a 4.0)) (+ -1.0 (* b (* b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+304) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (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+304) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (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+304) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+304:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+304)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+304)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+304], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999997e304

   1. Initial program 79.4%

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

      1. sub-neg 79.4%

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

      2. fma-def 79.4%

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

      3. fma-def 80.0%

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

      4. +-commutative 80.0%

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

      5. metadata-eval 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in a around inf 80.4%

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

   if 4.9999999999999997e304 < (*.f64 b b)

   1. Initial program 54.3%

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

      1. sub-neg 54.3%

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

      2. fma-def 54.3%

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

      3. fma-def 55.7%

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

      4. +-commutative 55.7%

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

      5. metadata-eval 55.7%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in a around 0 50.0%

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

      1. associate-+r+ 50.0%

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

      2. associate-*r* 50.0%

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

      3. distribute-rgt-out 74.3%

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

      4. metadata-eval 74.3%

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

      5. distribute-lft-in 74.3%

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

      6. unpow2 74.3%

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

      7. distribute-rgt-in 74.3%

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

      8. metadata-eval 74.3%

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

   6. Simplified 74.3%

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

   7. Taylor expanded in b around 0 74.3%

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

      1. unpow2 74.3%

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

      2. *-commutative 74.3%

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

      3. +-commutative 74.3%

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

      4. *-commutative 74.3%

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

      5. fma-udef 74.3%

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

      6. associate-*l* 74.3%

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

      7. fma-udef 74.3%

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

      8. *-commutative 74.3%

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

      9. fma-def 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in a around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. associate-*l* 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+304}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 6: 93.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+20}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+20) (+ -1.0 (pow a 4.0)) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+20) {
		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 * b) <= 2d+20) 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 * b) <= 2e+20) {
		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 * b) <= 2e+20:
		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 (Float64(b * b) <= 2e+20)
		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 * b) <= 2e+20)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+20], 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 (*.f64 b b) < 2e20

   1. Initial program 84.5%

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

      1. sub-neg 84.5%

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

      2. fma-def 84.6%

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

      3. fma-def 84.6%

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

      4. +-commutative 84.6%

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

      5. metadata-eval 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in a around inf 96.1%

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

   if 2e20 < (*.f64 b b)

   1. Initial program 59.9%

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

      1. sub-neg 59.9%

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

      2. fma-def 59.9%

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

      3. fma-def 61.5%

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

      4. +-commutative 61.5%

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

      5. metadata-eval 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in b around inf 91.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+20}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 7: 83.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+304}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+304)
   (+ -1.0 (* (* a a) (* a a)))
   (+ -1.0 (* b (* b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+304) {
		tmp = -1.0 + ((a * a) * (a * a));
	} else {
		tmp = -1.0 + (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+304) then
        tmp = (-1.0d0) + ((a * a) * (a * a))
    else
        tmp = (-1.0d0) + (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+304) {
		tmp = -1.0 + ((a * a) * (a * a));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+304:
		tmp = -1.0 + ((a * a) * (a * a))
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+304)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+304)
		tmp = -1.0 + ((a * a) * (a * a));
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+304], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999997e304

   1. Initial program 79.4%

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

      1. sub-neg 79.4%

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

      2. fma-def 79.4%

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

      3. fma-def 80.0%

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

      4. +-commutative 80.0%

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

      5. metadata-eval 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in a around inf 80.4%

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

      1. metadata-eval 80.4%

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

      2. pow-prod-up 80.3%

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

      3. pow-prod-down 80.3%

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

      4. pow2 80.3%

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

   6. Applied egg-rr 80.3%

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

   if 4.9999999999999997e304 < (*.f64 b b)

   1. Initial program 54.3%

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

      1. sub-neg 54.3%

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

      2. fma-def 54.3%

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

      3. fma-def 55.7%

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

      4. +-commutative 55.7%

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

      5. metadata-eval 55.7%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in a around 0 50.0%

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

      1. associate-+r+ 50.0%

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

      2. associate-*r* 50.0%

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

      3. distribute-rgt-out 74.3%

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

      4. metadata-eval 74.3%

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

      5. distribute-lft-in 74.3%

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

      6. unpow2 74.3%

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

      7. distribute-rgt-in 74.3%

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

      8. metadata-eval 74.3%

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

   6. Simplified 74.3%

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

   7. Taylor expanded in b around 0 74.3%

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

      1. unpow2 74.3%

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

      2. *-commutative 74.3%

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

      3. +-commutative 74.3%

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

      4. *-commutative 74.3%

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

      5. fma-udef 74.3%

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

      6. associate-*l* 74.3%

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

      7. fma-udef 74.3%

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

      8. *-commutative 74.3%

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

      9. fma-def 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in a around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. associate-*l* 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+304}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 8: 69.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+283}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+283) (+ -1.0 (* a (* a 4.0))) (+ -1.0 (* b (* b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+283) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = -1.0 + (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+283) then
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    else
        tmp = (-1.0d0) + (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+283) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+283:
		tmp = -1.0 + (a * (a * 4.0))
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+283)
		tmp = Float64(-1.0 + Float64(a * Float64(a * 4.0)));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+283)
		tmp = -1.0 + (a * (a * 4.0));
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+283], N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000004e283

   1. Initial program 79.5%

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

      1. sub-neg 79.5%

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

      2. fma-def 79.5%

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

      3. fma-def 80.1%

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

      4. +-commutative 80.1%

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

      5. metadata-eval 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in b around 0 66.3%

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

      1. associate-*r* 66.3%

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

      2. unpow2 66.3%

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

   6. Simplified 66.3%

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

   7. Taylor expanded in a around 0 60.8%

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

      1. unpow2 60.8%

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

      2. associate-*r* 60.8%

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

   9. Simplified 60.8%

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

   if 5.0000000000000004e283 < (*.f64 b b)

   1. Initial program 55.4%

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

      1. sub-neg 55.4%

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

      2. fma-def 55.4%

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

      3. fma-def 56.8%

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

      4. +-commutative 56.8%

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

      5. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in a around 0 51.4%

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

      1. associate-+r+ 51.4%

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

      2. associate-*r* 51.4%

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

      3. distribute-rgt-out 74.3%

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

      4. metadata-eval 74.3%

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

      5. distribute-lft-in 74.3%

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

      6. unpow2 74.3%

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

      7. distribute-rgt-in 74.3%

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

      8. metadata-eval 74.3%

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

   6. Simplified 74.3%

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

   7. Taylor expanded in b around 0 70.7%

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

      1. unpow2 70.7%

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

      2. *-commutative 70.7%

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

      3. +-commutative 70.7%

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

      4. *-commutative 70.7%

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

      5. fma-udef 70.7%

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

      6. associate-*l* 70.7%

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

      7. fma-udef 70.7%

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

      8. *-commutative 70.7%

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

      9. fma-def 70.7%

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

   9. Simplified 70.7%

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

   10. Taylor expanded in a around 0 95.2%

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

       1. *-commutative 95.2%

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

       2. unpow2 95.2%

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

       3. associate-*l* 95.2%

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

   12. Simplified 95.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+283}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 9: 51.4% accurate, 18.3× speedup

Math

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

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(b * Float64(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[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

   1. sub-neg 72.5%

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

   2. fma-def 72.5%

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

   3. fma-def 73.3%

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

   4. +-commutative 73.3%

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

   5. metadata-eval 73.3%

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

3. Simplified 73.3%

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

4. Taylor expanded in a around 0 56.4%

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

   1. associate-+r+ 56.4%

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

   2. associate-*r* 56.4%

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

   3. distribute-rgt-out 63.1%

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

   4. metadata-eval 63.1%

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

   5. distribute-lft-in 63.1%

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

   6. unpow2 63.1%

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

   7. distribute-rgt-in 63.1%

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

   8. metadata-eval 63.1%

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

6. Simplified 63.1%

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

7. Taylor expanded in b around 0 50.6%

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

   1. unpow2 50.6%

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

   2. *-commutative 50.6%

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

   3. +-commutative 50.6%

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

   4. *-commutative 50.6%

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

   5. fma-udef 50.6%

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

   6. associate-*l* 50.6%

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

   7. fma-udef 50.6%

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

   8. *-commutative 50.6%

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

   9. fma-def 50.6%

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

9. Simplified 50.6%

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

10. Taylor expanded in a around 0 54.9%

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

    1. *-commutative 54.9%

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

    2. unpow2 54.9%

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

    3. associate-*l* 54.9%

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

12. Simplified 54.9%

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

13. Final simplification 54.9%

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

Reproduce

herbie shell --seed 2023192 
(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))