Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.1% → 99.1%

Time: 8.6s

Alternatives: 9

Speedup: 1.2×

• 61.0% 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: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 10^{+77}:\\ \;\;\;\;\left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1e+77)
   (+ (+ (pow (sqrt (hypot a b)) 8.0) (* 4.0 (* a (* a (- 1.0 a))))) -1.0)
   (+ -1.0 (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1e+77) {
		tmp = (pow(sqrt(hypot(a, b)), 8.0) + (4.0 * (a * (a * (1.0 - a))))) + -1.0;
	} else {
		tmp = -1.0 + pow(a, 4.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1e+77) {
		tmp = (Math.pow(Math.sqrt(Math.hypot(a, b)), 8.0) + (4.0 * (a * (a * (1.0 - a))))) + -1.0;
	} else {
		tmp = -1.0 + Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1e+77:
		tmp = (math.pow(math.sqrt(math.hypot(a, b)), 8.0) + (4.0 * (a * (a * (1.0 - a))))) + -1.0
	else:
		tmp = -1.0 + math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1e+77)
		tmp = Float64(Float64((sqrt(hypot(a, b)) ^ 8.0) + Float64(4.0 * Float64(a * Float64(a * Float64(1.0 - a))))) + -1.0);
	else
		tmp = Float64(-1.0 + (a ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1e+77)
		tmp = ((sqrt(hypot(a, b)) ^ 8.0) + (4.0 * (a * (a * (1.0 - a))))) + -1.0;
	else
		tmp = -1.0 + (a ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1e+77], N[(N[(N[Power[N[Sqrt[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision]], $MachinePrecision], 8.0], $MachinePrecision] + N[(4.0 * N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.99999999999999983e76

   1. Initial program 83.5%

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

      1. sub-neg 83.5%

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

      2. fma-def 83.5%

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

      3. fma-def 83.5%

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

      4. +-commutative 83.5%

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

      5. metadata-eval 83.5%

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

   3. Simplified 83.5%

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

      1. fma-def 83.5%

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

      2. metadata-eval 83.5%

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

      3. sqrt-pow2 83.6%

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

      4. hypot-udef 83.6%

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

      5. add-sqr-sqrt 83.4%

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

      6. unpow-prod-down 83.3%

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

   5. Applied egg-rr 83.3%

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

      1. pow-sqr 83.3%

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

      2. metadata-eval 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in b around 0 98.7%

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

      1. unpow2 98.7%

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

      2. associate-*r* 98.7%

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

   10. Simplified 98.7%

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

   if 9.99999999999999983e76 < a

   1. Initial program 11.1%

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

      1. sub-neg 11.1%

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

      2. fma-def 11.1%

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

      3. fma-def 19.4%

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

      4. +-commutative 19.4%

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

      5. metadata-eval 19.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in a around inf 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{+77}:\\ \;\;\;\;\left({\left(\sqrt{\mathsf{hypot}\left(a, b\right)}\right)}^{8} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(1.0 - a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(-1.0 + t$95$0), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $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 99.9%

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

   if +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 4.4%

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

      4. +-commutative 4.4%

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

      5. metadata-eval 4.4%

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

   3. Simplified 4.4%

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

   4. Taylor expanded in a around inf 94.5%

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

4. Final simplification 98.4%

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

Alternative 3: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00135:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.00135)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (+ (pow b 4.0) (* (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00135) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.00135d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.00135) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.00135:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00135)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.00135)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00135], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0013500000000000001

   1. Initial program 78.6%

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

      1. sub-neg 78.6%

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

      2. fma-def 78.6%

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

      3. fma-def 79.1%

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

      4. +-commutative 79.1%

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

      5. metadata-eval 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in a around inf 81.9%

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

   if 0.0013500000000000001 < b

   1. Initial program 55.9%

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

      1. sub-neg 55.9%

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

      2. fma-def 55.9%

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

      3. fma-def 59.3%

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

      4. +-commutative 59.3%

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

      5. metadata-eval 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in a around 0 57.6%

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

      1. associate-+r+ 57.6%

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

      2. associate-*r* 57.6%

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

      3. distribute-rgt-out 67.8%

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

      4. metadata-eval 67.8%

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

      5. distribute-lft-in 67.8%

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

      6. +-commutative 67.8%

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

      7. unpow2 67.8%

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

      8. distribute-lft-in 67.8%

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

      9. metadata-eval 67.8%

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

   6. Simplified 67.8%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.00135:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 4: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \lor \neg \left(a \leq 14000\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + 4 \cdot \frac{1}{\frac{b \cdot \left(a + -3\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a - 9\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.0) (not (<= a 14000.0)))
   (+ -1.0 (pow a 4.0))
   (+
    -1.0
    (* 4.0 (/ 1.0 (/ (* b (+ a -3.0)) (* b (* (* b b) (- (* a a) 9.0)))))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.0) || !(a <= 14000.0)) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (4.0 * (1.0 / ((b * (a + -3.0)) / (b * ((b * b) * ((a * a) - 9.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-3.0d0)) .or. (.not. (a <= 14000.0d0))) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (4.0d0 * (1.0d0 / ((b * (a + (-3.0d0))) / (b * ((b * b) * ((a * a) - 9.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -3.0) || !(a <= 14000.0)) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (4.0 * (1.0 / ((b * (a + -3.0)) / (b * ((b * b) * ((a * a) - 9.0))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -3.0) or not (a <= 14000.0):
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + (4.0 * (1.0 / ((b * (a + -3.0)) / (b * ((b * b) * ((a * a) - 9.0))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.0) || !(a <= 14000.0))
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(4.0 * Float64(1.0 / Float64(Float64(b * Float64(a + -3.0)) / Float64(b * Float64(Float64(b * b) * Float64(Float64(a * a) - 9.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -3.0) || ~((a <= 14000.0)))
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (4.0 * (1.0 / ((b * (a + -3.0)) / (b * ((b * b) * ((a * a) - 9.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.0], N[Not[LessEqual[a, 14000.0]], $MachinePrecision]], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(4.0 * N[(1.0 / N[(N[(b * N[(a + -3.0), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] - 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3 or 14000 < a

   1. Initial program 49.1%

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

      1. sub-neg 49.1%

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

      2. fma-def 49.1%

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

      3. fma-def 51.3%

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

      4. +-commutative 51.3%

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

      5. metadata-eval 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in a around inf 86.8%

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

   if -3 < a < 14000

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. fma-def 99.9%

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

      3. fma-def 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-def 99.9%

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

      2. metadata-eval 99.9%

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

      3. sqrt-pow2 100.0%

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

      4. hypot-udef 100.0%

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

      5. add-sqr-sqrt 99.8%

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

      6. unpow-prod-down 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. pow-sqr 99.8%

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

      2. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in b around inf 78.2%

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

      1. +-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. associate-*l* 78.2%

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

   10. Simplified 78.2%

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

       1. distribute-lft-in 78.2%

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

       2. flip-+ 75.0%

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

       3. *-commutative 75.0%

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

       4. *-commutative 75.0%

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

       5. *-commutative 75.0%

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

   12. Applied egg-rr 75.0%

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

       1. swap-sqr 60.2%

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

       2. associate-*r* 60.2%

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

       3. *-commutative 60.2%

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

       4. associate-*l* 68.4%

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

       5. swap-sqr 68.4%

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

       6. metadata-eval 68.4%

          \[\leadsto 4 \cdot \left(b \cdot \frac{a \cdot \left(b \cdot \left(b \cdot a\right)\right) - \left(b \cdot b\right) \cdot \color{blue}{9}}{a \cdot b - b \cdot 3}\right) + -1 \]

       7. *-commutative 68.4%

          \[\leadsto 4 \cdot \left(b \cdot \frac{a \cdot \left(b \cdot \left(b \cdot a\right)\right) - \left(b \cdot b\right) \cdot 9}{\color{blue}{b \cdot a} - b \cdot 3}\right) + -1 \]

   14. Simplified 68.4%

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

       1. associate-*r* 75.0%

          \[\leadsto 4 \cdot \left(b \cdot \frac{\color{blue}{\left(a \cdot b\right) \cdot \left(b \cdot a\right)} - \left(b \cdot b\right) \cdot 9}{b \cdot a - b \cdot 3}\right) + -1 \]

       2. *-commutative 75.0%

          \[\leadsto 4 \cdot \left(b \cdot \frac{\left(a \cdot b\right) \cdot \color{blue}{\left(a \cdot b\right)} - \left(b \cdot b\right) \cdot 9}{b \cdot a - b \cdot 3}\right) + -1 \]

       3. metadata-eval 75.0%

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

       4. swap-sqr 75.0%

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

       5. *-commutative 75.0%

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

       6. associate-*r/ 84.0%

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

   16. Applied egg-rr 87.2%

       \[\leadsto 4 \cdot \color{blue}{\frac{1}{\frac{b \cdot \left(a + -3\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a - 9\right)\right)}}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \lor \neg \left(a \leq 14000\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + 4 \cdot \frac{1}{\frac{b \cdot \left(a + -3\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a - 9\right)\right)}}\\ \end{array} \]

Alternative 5: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00135:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.00135) (+ -1.0 (pow a 4.0)) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00135) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.00135d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.00135) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.00135:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00135)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.00135)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00135], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0013500000000000001

   1. Initial program 78.6%

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

      1. sub-neg 78.6%

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

      2. fma-def 78.6%

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

      3. fma-def 79.1%

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

      4. +-commutative 79.1%

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

      5. metadata-eval 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in a around inf 81.9%

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

   if 0.0013500000000000001 < b

   1. Initial program 55.9%

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

      1. sub-neg 55.9%

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

      2. fma-def 55.9%

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

      3. fma-def 59.3%

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

      4. +-commutative 59.3%

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

      5. metadata-eval 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in b around inf 98.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.00135:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 6: 73.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+87}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+301}:\\ \;\;\;\;-1 + 4 \cdot \frac{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a - 9\right)\right)}{b}}{a + -3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+87)
   (+ -1.0 (* 4.0 (* a a)))
   (if (<= (* b b) 5e+301)
     (+ -1.0 (* 4.0 (/ (/ (* b (* (* b b) (- (* a a) 9.0))) b) (+ a -3.0))))
     (+ -1.0 (* (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+87) {
		tmp = -1.0 + (4.0 * (a * a));
	} else if ((b * b) <= 5e+301) {
		tmp = -1.0 + (4.0 * (((b * ((b * b) * ((a * a) - 9.0))) / b) / (a + -3.0)));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+87) then
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    else if ((b * b) <= 5d+301) then
        tmp = (-1.0d0) + (4.0d0 * (((b * ((b * b) * ((a * a) - 9.0d0))) / b) / (a + (-3.0d0))))
    else
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+87) {
		tmp = -1.0 + (4.0 * (a * a));
	} else if ((b * b) <= 5e+301) {
		tmp = -1.0 + (4.0 * (((b * ((b * b) * ((a * a) - 9.0))) / b) / (a + -3.0)));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+87:
		tmp = -1.0 + (4.0 * (a * a))
	elif (b * b) <= 5e+301:
		tmp = -1.0 + (4.0 * (((b * ((b * b) * ((a * a) - 9.0))) / b) / (a + -3.0)))
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+87)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * a)));
	elseif (Float64(b * b) <= 5e+301)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(Float64(Float64(b * Float64(Float64(b * b) * Float64(Float64(a * a) - 9.0))) / b) / Float64(a + -3.0))));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+87)
		tmp = -1.0 + (4.0 * (a * a));
	elseif ((b * b) <= 5e+301)
		tmp = -1.0 + (4.0 * (((b * ((b * b) * ((a * a) - 9.0))) / b) / (a + -3.0)));
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+87], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 5e+301], N[(-1.0 + N[(4.0 * N[(N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] - 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / N[(a + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 1.9999999999999999e87

   1. Initial program 86.8%

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

      1. sub-neg 86.8%

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

      2. fma-def 86.8%

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

      3. fma-def 86.8%

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

      4. +-commutative 86.8%

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

      5. metadata-eval 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in b around 0 80.8%

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

      1. associate-*r* 80.8%

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

      2. unpow2 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in a around 0 70.9%

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

      1. unpow2 70.9%

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

   9. Simplified 70.9%

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

   if 1.9999999999999999e87 < (*.f64 b b) < 5.0000000000000004e301

   1. Initial program 78.7%

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

      1. sub-neg 78.7%

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

      2. fma-def 78.7%

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

      3. fma-def 78.7%

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

      4. +-commutative 78.7%

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

      5. metadata-eval 78.7%

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

   3. Simplified 78.7%

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

      1. fma-def 78.7%

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

      2. metadata-eval 78.7%

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

      3. sqrt-pow2 78.8%

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

      4. hypot-udef 78.8%

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

      5. add-sqr-sqrt 78.6%

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

      6. unpow-prod-down 78.6%

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

   5. Applied egg-rr 78.6%

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

      1. pow-sqr 78.5%

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

      2. metadata-eval 78.5%

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

   7. Simplified 78.5%

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

   8. Taylor expanded in b around inf 7.8%

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

      1. +-commutative 7.8%

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

      2. unpow2 7.8%

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

      3. associate-*l* 7.8%

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

   10. Simplified 7.8%

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

       1. distribute-lft-in 7.8%

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

       2. flip-+ 4.8%

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

       3. *-commutative 4.8%

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

       4. *-commutative 4.8%

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

       5. *-commutative 4.8%

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

   12. Applied egg-rr 4.8%

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

       1. swap-sqr 4.8%

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

       2. associate-*r* 4.8%

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

       3. *-commutative 4.8%

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

       4. associate-*l* 4.8%

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

       5. swap-sqr 4.8%

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

       6. metadata-eval 4.8%

          \[\leadsto 4 \cdot \left(b \cdot \frac{a \cdot \left(b \cdot \left(b \cdot a\right)\right) - \left(b \cdot b\right) \cdot \color{blue}{9}}{a \cdot b - b \cdot 3}\right) + -1 \]

       7. *-commutative 4.8%

          \[\leadsto 4 \cdot \left(b \cdot \frac{a \cdot \left(b \cdot \left(b \cdot a\right)\right) - \left(b \cdot b\right) \cdot 9}{\color{blue}{b \cdot a} - b \cdot 3}\right) + -1 \]

   14. Simplified 4.8%

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

       1. associate-*r* 4.8%

          \[\leadsto 4 \cdot \left(b \cdot \frac{\color{blue}{\left(a \cdot b\right) \cdot \left(b \cdot a\right)} - \left(b \cdot b\right) \cdot 9}{b \cdot a - b \cdot 3}\right) + -1 \]

       2. *-commutative 4.8%

          \[\leadsto 4 \cdot \left(b \cdot \frac{\left(a \cdot b\right) \cdot \color{blue}{\left(a \cdot b\right)} - \left(b \cdot b\right) \cdot 9}{b \cdot a - b \cdot 3}\right) + -1 \]

       3. metadata-eval 4.8%

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

       4. swap-sqr 4.8%

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

       5. *-commutative 4.8%

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

       6. associate-*r/ 40.8%

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

   16. Applied egg-rr 43.9%

       \[\leadsto 4 \cdot \color{blue}{\frac{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a - 9\right)\right)}{b}}{a + -3}} + -1 \]

   if 5.0000000000000004e301 < (*.f64 b b)

   1. Initial program 41.4%

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

      1. sub-neg 41.4%

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

      2. fma-def 41.4%

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

      3. fma-def 45.7%

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

      4. +-commutative 45.7%

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

      5. metadata-eval 45.7%

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

   3. Simplified 45.7%

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

      1. fma-def 45.7%

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

      2. metadata-eval 45.7%

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

      3. sqrt-pow2 45.7%

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

      4. hypot-udef 45.7%

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

      5. add-sqr-sqrt 45.7%

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

      6. unpow-prod-down 45.7%

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

   5. Applied egg-rr 45.7%

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

      1. pow-sqr 45.7%

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

      2. metadata-eval 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in b around inf 57.1%

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

      1. +-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. associate-*l* 57.1%

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

   10. Simplified 57.1%

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

   11. Taylor expanded in a around 0 100.0%

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

       1. unpow2 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+87}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+301}:\\ \;\;\;\;-1 + 4 \cdot \frac{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a - 9\right)\right)}{b}}{a + -3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 7: 69.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+274}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+274) (+ -1.0 (* 4.0 (* a a))) (+ -1.0 (* (* b b) 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+274) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+274) then
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    else
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+274) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+274:
		tmp = -1.0 + (4.0 * (a * a))
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+274)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+274)
		tmp = -1.0 + (4.0 * (a * a));
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+274], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999998e274

   1. Initial program 84.8%

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

      1. sub-neg 84.8%

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

      2. fma-def 84.8%

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

      3. fma-def 84.8%

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

      4. +-commutative 84.8%

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

      5. metadata-eval 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in b around 0 72.8%

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

      1. associate-*r* 72.8%

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

      2. unpow2 72.8%

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

   6. Simplified 72.8%

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

   7. Taylor expanded in a around 0 63.3%

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

      1. unpow2 63.3%

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

   9. Simplified 63.3%

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

   if 4.9999999999999998e274 < (*.f64 b b)

   1. Initial program 46.1%

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

      1. sub-neg 46.1%

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

      2. fma-def 46.1%

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

      3. fma-def 50.0%

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

      4. +-commutative 50.0%

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

      5. metadata-eval 50.0%

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

   3. Simplified 50.0%

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

      1. fma-def 50.0%

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

      2. metadata-eval 50.0%

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

      3. sqrt-pow2 50.0%

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

      4. hypot-udef 50.0%

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

      5. add-sqr-sqrt 50.0%

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

      6. unpow-prod-down 50.0%

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

   5. Applied egg-rr 50.0%

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

      1. pow-sqr 50.0%

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

      2. metadata-eval 50.0%

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

   7. Simplified 50.0%

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

   8. Taylor expanded in b around inf 53.5%

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

      1. +-commutative 53.5%

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

      2. unpow2 53.5%

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

      3. associate-*l* 53.5%

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

   10. Simplified 53.5%

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

   11. Taylor expanded in a around 0 92.9%

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

       1. unpow2 92.9%

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

   13. Simplified 92.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+274}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 8: 51.6% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. sub-neg 73.3%

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

   2. fma-def 73.3%

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

   3. fma-def 74.5%

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

   4. +-commutative 74.5%

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

   5. metadata-eval 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in b around 0 62.0%

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

   1. associate-*r* 62.0%

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

   2. unpow2 62.0%

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

6. Simplified 62.0%

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

7. Taylor expanded in a around 0 56.3%

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

   1. unpow2 56.3%

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

9. Simplified 56.3%

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

10. Final simplification 56.3%

    \[\leadsto -1 + 4 \cdot \left(a \cdot a\right) \]

Alternative 9: 24.5% 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 73.3%

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

   1. sub-neg 73.3%

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

   2. fma-def 73.3%

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

   3. fma-def 74.5%

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

   4. +-commutative 74.5%

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

   5. metadata-eval 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in b around inf 70.9%

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

5. Taylor expanded in b around 0 28.1%

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

6. Final simplification 28.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023238 
(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))