Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.5% → 98.4%

Time: 6.8s

Alternatives: 7

Speedup: 6.7×

• 60.5% 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: 74.5% 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: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b)
	tmp = 0.0
	if (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))))) <= Inf)
		tmp = Float64(Float64((hypot(a, b) ^ 4.0) + Float64(4.0 * Float64(Float64(b * Float64(b * Float64(a + 3.0))) + Float64(a * Float64(a * Float64(1.0 - a)))))) + -1.0);
	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)
	tmp = 0.0;
	if (((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= Inf)
		tmp = ((hypot(a, b) ^ 4.0) + (4.0 * ((b * (b * (a + 3.0))) + (a * (a * (1.0 - a)))))) + -1.0;
	else
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[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], Infinity], N[(N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(N[(b * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-def 99.9%

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

      2. metadata-eval 99.9%

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

      3. sqrt-pow2 100.0%

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

      4. hypot-udef 100.0%

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

      5. expm1-log1p-u 98.1%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + 4 \cdot \left(b \cdot \left(b \cdot \left(a + 3\right)\right) + a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      6. expm1-udef 98.1%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + 4 \cdot \left(b \cdot \left(b \cdot \left(a + 3\right)\right) + a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

   7. Applied egg-rr 98.1%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + 4 \cdot \left(b \cdot \left(b \cdot \left(a + 3\right)\right) + a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]
   8. Step-by-step derivation

      1. expm1-def 98.1%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + 4 \cdot \left(b \cdot \left(b \cdot \left(a + 3\right)\right) + a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      2. expm1-log1p 100.0%

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

   9. Simplified 100.0%

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

   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 7.9%

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

      4. +-commutative 7.9%

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

      5. metadata-eval 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in b around 0 41.8%

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

      1. associate-*r* 41.8%

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

      2. unpow2 41.8%

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

   6. Simplified 41.8%

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

      1. metadata-eval 41.8%

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

      2. pow-prod-up 41.8%

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

      3. pow2 41.8%

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

      4. pow2 41.8%

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

      5. fma-def 41.8%

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

      6. *-commutative 41.8%

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

   8. Applied egg-rr 41.8%

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

      1. fma-udef 41.8%

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

      2. associate-*r* 41.8%

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

      3. distribute-rgt-out 92.6%

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

   10. Applied egg-rr 92.6%

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

4. Final simplification 98.2%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

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)
     (+ t_0 -1.0)
     (+ -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 = t_0 + -1.0;
	} 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 = t_0 + -1.0;
	} 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 = t_0 + -1.0
	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(t_0 + -1.0);
	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 = t_0 + -1.0;
	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[(t$95$0 + -1.0), $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 99.9%

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

   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 7.9%

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

      4. +-commutative 7.9%

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

      5. metadata-eval 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in b around 0 41.8%

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

      1. associate-*r* 41.8%

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

      2. unpow2 41.8%

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

   6. Simplified 41.8%

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

      1. metadata-eval 41.8%

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

      2. pow-prod-up 41.8%

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

      3. pow2 41.8%

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

      4. pow2 41.8%

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

      5. fma-def 41.8%

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

      6. *-commutative 41.8%

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

   8. Applied egg-rr 41.8%

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

      1. fma-udef 41.8%

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

      2. associate-*r* 41.8%

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

      3. distribute-rgt-out 92.6%

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

   10. Applied egg-rr 92.6%

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

4. Final simplification 98.1%

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

Alternative 3: 94.2% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 10.0)
   (+ -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) <= 10.0) {
		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) <= 10.0d0) 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) <= 10.0) {
		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) <= 10.0:
		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) <= 10.0)
		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) <= 10.0)
		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], 10.0], 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) < 10

   1. Initial program 90.5%

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

      1. sub-neg 90.5%

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

      2. fma-def 90.5%

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

      3. fma-def 90.5%

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

      4. +-commutative 90.5%

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

      5. metadata-eval 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in b around 0 90.0%

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

      1. associate-*r* 90.0%

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

      2. unpow2 90.0%

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

   6. Simplified 90.0%

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

      1. metadata-eval 90.0%

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

      2. pow-prod-up 89.9%

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

      3. pow2 89.9%

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

      4. pow2 89.9%

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

      5. fma-def 89.9%

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

      6. *-commutative 89.9%

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

   8. Applied egg-rr 89.9%

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

      1. fma-udef 89.9%

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

      2. associate-*r* 89.9%

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

      3. distribute-rgt-out 99.2%

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

   10. Applied egg-rr 99.2%

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

   if 10 < (*.f64 b b)

   1. Initial program 60.0%

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

      1. sub-neg 60.0%

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

      2. fma-def 60.1%

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

      3. fma-def 64.0%

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

      4. +-commutative 64.0%

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

      5. metadata-eval 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in b around inf 89.5%

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

4. Final simplification 94.4%

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

Alternative 4: 84.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.5 \cdot 10^{+307}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1.5e+307)
   (+ -1.0 (* (* a a) (+ (* a a) (* 4.0 (- 1.0 a)))))
   (+ -1.0 (* (* b b) 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1.5e+307) {
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - 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) <= 1.5d+307) then
        tmp = (-1.0d0) + ((a * a) * ((a * a) + (4.0d0 * (1.0d0 - 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) <= 1.5e+307) {
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1.5e+307:
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))))
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1.5e+307)
		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) * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1.5e+307)
		tmp = -1.0 + ((a * a) * ((a * a) + (4.0 * (1.0 - a))));
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1.5e+307], 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] * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.4999999999999999e307

   1. Initial program 82.0%

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

      1. sub-neg 82.0%

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

      2. fma-def 82.1%

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

      3. fma-def 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. Taylor expanded in b around 0 64.4%

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

      1. associate-*r* 64.4%

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

      2. unpow2 64.4%

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

   6. Simplified 64.4%

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

      1. metadata-eval 64.4%

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

      2. pow-prod-up 64.4%

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

      3. pow2 64.4%

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

      4. pow2 64.4%

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

      5. fma-def 64.4%

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

      6. *-commutative 64.4%

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

   8. Applied egg-rr 64.4%

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

      1. fma-udef 64.4%

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

      2. associate-*r* 64.4%

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

      3. distribute-rgt-out 76.7%

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

   10. Applied egg-rr 76.7%

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

   if 1.4999999999999999e307 < (*.f64 b b)

   1. Initial program 50.0%

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

      1. sub-neg 50.0%

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

      2. fma-def 50.0%

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

      3. fma-def 53.7%

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

      4. +-commutative 53.7%

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

      5. metadata-eval 53.7%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in a around 0 37.0%

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

      1. associate-+r+ 37.0%

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

      2. associate-*r* 37.0%

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

      3. distribute-rgt-out 63.0%

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

      4. unpow2 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in b around 0 63.0%

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

      1. +-commutative 63.0%

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

      2. *-commutative 63.0%

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

      3. fma-udef 63.0%

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

      4. *-commutative 63.0%

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

      5. unpow2 63.0%

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

   9. Simplified 63.0%

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

   10. Taylor expanded in a around 0 100.0%

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

       1. unpow2 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.5 \cdot 10^{+307}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4 \cdot \left(1 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 5: 68.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5.4 \cdot 10^{+232}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{elif}\;b \cdot b \leq 1.45 \cdot 10^{+307}:\\ \;\;\;\;-1 + a \cdot \left(\left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5.4e+232)
   (+ -1.0 (* (* a a) 4.0))
   (if (<= (* b b) 1.45e+307)
     (+ -1.0 (* a (* (* b b) 4.0)))
     (+ -1.0 (* (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5.4e+232) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else if ((b * b) <= 1.45e+307) {
		tmp = -1.0 + (a * ((b * b) * 4.0));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5.4d+232) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else if ((b * b) <= 1.45d+307) then
        tmp = (-1.0d0) + (a * ((b * b) * 4.0d0))
    else
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5.4e+232) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else if ((b * b) <= 1.45e+307) {
		tmp = -1.0 + (a * ((b * b) * 4.0));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5.4e+232:
		tmp = -1.0 + ((a * a) * 4.0)
	elif (b * b) <= 1.45e+307:
		tmp = -1.0 + (a * ((b * b) * 4.0))
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5.4e+232)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	elseif (Float64(b * b) <= 1.45e+307)
		tmp = Float64(-1.0 + Float64(a * Float64(Float64(b * b) * 4.0)));
	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) <= 5.4e+232)
		tmp = -1.0 + ((a * a) * 4.0);
	elseif ((b * b) <= 1.45e+307)
		tmp = -1.0 + (a * ((b * b) * 4.0));
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5.4e+232], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 1.45e+307], N[(-1.0 + N[(a * N[(N[(b * b), $MachinePrecision] * 4.0), $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) < 5.4000000000000002e232

   1. Initial program 83.6%

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

      1. sub-neg 83.6%

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

      2. fma-def 83.7%

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

      3. fma-def 84.2%

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

      4. +-commutative 84.2%

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

      5. metadata-eval 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in b around 0 71.4%

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

      1. associate-*r* 71.4%

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

      2. unpow2 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in a around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. unpow2 61.1%

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

   9. Simplified 61.1%

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

   if 5.4000000000000002e232 < (*.f64 b b) < 1.44999999999999999e307

   1. Initial program 69.6%

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

      1. sub-neg 69.6%

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

      2. fma-def 69.6%

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

      3. fma-def 78.3%

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

      4. +-commutative 78.3%

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

      5. metadata-eval 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in a around 0 91.3%

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

      1. associate-+r+ 91.3%

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

      2. associate-*r* 91.3%

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

      3. distribute-rgt-out 91.3%

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

      4. unpow2 91.3%

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

   6. Simplified 91.3%

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

   7. Taylor expanded in a around inf 33.0%

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

      1. *-commutative 33.0%

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

      2. associate-*l* 33.0%

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

      3. unpow2 33.0%

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

   9. Simplified 33.0%

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

   if 1.44999999999999999e307 < (*.f64 b b)

   1. Initial program 50.0%

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

      1. sub-neg 50.0%

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

      2. fma-def 50.0%

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

      3. fma-def 53.7%

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

      4. +-commutative 53.7%

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

      5. metadata-eval 53.7%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in a around 0 37.0%

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

      1. associate-+r+ 37.0%

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

      2. associate-*r* 37.0%

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

      3. distribute-rgt-out 63.0%

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

      4. unpow2 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in b around 0 63.0%

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

      1. +-commutative 63.0%

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

      2. *-commutative 63.0%

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

      3. fma-udef 63.0%

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

      4. *-commutative 63.0%

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

      5. unpow2 63.0%

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

   9. Simplified 63.0%

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

   10. Taylor expanded in a around 0 100.0%

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

       1. unpow2 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5.4 \cdot 10^{+232}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{elif}\;b \cdot b \leq 1.45 \cdot 10^{+307}:\\ \;\;\;\;-1 + a \cdot \left(\left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 6: 69.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.25 \cdot 10^{+307}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1.25e+307)
   (+ -1.0 (* (* a a) 4.0))
   (+ -1.0 (* (* b b) 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1.25e+307) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1.25d+307) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1.25e+307) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1.25e+307:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1.25e+307)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	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) <= 1.25e+307)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1.25e+307], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $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) < 1.25e307

   1. Initial program 82.0%

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

      1. sub-neg 82.0%

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

      2. fma-def 82.1%

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

      3. fma-def 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. Taylor expanded in b around 0 64.4%

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

      1. associate-*r* 64.4%

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

      2. unpow2 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in a around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. unpow2 55.9%

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

   9. Simplified 55.9%

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

   if 1.25e307 < (*.f64 b b)

   1. Initial program 50.0%

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

      1. sub-neg 50.0%

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

      2. fma-def 50.0%

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

      3. fma-def 53.7%

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

      4. +-commutative 53.7%

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

      5. metadata-eval 53.7%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in a around 0 37.0%

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

      1. associate-+r+ 37.0%

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

      2. associate-*r* 37.0%

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

      3. distribute-rgt-out 63.0%

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

      4. unpow2 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in b around 0 63.0%

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

      1. +-commutative 63.0%

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

      2. *-commutative 63.0%

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

      3. fma-udef 63.0%

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

      4. *-commutative 63.0%

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

      5. unpow2 63.0%

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

   9. Simplified 63.0%

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

   10. Taylor expanded in a around 0 100.0%

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

       1. unpow2 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.25 \cdot 10^{+307}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 7: 51.0% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. sub-neg 75.3%

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

   2. fma-def 75.3%

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

   3. fma-def 77.2%

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

   4. +-commutative 77.2%

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

   5. metadata-eval 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in a around 0 55.3%

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

   1. associate-+r+ 55.3%

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

   2. associate-*r* 55.3%

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

   3. distribute-rgt-out 60.8%

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

   4. unpow2 60.8%

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

6. Simplified 60.8%

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

7. Taylor expanded in b around 0 44.8%

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

   1. +-commutative 44.8%

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

   2. *-commutative 44.8%

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

   3. fma-udef 44.8%

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

   4. *-commutative 44.8%

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

   5. unpow2 44.8%

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

9. Simplified 44.8%

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

10. Taylor expanded in a around 0 48.8%

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

    1. unpow2 48.8%

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

12. Simplified 48.8%

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

13. Final simplification 48.8%

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

Reproduce

herbie shell --seed 2023174 
(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))