Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.7% → 98.9%

Time: 7.3s

Alternatives: 11

Speedup: 8.5×

• 59.8% 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: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + b \cdot \left(b \cdot 12\right)\right) + -1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+ (+ (pow (sqrt (hypot a b)) 8.0) (* b (* b 12.0))) -1.0))

C

double code(double a, double b) {
	return (pow(sqrt(hypot(a, b)), 8.0) + (b * (b * 12.0))) + -1.0;
}

Java

public static double code(double a, double b) {
	return (Math.pow(Math.sqrt(Math.hypot(a, b)), 8.0) + (b * (b * 12.0))) + -1.0;
}

Python

def code(a, b):
	return (math.pow(math.sqrt(math.hypot(a, b)), 8.0) + (b * (b * 12.0))) + -1.0

Julia

function code(a, b)
	return Float64(Float64((sqrt(hypot(a, b)) ^ 8.0) + Float64(b * Float64(b * 12.0))) + -1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((sqrt(hypot(a, b)) ^ 8.0) + (b * (b * 12.0))) + -1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[Sqrt[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision]], $MachinePrecision], 8.0], $MachinePrecision] + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 74.8%

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

   1. sub-neg 74.8%

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

   2. fma-def 74.8%

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

   3. fma-def 77.6%

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

   4. +-commutative 77.6%

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

   5. metadata-eval 77.6%

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

3. Simplified 77.6%

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

   1. fma-def 77.6%

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

   2. metadata-eval 77.6%

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

   3. sqrt-pow2 77.7%

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

   4. hypot-udef 77.7%

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

   5. add-sqr-sqrt 77.4%

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

   6. unpow-prod-down 77.4%

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

5. Applied egg-rr 77.4%

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

   1. pow-sqr 77.4%

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

   2. metadata-eval 77.4%

      \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{\color{blue}{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

7. Simplified 77.4%

   \[\leadsto \left(\color{blue}{{\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

8. Taylor expanded in a around 0 99.1%

   \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + 4 \cdot \color{blue}{\left(3 \cdot {b}^{2}\right)}\right) + -1 \]
9. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + 4 \cdot \left(3 \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) + -1 \]

   2. associate-*r* 99.1%

      \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + 4 \cdot \color{blue}{\left(\left(3 \cdot b\right) \cdot b\right)}\right) + -1 \]

10. Simplified 99.1%

    \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + 4 \cdot \color{blue}{\left(\left(3 \cdot b\right) \cdot b\right)}\right) + -1 \]

11. Taylor expanded in b around 0 99.1%

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

    1. unpow2 99.1%

       \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + 12 \cdot \color{blue}{\left(b \cdot b\right)}\right) + -1 \]

    2. *-commutative 99.1%

       \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + \color{blue}{\left(b \cdot b\right) \cdot 12}\right) + -1 \]

    3. associate-*r* 99.1%

       \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + \color{blue}{b \cdot \left(b \cdot 12\right)}\right) + -1 \]

13. Simplified 99.1%

    \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + \color{blue}{b \cdot \left(b \cdot 12\right)}\right) + -1 \]

14. Final simplification 99.1%

    \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + b \cdot \left(b \cdot 12\right)\right) + -1 \]

Alternative 2: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - 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) + (4.0 * (1.0 - 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(a * a) * Float64(1.0 - 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(Float64(a * a) + Float64(4.0 * Float64(1.0 - a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - 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) + (4.0 * (1.0 - 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[(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$0, Infinity], N[(-1.0 + t$95$0), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(4.0 * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $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 10.9%

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

      4. +-commutative 10.9%

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

      5. metadata-eval 10.9%

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

   3. Simplified 10.9%

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

   4. Taylor expanded in b around 0 25.4%

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

      1. associate-*r* 25.4%

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

      2. unpow2 25.4%

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

   6. Simplified 25.4%

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

      1. sqr-pow 25.4%

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

      2. metadata-eval 25.4%

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

      3. pow2 25.4%

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

      4. metadata-eval 25.4%

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

      5. pow2 25.4%

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

      6. fma-def 25.4%

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

      7. *-commutative 25.4%

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

   8. Applied egg-rr 25.4%

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

      1. fma-udef 25.4%

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

      2. associate-*r* 25.4%

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

      3. distribute-rgt-out 94.2%

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

   10. Applied egg-rr 94.2%

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

4. Final simplification 98.4%

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

Alternative 3: 94.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10000000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(12 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 10000000.0)
   (+ -1.0 (* (* a a) (+ (* a a) (* 4.0 (- 1.0 a)))))
   (+ -1.0 (+ (* 12.0 (* b b)) (pow b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 10000000.0) {
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	} else {
		tmp = -1.0 + ((12.0 * (b * b)) + pow(b, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 10000000.0d0) then
        tmp = (-1.0d0) + ((a * a) * ((a * a) + (4.0d0 * (1.0d0 - a))))
    else
        tmp = (-1.0d0) + ((12.0d0 * (b * b)) + (b ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 10000000.0) {
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	} else {
		tmp = -1.0 + ((12.0 * (b * b)) + Math.pow(b, 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 10000000.0:
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))))
	else:
		tmp = -1.0 + ((12.0 * (b * b)) + math.pow(b, 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 10000000.0)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(Float64(a * a) + Float64(4.0 * Float64(1.0 - a)))));
	else
		tmp = Float64(-1.0 + Float64(Float64(12.0 * Float64(b * b)) + (b ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 10000000.0)
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	else
		tmp = -1.0 + ((12.0 * (b * b)) + (b ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 10000000.0], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(4.0 * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e7

   1. Initial program 85.7%

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

      1. sub-neg 85.7%

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

      2. fma-def 85.7%

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

      3. fma-def 85.7%

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

      4. +-commutative 85.7%

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

      5. metadata-eval 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in b around 0 84.5%

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

      1. associate-*r* 84.5%

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

      2. unpow2 84.5%

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

   6. Simplified 84.5%

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

      1. sqr-pow 84.4%

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

      2. metadata-eval 84.4%

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

      3. pow2 84.4%

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

      4. metadata-eval 84.4%

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

      5. pow2 84.4%

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

      6. fma-def 84.4%

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

      7. *-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. fma-udef 84.4%

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

      2. associate-*r* 84.4%

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

      3. distribute-rgt-out 98.6%

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

   10. Applied egg-rr 98.6%

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

   if 1e7 < (*.f64 b b)

   1. Initial program 64.1%

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

      1. sub-neg 64.1%

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

      2. fma-def 64.1%

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

      3. fma-def 69.6%

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

      4. +-commutative 69.6%

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

      5. metadata-eval 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in a around 0 64.8%

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

      1. associate-+r+ 64.8%

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

      2. associate-*r* 64.8%

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

      3. distribute-rgt-out 77.2%

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

      4. metadata-eval 77.2%

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

      5. distribute-lft-in 77.2%

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

      6. unpow2 77.2%

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

      7. distribute-rgt-in 77.2%

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

      8. metadata-eval 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in a around 0 90.5%

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

      1. unpow2 90.5%

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

   9. Simplified 90.5%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10000000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(12 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \end{array} \]

Alternative 4: 94.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000000000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1000000000000.0)
   (+ -1.0 (* (* a a) (+ (* a a) (* 4.0 (- 1.0 a)))))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000000000.0) {
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1000000000000.0d0) then
        tmp = (-1.0d0) + ((a * a) * ((a * a) + (4.0d0 * (1.0d0 - a))))
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000000000.0) {
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1000000000000.0:
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1000000000000.0)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(Float64(a * a) + Float64(4.0 * Float64(1.0 - a)))));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1000000000000.0)
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1000000000000.0], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(4.0 * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e12

   1. Initial program 85.2%

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

      1. sub-neg 85.2%

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

      2. fma-def 85.2%

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

      3. fma-def 85.2%

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

      4. +-commutative 85.2%

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

      5. metadata-eval 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in b around 0 83.2%

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

      1. associate-*r* 83.2%

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

      2. unpow2 83.2%

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

   6. Simplified 83.2%

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

      1. sqr-pow 83.1%

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

      2. metadata-eval 83.1%

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

      3. pow2 83.1%

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

      4. metadata-eval 83.1%

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

      5. pow2 83.1%

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

      6. fma-def 83.1%

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

      7. *-commutative 83.1%

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

   8. Applied egg-rr 83.1%

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

      1. fma-udef 83.1%

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

      2. associate-*r* 83.1%

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

      3. distribute-rgt-out 97.9%

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

   10. Applied egg-rr 97.9%

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

   if 1e12 < (*.f64 b b)

   1. Initial program 64.4%

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

      1. sub-neg 64.4%

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

      2. fma-def 64.4%

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

      3. fma-def 69.9%

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

      4. +-commutative 69.9%

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

      5. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in b around inf 91.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000000000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 5: 94.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10000000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(12 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 10000000.0)
   (+ -1.0 (* (* a a) (+ (* a a) (* 4.0 (- 1.0 a)))))
   (+ -1.0 (* (* b b) (+ 12.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 10000000.0) {
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	} else {
		tmp = -1.0 + ((b * b) * (12.0 + (b * b)));
	}
	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) <= 10000000.0d0) then
        tmp = (-1.0d0) + ((a * a) * ((a * a) + (4.0d0 * (1.0d0 - a))))
    else
        tmp = (-1.0d0) + ((b * b) * (12.0d0 + (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 10000000.0) {
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	} else {
		tmp = -1.0 + ((b * b) * (12.0 + (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 10000000.0:
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))))
	else:
		tmp = -1.0 + ((b * b) * (12.0 + (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 10000000.0)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(Float64(a * a) + Float64(4.0 * Float64(1.0 - a)))));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(12.0 + Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 10000000.0)
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	else
		tmp = -1.0 + ((b * b) * (12.0 + (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 10000000.0], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(4.0 * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(12.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e7

   1. Initial program 85.7%

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

      1. sub-neg 85.7%

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

      2. fma-def 85.7%

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

      3. fma-def 85.7%

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

      4. +-commutative 85.7%

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

      5. metadata-eval 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in b around 0 84.5%

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

      1. associate-*r* 84.5%

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

      2. unpow2 84.5%

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

   6. Simplified 84.5%

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

      1. sqr-pow 84.4%

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

      2. metadata-eval 84.4%

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

      3. pow2 84.4%

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

      4. metadata-eval 84.4%

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

      5. pow2 84.4%

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

      6. fma-def 84.4%

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

      7. *-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. fma-udef 84.4%

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

      2. associate-*r* 84.4%

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

      3. distribute-rgt-out 98.6%

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

   10. Applied egg-rr 98.6%

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

   if 1e7 < (*.f64 b b)

   1. Initial program 64.1%

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

      1. sub-neg 64.1%

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

      2. fma-def 64.1%

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

      3. fma-def 69.6%

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

      4. +-commutative 69.6%

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

      5. metadata-eval 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in a around 0 64.8%

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

      1. associate-+r+ 64.8%

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

      2. associate-*r* 64.8%

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

      3. distribute-rgt-out 77.2%

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

      4. metadata-eval 77.2%

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

      5. distribute-lft-in 77.2%

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

      6. unpow2 77.2%

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

      7. distribute-rgt-in 77.2%

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

      8. metadata-eval 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in a around 0 90.5%

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

      1. unpow2 90.5%

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

   9. Simplified 90.5%

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

       1. +-commutative 90.5%

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

       2. sqr-pow 90.3%

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

       3. metadata-eval 90.3%

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

       4. pow2 90.3%

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

       5. metadata-eval 90.3%

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

       6. pow2 90.3%

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

       7. distribute-rgt-out 90.3%

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

   11. Applied egg-rr 90.3%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10000000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(12 + b \cdot b\right)\\ \end{array} \]

Alternative 6: 85.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+153} \lor \neg \left(a \leq 6.8 \cdot 10^{+153}\right):\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(12 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -7e+153) (not (<= a 6.8e+153)))
   (+ -1.0 (* (* a a) 4.0))
   (+ -1.0 (* (* b b) (+ 12.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -7e+153) || !(a <= 6.8e+153)) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + ((b * b) * (12.0 + (b * b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-7d+153)) .or. (.not. (a <= 6.8d+153))) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * (12.0d0 + (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -7e+153) || !(a <= 6.8e+153)) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + ((b * b) * (12.0 + (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -7e+153) or not (a <= 6.8e+153):
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = -1.0 + ((b * b) * (12.0 + (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -7e+153) || !(a <= 6.8e+153))
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(12.0 + Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -7e+153) || ~((a <= 6.8e+153)))
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = -1.0 + ((b * b) * (12.0 + (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -7e+153], N[Not[LessEqual[a, 6.8e+153]], $MachinePrecision]], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(12.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.9999999999999998e153 or 6.7999999999999995e153 < a

   1. Initial program 27.1%

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

      1. sub-neg 27.1%

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

      2. fma-def 27.1%

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

      3. fma-def 27.1%

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

      4. +-commutative 27.1%

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

      5. metadata-eval 27.1%

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

   3. Simplified 27.1%

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

   4. Taylor expanded in b around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. unpow2 45.8%

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

   6. Simplified 45.8%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if -6.9999999999999998e153 < a < 6.7999999999999995e153

   1. Initial program 89.1%

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

      1. sub-neg 89.1%

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

      2. fma-def 89.1%

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

      3. fma-def 92.7%

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

      4. +-commutative 92.7%

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

      5. metadata-eval 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in a around 0 70.3%

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

      1. associate-+r+ 70.3%

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

      2. associate-*r* 70.3%

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

      3. distribute-rgt-out 78.5%

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

      4. metadata-eval 78.5%

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

      5. distribute-lft-in 78.5%

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

      6. unpow2 78.5%

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

      7. distribute-rgt-in 78.5%

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

      8. metadata-eval 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in a around 0 83.0%

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

      1. unpow2 83.0%

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

   9. Simplified 83.0%

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

       1. +-commutative 83.0%

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

       2. sqr-pow 82.9%

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

       3. metadata-eval 82.9%

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

       4. pow2 82.9%

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

       5. metadata-eval 82.9%

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

       6. pow2 82.9%

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

       7. distribute-rgt-out 82.9%

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

   11. Applied egg-rr 82.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+153} \lor \neg \left(a \leq 6.8 \cdot 10^{+153}\right):\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(12 + b \cdot b\right)\\ \end{array} \]

Alternative 7: 57.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+89}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\left(a + 3\right) \cdot \left(\left(b \cdot b\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e+89) (+ -1.0 (* (* a a) 4.0)) (* (+ a 3.0) (* (* b b) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+89) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = (a + 3.0) * ((b * b) * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.5d+89) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = (a + 3.0d0) * ((b * b) * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+89) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = (a + 3.0) * ((b * b) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.5e+89:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = (a + 3.0) * ((b * b) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e+89)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = Float64(Float64(a + 3.0) * Float64(Float64(b * b) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.5e+89)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = (a + 3.0) * ((b * b) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e+89], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(a + 3.0), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.50000000000000006e89

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. fma-def 76.9%

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

      3. fma-def 78.9%

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

      4. +-commutative 78.9%

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

      5. metadata-eval 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in b around 0 60.8%

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

      1. associate-*r* 60.8%

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

      2. unpow2 60.8%

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

   6. Simplified 60.8%

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

   7. Taylor expanded in a around 0 57.5%

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

      1. unpow2 57.5%

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

   9. Simplified 57.5%

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

   if 1.50000000000000006e89 < b

   1. Initial program 65.2%

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

      1. sub-neg 65.2%

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

      2. fma-def 65.2%

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

      3. fma-def 71.7%

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

      4. +-commutative 71.7%

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

      5. metadata-eval 71.7%

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

   3. Simplified 71.7%

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

      1. fma-def 71.7%

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

      2. metadata-eval 71.7%

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

      3. sqrt-pow2 71.7%

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

      4. hypot-udef 71.7%

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

      5. add-sqr-sqrt 71.7%

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

      6. unpow-prod-down 71.7%

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

   5. Applied egg-rr 71.7%

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

      1. pow-sqr 71.7%

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

      2. metadata-eval 71.7%

         \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{\color{blue}{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

   7. Simplified 71.7%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

   8. Taylor expanded in b around inf 71.0%

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

      1. +-commutative 71.0%

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

      2. unpow2 71.0%

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

      3. associate-*l* 71.0%

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

   10. Simplified 71.0%

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

   11. Taylor expanded in b around inf 71.0%

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

       1. unpow2 71.0%

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

       2. associate-*r* 71.0%

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

       3. *-commutative 71.0%

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

   13. Simplified 71.0%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+89}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\left(a + 3\right) \cdot \left(\left(b \cdot b\right) \cdot 4\right)\\ \end{array} \]

Alternative 8: 60.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+127}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + 4 \cdot \left(b \cdot \left(b \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+127)
   (+ -1.0 (* (* a a) 4.0))
   (+ -1.0 (* 4.0 (* b (* b 3.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+127) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + (4.0 * (b * (b * 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.35d+127) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = (-1.0d0) + (4.0d0 * (b * (b * 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+127) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + (4.0 * (b * (b * 3.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.35e+127:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = -1.0 + (4.0 * (b * (b * 3.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+127)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = Float64(-1.0 + Float64(4.0 * Float64(b * Float64(b * 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e+127)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = -1.0 + (4.0 * (b * (b * 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.35e+127], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(4.0 * N[(b * N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3500000000000001e127

   1. Initial program 75.7%

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

      1. sub-neg 75.7%

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

      2. fma-def 75.7%

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

      3. fma-def 77.5%

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

      4. +-commutative 77.5%

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

      5. metadata-eval 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in b around 0 58.1%

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

      1. associate-*r* 58.1%

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

      2. unpow2 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in a around 0 57.2%

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

      1. unpow2 57.2%

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

   9. Simplified 57.2%

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

   if 1.3500000000000001e127 < b

   1. Initial program 69.4%

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

      1. sub-neg 69.4%

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

      2. fma-def 69.4%

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

      3. fma-def 77.8%

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

      4. +-commutative 77.8%

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

      5. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

      1. fma-def 77.8%

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

      2. metadata-eval 77.8%

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

      3. sqrt-pow2 77.8%

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

      4. hypot-udef 77.8%

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

      5. add-sqr-sqrt 77.8%

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

      6. unpow-prod-down 77.8%

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

   5. Applied egg-rr 77.8%

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

      1. pow-sqr 77.8%

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

      2. metadata-eval 77.8%

         \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{\color{blue}{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

   7. Simplified 77.8%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

   8. Taylor expanded in b around inf 76.1%

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

      1. +-commutative 76.1%

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

      2. unpow2 76.1%

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

      3. associate-*l* 76.1%

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

   10. Simplified 76.1%

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

   11. Taylor expanded in a around 0 84.8%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+127}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + 4 \cdot \left(b \cdot \left(b \cdot 3\right)\right)\\ \end{array} \]

Alternative 9: 33.9% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3100.0) -1.0 (* a (* (* b b) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3100.0) {
		tmp = -1.0;
	} else {
		tmp = a * ((b * 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 <= 3100.0d0) then
        tmp = -1.0d0
    else
        tmp = a * ((b * b) * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3100.0) {
		tmp = -1.0;
	} else {
		tmp = a * ((b * b) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3100.0:
		tmp = -1.0
	else:
		tmp = a * ((b * b) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3100.0)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(Float64(b * b) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3100.0)
		tmp = -1.0;
	else
		tmp = a * ((b * b) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3100.0], -1.0, N[(a * N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3100

   1. Initial program 77.4%

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

      1. sub-neg 77.4%

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

      2. fma-def 77.4%

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

      3. fma-def 79.5%

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

      4. +-commutative 79.5%

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

      5. metadata-eval 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in a around 0 56.5%

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

      1. associate-+r+ 56.5%

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

      2. associate-*r* 56.5%

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

      3. distribute-rgt-out 60.7%

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

      4. metadata-eval 60.7%

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

      5. distribute-lft-in 60.7%

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

      6. unpow2 60.7%

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

      7. distribute-rgt-in 60.7%

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

      8. metadata-eval 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in b around 0 35.7%

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

   if 3100 < b

   1. Initial program 67.5%

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

      1. sub-neg 67.5%

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

      2. fma-def 67.5%

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

      3. fma-def 72.1%

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

      4. +-commutative 72.1%

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

      5. metadata-eval 72.1%

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

   3. Simplified 72.1%

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

      1. fma-def 72.1%

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

      2. metadata-eval 72.1%

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

      3. sqrt-pow2 72.3%

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

      4. hypot-udef 72.3%

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

      5. add-sqr-sqrt 71.9%

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

      6. unpow-prod-down 71.8%

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

   5. Applied egg-rr 71.8%

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

      1. pow-sqr 71.9%

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

      2. metadata-eval 71.9%

         \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{\color{blue}{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

   7. Simplified 71.9%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

   8. Taylor expanded in b around inf 51.7%

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

      1. +-commutative 51.7%

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

      2. unpow2 51.7%

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

      3. associate-*l* 51.7%

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

   10. Simplified 51.7%

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

   11. Taylor expanded in a around inf 38.1%

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

       1. unpow2 38.1%

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

       2. associate-*r* 38.1%

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

       3. *-commutative 38.1%

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

       4. associate-*l* 38.1%

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

   13. Simplified 38.1%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot 4\right)\\ \end{array} \]

Alternative 10: 54.0% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{+88}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.4e+88) (+ -1.0 (* (* a a) 4.0)) (* a (* (* b b) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.4e+88) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = a * ((b * 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 <= 6.4d+88) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = a * ((b * b) * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.4e+88) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = a * ((b * b) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.4e+88:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = a * ((b * b) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.4e+88)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = Float64(a * Float64(Float64(b * b) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.4e+88)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = a * ((b * b) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.4e+88], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.3999999999999997e88

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. fma-def 76.9%

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

      3. fma-def 78.9%

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

      4. +-commutative 78.9%

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

      5. metadata-eval 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in b around 0 60.8%

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

      1. associate-*r* 60.8%

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

      2. unpow2 60.8%

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

   6. Simplified 60.8%

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

   7. Taylor expanded in a around 0 57.5%

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

      1. unpow2 57.5%

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

   9. Simplified 57.5%

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

   if 6.3999999999999997e88 < b

   1. Initial program 65.2%

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

      1. sub-neg 65.2%

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

      2. fma-def 65.2%

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

      3. fma-def 71.7%

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

      4. +-commutative 71.7%

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

      5. metadata-eval 71.7%

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

   3. Simplified 71.7%

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

      1. fma-def 71.7%

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

      2. metadata-eval 71.7%

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

      3. sqrt-pow2 71.7%

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

      4. hypot-udef 71.7%

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

      5. add-sqr-sqrt 71.7%

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

      6. unpow-prod-down 71.7%

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

   5. Applied egg-rr 71.7%

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

      1. pow-sqr 71.7%

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

      2. metadata-eval 71.7%

         \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{\color{blue}{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

   7. Simplified 71.7%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8}} + 4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

   8. Taylor expanded in b around inf 71.0%

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

      1. +-commutative 71.0%

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

      2. unpow2 71.0%

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

      3. associate-*l* 71.0%

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

   10. Simplified 71.0%

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

   11. Taylor expanded in a around inf 52.7%

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

       1. unpow2 52.7%

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

       2. associate-*r* 52.7%

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

       3. *-commutative 52.7%

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

       4. associate-*l* 52.7%

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

   13. Simplified 52.7%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{+88}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot 4\right)\\ \end{array} \]

Alternative 11: 25.7% accurate, 128.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.8%

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

   1. sub-neg 74.8%

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

   2. fma-def 74.8%

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

   3. fma-def 77.6%

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

   4. +-commutative 77.6%

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

   5. metadata-eval 77.6%

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

3. Simplified 77.6%

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

4. Taylor expanded in a around 0 59.9%

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

   1. associate-+r+ 59.9%

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

   2. associate-*r* 59.9%

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

   3. distribute-rgt-out 66.1%

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

   4. metadata-eval 66.1%

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

   5. distribute-lft-in 66.1%

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

   6. unpow2 66.1%

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

   7. distribute-rgt-in 66.1%

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

   8. metadata-eval 66.1%

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

6. Simplified 66.1%

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

7. Taylor expanded in b around 0 26.8%

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

8. Final simplification 26.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023200 
(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))