Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.2% → 98.6%

Time: 6.6s

Alternatives: 11

Speedup: 1.1×

• 61.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= ((double) INFINITY)) {
		tmp = fma(4.0, fma(a, fma(a, a, a), (b * (b * fma(a, -3.0, 1.0)))), pow(hypot(a, b), 4.0)) + -1.0;
	} else {
		tmp = pow(a, 3.0) * (a + 4.0);
	}
	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(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= Inf)
		tmp = Float64(fma(4.0, fma(a, fma(a, a, a), Float64(b * Float64(b * fma(a, -3.0, 1.0)))), (hypot(a, b) ^ 4.0)) + -1.0);
	else
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	end
	return 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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[a, 3.0], $MachinePrecision] * N[(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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

   3. Simplified 100.0%

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

   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 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in a around inf 28.1%

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

   5. Taylor expanded in b around 0 23.6%

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

      1. unpow2 23.6%

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

   7. Simplified 23.6%

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

      1. metadata-eval 23.6%

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

      2. pow-sqr 23.6%

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

      3. pow2 23.6%

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

      4. pow2 23.6%

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

      5. distribute-rgt-out 23.6%

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

   9. Applied egg-rr 23.6%

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

   10. Taylor expanded in a around inf 23.6%

       \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   11. Step-by-step derivation

       1. *-commutative 23.6%

          \[\leadsto \color{blue}{{a}^{3} \cdot 4} + {a}^{4} \]

       2. metadata-eval 23.6%

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

       3. pow-plus 23.6%

          \[\leadsto {a}^{3} \cdot 4 + \color{blue}{{a}^{3} \cdot a} \]

       4. distribute-lft-out 95.5%

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

   12. Simplified 95.5%

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

4. Final simplification 98.9%

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

Alternative 2: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = math.pow(a, 3.0) * (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(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = (a ^ 3.0) * (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[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[Power[a, 3.0], $MachinePrecision] * N[(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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.9%

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

   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 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in a around inf 28.1%

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

   5. Taylor expanded in b around 0 23.6%

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

      1. unpow2 23.6%

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

   7. Simplified 23.6%

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

      1. metadata-eval 23.6%

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

      2. pow-sqr 23.6%

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

      3. pow2 23.6%

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

      4. pow2 23.6%

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

      5. distribute-rgt-out 23.6%

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

   9. Applied egg-rr 23.6%

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

   10. Taylor expanded in a around inf 23.6%

       \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   11. Step-by-step derivation

       1. *-commutative 23.6%

          \[\leadsto \color{blue}{{a}^{3} \cdot 4} + {a}^{4} \]

       2. metadata-eval 23.6%

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

       3. pow-plus 23.6%

          \[\leadsto {a}^{3} \cdot 4 + \color{blue}{{a}^{3} \cdot a} \]

       4. distribute-lft-out 95.5%

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

   12. Simplified 95.5%

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

4. Final simplification 98.8%

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

Alternative 3: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -1.9 \cdot 10^{-237}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 0.00082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {a}^{3} + \left(a \cdot a\right) \cdot \left(a \cdot a + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e+76)
   (pow a 4.0)
   (if (<= a -1.9e-237)
     (pow b 4.0)
     (if (<= a 0.00082)
       -1.0
       (+ (* 4.0 (pow a 3.0)) (* (* a a) (+ (* a a) 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+76) {
		tmp = pow(a, 4.0);
	} else if (a <= -1.9e-237) {
		tmp = pow(b, 4.0);
	} else if (a <= 0.00082) {
		tmp = -1.0;
	} else {
		tmp = (4.0 * pow(a, 3.0)) + ((a * a) * ((a * a) + 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.4d+76)) then
        tmp = a ** 4.0d0
    else if (a <= (-1.9d-237)) then
        tmp = b ** 4.0d0
    else if (a <= 0.00082d0) then
        tmp = -1.0d0
    else
        tmp = (4.0d0 * (a ** 3.0d0)) + ((a * a) * ((a * a) + 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+76) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -1.9e-237) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 0.00082) {
		tmp = -1.0;
	} else {
		tmp = (4.0 * Math.pow(a, 3.0)) + ((a * a) * ((a * a) + 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.4e+76:
		tmp = math.pow(a, 4.0)
	elif a <= -1.9e-237:
		tmp = math.pow(b, 4.0)
	elif a <= 0.00082:
		tmp = -1.0
	else:
		tmp = (4.0 * math.pow(a, 3.0)) + ((a * a) * ((a * a) + 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.4e+76)
		tmp = a ^ 4.0;
	elseif (a <= -1.9e-237)
		tmp = b ^ 4.0;
	elseif (a <= 0.00082)
		tmp = -1.0;
	else
		tmp = Float64(Float64(4.0 * (a ^ 3.0)) + Float64(Float64(a * a) * Float64(Float64(a * a) + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.4e+76)
		tmp = a ^ 4.0;
	elseif (a <= -1.9e-237)
		tmp = b ^ 4.0;
	elseif (a <= 0.00082)
		tmp = -1.0;
	else
		tmp = (4.0 * (a ^ 3.0)) + ((a * a) * ((a * a) + 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.4e+76], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -1.9e-237], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 0.00082], -1.0, N[(N[(4.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.3999999999999998e76

   1. Initial program 17.9%

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

      1. associate--l+ 17.9%

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

      2. fma-def 17.9%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in a around inf 100.0%

      \[\leadsto \color{blue}{{a}^{4}} \]

   if -5.3999999999999998e76 < a < -1.90000000000000012e-237

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around inf 70.2%

      \[\leadsto \color{blue}{{b}^{4}} \]

   if -1.90000000000000012e-237 < a < 8.1999999999999998e-4

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 62.4%

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

      1. associate--l+ 62.4%

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

      2. associate-*r* 62.4%

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

      3. unpow2 62.4%

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

   6. Simplified 62.4%

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

   7. Taylor expanded in a around 0 62.4%

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

   if 8.1999999999999998e-4 < a

   1. Initial program 73.7%

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

      1. associate--l+ 73.7%

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

      2. fma-def 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in a around inf 93.7%

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

   5. Taylor expanded in b around 0 88.0%

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

      1. unpow2 88.0%

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

   7. Simplified 88.0%

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

      1. metadata-eval 88.0%

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

      2. pow-sqr 87.9%

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

      3. pow2 87.9%

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

      4. pow2 87.9%

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

      5. distribute-rgt-out 87.9%

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -1.9 \cdot 10^{-237}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 0.00082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {a}^{3} + \left(a \cdot a\right) \cdot \left(a \cdot a + 4\right)\\ \end{array} \]

Alternative 4: 85.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+24}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+24)
   (+ (pow a 4.0) (+ -1.0 (* (+ a 1.0) (* (* a a) 4.0))))
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+24) {
		tmp = pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	} else {
		tmp = 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) <= 5d+24) then
        tmp = (a ** 4.0d0) + ((-1.0d0) + ((a + 1.0d0) * ((a * a) * 4.0d0)))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+24) {
		tmp = Math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+24:
		tmp = math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+24)
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a + 1.0) * Float64(Float64(a * a) * 4.0))));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+24)
		tmp = (a ^ 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+24], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000045e24

   1. Initial program 82.4%

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

      1. associate--l+ 82.4%

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

      2. fma-def 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in b around 0 81.8%

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

      1. associate--l+ 81.8%

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

      2. associate-*r* 81.8%

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

      3. unpow2 81.8%

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

   6. Simplified 81.8%

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

   if 5.00000000000000045e24 < (*.f64 b b)

   1. Initial program 66.3%

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

      1. associate--l+ 66.3%

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

      2. fma-def 66.3%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in b around inf 92.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+24}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 68.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-246}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 0.00082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e+76)
   (pow a 4.0)
   (if (<= a -2.8e-246)
     (pow b 4.0)
     (if (<= a 0.00082) -1.0 (* (pow a 3.0) (+ a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+76) {
		tmp = pow(a, 4.0);
	} else if (a <= -2.8e-246) {
		tmp = pow(b, 4.0);
	} else if (a <= 0.00082) {
		tmp = -1.0;
	} else {
		tmp = pow(a, 3.0) * (a + 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.4d+76)) then
        tmp = a ** 4.0d0
    else if (a <= (-2.8d-246)) then
        tmp = b ** 4.0d0
    else if (a <= 0.00082d0) then
        tmp = -1.0d0
    else
        tmp = (a ** 3.0d0) * (a + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+76) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -2.8e-246) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 0.00082) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(a, 3.0) * (a + 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.4e+76:
		tmp = math.pow(a, 4.0)
	elif a <= -2.8e-246:
		tmp = math.pow(b, 4.0)
	elif a <= 0.00082:
		tmp = -1.0
	else:
		tmp = math.pow(a, 3.0) * (a + 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.4e+76)
		tmp = a ^ 4.0;
	elseif (a <= -2.8e-246)
		tmp = b ^ 4.0;
	elseif (a <= 0.00082)
		tmp = -1.0;
	else
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.4e+76)
		tmp = a ^ 4.0;
	elseif (a <= -2.8e-246)
		tmp = b ^ 4.0;
	elseif (a <= 0.00082)
		tmp = -1.0;
	else
		tmp = (a ^ 3.0) * (a + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.4e+76], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -2.8e-246], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 0.00082], -1.0, N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.3999999999999998e76

   1. Initial program 17.9%

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

      1. associate--l+ 17.9%

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

      2. fma-def 17.9%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in a around inf 100.0%

      \[\leadsto \color{blue}{{a}^{4}} \]

   if -5.3999999999999998e76 < a < -2.7999999999999999e-246

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around inf 70.2%

      \[\leadsto \color{blue}{{b}^{4}} \]

   if -2.7999999999999999e-246 < a < 8.1999999999999998e-4

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 62.4%

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

      1. associate--l+ 62.4%

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

      2. associate-*r* 62.4%

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

      3. unpow2 62.4%

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

   6. Simplified 62.4%

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

   7. Taylor expanded in a around 0 62.4%

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

   if 8.1999999999999998e-4 < a

   1. Initial program 73.7%

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

      1. associate--l+ 73.7%

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

      2. fma-def 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in a around inf 93.7%

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

   5. Taylor expanded in b around 0 88.0%

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

      1. unpow2 88.0%

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

   7. Simplified 88.0%

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

      1. metadata-eval 88.0%

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

      2. pow-sqr 87.9%

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

      3. pow2 87.9%

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

      4. pow2 87.9%

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

      5. distribute-rgt-out 87.9%

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

   9. Applied egg-rr 87.9%

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

   10. Taylor expanded in a around inf 87.6%

       \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   11. Step-by-step derivation

       1. *-commutative 87.6%

          \[\leadsto \color{blue}{{a}^{3} \cdot 4} + {a}^{4} \]

       2. metadata-eval 87.6%

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

       3. pow-plus 87.5%

          \[\leadsto {a}^{3} \cdot 4 + \color{blue}{{a}^{3} \cdot a} \]

       4. distribute-lft-out 87.5%

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

   12. Simplified 87.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-246}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 0.00082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \end{array} \]

Alternative 6: 64.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -1.85 \cdot 10^{-196}:\\ \;\;\;\;a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\\ \mathbf{elif}\;a \leq 0.00082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e+76)
   (pow a 4.0)
   (if (<= a -1.85e-196)
     (* a (* a (* b (* b 2.0))))
     (if (<= a 0.00082) -1.0 (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+76) {
		tmp = pow(a, 4.0);
	} else if (a <= -1.85e-196) {
		tmp = a * (a * (b * (b * 2.0)));
	} else if (a <= 0.00082) {
		tmp = -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.4d+76)) then
        tmp = a ** 4.0d0
    else if (a <= (-1.85d-196)) then
        tmp = a * (a * (b * (b * 2.0d0)))
    else if (a <= 0.00082d0) then
        tmp = -1.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+76) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -1.85e-196) {
		tmp = a * (a * (b * (b * 2.0)));
	} else if (a <= 0.00082) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.4e+76:
		tmp = math.pow(a, 4.0)
	elif a <= -1.85e-196:
		tmp = a * (a * (b * (b * 2.0)))
	elif a <= 0.00082:
		tmp = -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.4e+76)
		tmp = a ^ 4.0;
	elseif (a <= -1.85e-196)
		tmp = Float64(a * Float64(a * Float64(b * Float64(b * 2.0))));
	elseif (a <= 0.00082)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.4e+76)
		tmp = a ^ 4.0;
	elseif (a <= -1.85e-196)
		tmp = a * (a * (b * (b * 2.0)));
	elseif (a <= 0.00082)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.4e+76], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -1.85e-196], N[(a * N[(a * N[(b * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00082], -1.0, N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.3999999999999998e76 or 8.1999999999999998e-4 < a

   1. Initial program 48.7%

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

      1. associate--l+ 48.7%

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

      2. fma-def 48.7%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in a around inf 92.6%

      \[\leadsto \color{blue}{{a}^{4}} \]

   if -5.3999999999999998e76 < a < -1.85000000000000005e-196

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 52.5%

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

   5. Taylor expanded in b around inf 40.3%

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

      1. unpow2 40.3%

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

      2. unpow2 40.3%

         \[\leadsto 2 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

      3. *-commutative 40.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)} \]

      4. associate-*l* 40.3%

         \[\leadsto \color{blue}{\left(2 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)} \]

      5. *-commutative 40.3%

         \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)} \]

      6. associate-*l* 48.9%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(2 \cdot \left(b \cdot b\right)\right)\right)} \]

      7. associate-*r* 48.9%

         \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(2 \cdot b\right) \cdot b\right)}\right) \]

   7. Simplified 48.9%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\left(2 \cdot b\right) \cdot b\right)\right)} \]

   if -1.85000000000000005e-196 < a < 8.1999999999999998e-4

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 61.1%

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

      1. associate--l+ 61.1%

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

      2. associate-*r* 61.1%

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

      3. unpow2 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in a around 0 61.1%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -1.85 \cdot 10^{-196}:\\ \;\;\;\;a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\\ \mathbf{elif}\;a \leq 0.00082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7: 68.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-245}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 0.00082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.6e+76)
   (pow a 4.0)
   (if (<= a -5.2e-245) (pow b 4.0) (if (<= a 0.00082) -1.0 (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.6e+76) {
		tmp = pow(a, 4.0);
	} else if (a <= -5.2e-245) {
		tmp = pow(b, 4.0);
	} else if (a <= 0.00082) {
		tmp = -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.6d+76)) then
        tmp = a ** 4.0d0
    else if (a <= (-5.2d-245)) then
        tmp = b ** 4.0d0
    else if (a <= 0.00082d0) then
        tmp = -1.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.6e+76) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -5.2e-245) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 0.00082) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.6e+76:
		tmp = math.pow(a, 4.0)
	elif a <= -5.2e-245:
		tmp = math.pow(b, 4.0)
	elif a <= 0.00082:
		tmp = -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.6e+76)
		tmp = a ^ 4.0;
	elseif (a <= -5.2e-245)
		tmp = b ^ 4.0;
	elseif (a <= 0.00082)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.6e+76)
		tmp = a ^ 4.0;
	elseif (a <= -5.2e-245)
		tmp = b ^ 4.0;
	elseif (a <= 0.00082)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.6e+76], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -5.2e-245], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 0.00082], -1.0, N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.5999999999999997e76 or 8.1999999999999998e-4 < a

   1. Initial program 48.7%

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

      1. associate--l+ 48.7%

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

      2. fma-def 48.7%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in a around inf 92.6%

      \[\leadsto \color{blue}{{a}^{4}} \]

   if -5.5999999999999997e76 < a < -5.20000000000000013e-245

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around inf 70.2%

      \[\leadsto \color{blue}{{b}^{4}} \]

   if -5.20000000000000013e-245 < a < 8.1999999999999998e-4

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 62.4%

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

      1. associate--l+ 62.4%

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

      2. associate-*r* 62.4%

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

      3. unpow2 62.4%

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

   6. Simplified 62.4%

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

   7. Taylor expanded in a around 0 62.4%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-245}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 0.00082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 8: 56.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot 4\right)\\ t_1 := 2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;a \leq -4.4 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a 4.0))) (t_1 (* 2.0 (* (* a a) (* b b)))))
   (if (<= a -4.4e+145)
     t_0
     (if (<= a -4.8e-137)
       t_1
       (if (<= a 4.5e-5) -1.0 (if (<= a 6.6e+153) t_1 t_0))))))

C

double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = 2.0 * ((a * a) * (b * b));
	double tmp;
	if (a <= -4.4e+145) {
		tmp = t_0;
	} else if (a <= -4.8e-137) {
		tmp = t_1;
	} else if (a <= 4.5e-5) {
		tmp = -1.0;
	} else if (a <= 6.6e+153) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (a * 4.0d0)
    t_1 = 2.0d0 * ((a * a) * (b * b))
    if (a <= (-4.4d+145)) then
        tmp = t_0
    else if (a <= (-4.8d-137)) then
        tmp = t_1
    else if (a <= 4.5d-5) then
        tmp = -1.0d0
    else if (a <= 6.6d+153) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = 2.0 * ((a * a) * (b * b));
	double tmp;
	if (a <= -4.4e+145) {
		tmp = t_0;
	} else if (a <= -4.8e-137) {
		tmp = t_1;
	} else if (a <= 4.5e-5) {
		tmp = -1.0;
	} else if (a <= 6.6e+153) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * 4.0)
	t_1 = 2.0 * ((a * a) * (b * b))
	tmp = 0
	if a <= -4.4e+145:
		tmp = t_0
	elif a <= -4.8e-137:
		tmp = t_1
	elif a <= 4.5e-5:
		tmp = -1.0
	elif a <= 6.6e+153:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * 4.0))
	t_1 = Float64(2.0 * Float64(Float64(a * a) * Float64(b * b)))
	tmp = 0.0
	if (a <= -4.4e+145)
		tmp = t_0;
	elseif (a <= -4.8e-137)
		tmp = t_1;
	elseif (a <= 4.5e-5)
		tmp = -1.0;
	elseif (a <= 6.6e+153)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * 4.0);
	t_1 = 2.0 * ((a * a) * (b * b));
	tmp = 0.0;
	if (a <= -4.4e+145)
		tmp = t_0;
	elseif (a <= -4.8e-137)
		tmp = t_1;
	elseif (a <= 4.5e-5)
		tmp = -1.0;
	elseif (a <= 6.6e+153)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(a * a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.4e+145], t$95$0, If[LessEqual[a, -4.8e-137], t$95$1, If[LessEqual[a, 4.5e-5], -1.0, If[LessEqual[a, 6.6e+153], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.40000000000000017e145 or 6.59999999999999989e153 < a

   1. Initial program 32.8%

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

      1. associate--l+ 32.8%

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

      2. fma-def 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in a around inf 44.8%

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

   5. Taylor expanded in b around 0 44.8%

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

      1. unpow2 44.8%

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

   7. Simplified 44.8%

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

   8. Taylor expanded in a around 0 98.7%

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

      1. unpow2 98.7%

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

      2. *-commutative 98.7%

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

      3. associate-*r* 98.7%

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

   10. Simplified 98.7%

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

   if -4.40000000000000017e145 < a < -4.8000000000000001e-137 or 4.50000000000000028e-5 < a < 6.59999999999999989e153

   1. Initial program 80.0%

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

      1. associate--l+ 80.0%

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

      2. fma-def 80.0%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in a around inf 71.1%

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

   5. Taylor expanded in b around inf 40.0%

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

      1. unpow2 40.0%

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

      2. unpow2 40.0%

         \[\leadsto 2 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

      3. *-commutative 40.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)} \]

   if -4.8000000000000001e-137 < a < 4.50000000000000028e-5

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 58.5%

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

      1. associate--l+ 58.5%

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

      2. associate-*r* 58.5%

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

      3. unpow2 58.5%

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

   6. Simplified 58.5%

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

   7. Taylor expanded in a around 0 58.5%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{+145}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-137}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 9: 55.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot 4\right)\\ t_1 := a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\\ \mathbf{if}\;a \leq -4.4 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -3.9 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a 4.0))) (t_1 (* a (* a (* b (* b 2.0))))))
   (if (<= a -4.4e+145)
     t_0
     (if (<= a -3.9e-196)
       t_1
       (if (<= a 4.7e-7) -1.0 (if (<= a 6.6e+153) t_1 t_0))))))

C

double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = a * (a * (b * (b * 2.0)));
	double tmp;
	if (a <= -4.4e+145) {
		tmp = t_0;
	} else if (a <= -3.9e-196) {
		tmp = t_1;
	} else if (a <= 4.7e-7) {
		tmp = -1.0;
	} else if (a <= 6.6e+153) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (a * 4.0d0)
    t_1 = a * (a * (b * (b * 2.0d0)))
    if (a <= (-4.4d+145)) then
        tmp = t_0
    else if (a <= (-3.9d-196)) then
        tmp = t_1
    else if (a <= 4.7d-7) then
        tmp = -1.0d0
    else if (a <= 6.6d+153) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = a * (a * (b * (b * 2.0)));
	double tmp;
	if (a <= -4.4e+145) {
		tmp = t_0;
	} else if (a <= -3.9e-196) {
		tmp = t_1;
	} else if (a <= 4.7e-7) {
		tmp = -1.0;
	} else if (a <= 6.6e+153) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * 4.0)
	t_1 = a * (a * (b * (b * 2.0)))
	tmp = 0
	if a <= -4.4e+145:
		tmp = t_0
	elif a <= -3.9e-196:
		tmp = t_1
	elif a <= 4.7e-7:
		tmp = -1.0
	elif a <= 6.6e+153:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * 4.0))
	t_1 = Float64(a * Float64(a * Float64(b * Float64(b * 2.0))))
	tmp = 0.0
	if (a <= -4.4e+145)
		tmp = t_0;
	elseif (a <= -3.9e-196)
		tmp = t_1;
	elseif (a <= 4.7e-7)
		tmp = -1.0;
	elseif (a <= 6.6e+153)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * 4.0);
	t_1 = a * (a * (b * (b * 2.0)));
	tmp = 0.0;
	if (a <= -4.4e+145)
		tmp = t_0;
	elseif (a <= -3.9e-196)
		tmp = t_1;
	elseif (a <= 4.7e-7)
		tmp = -1.0;
	elseif (a <= 6.6e+153)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(a * N[(b * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.4e+145], t$95$0, If[LessEqual[a, -3.9e-196], t$95$1, If[LessEqual[a, 4.7e-7], -1.0, If[LessEqual[a, 6.6e+153], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.40000000000000017e145 or 6.59999999999999989e153 < a

   1. Initial program 32.8%

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

      1. associate--l+ 32.8%

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

      2. fma-def 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in a around inf 44.8%

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

   5. Taylor expanded in b around 0 44.8%

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

      1. unpow2 44.8%

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

   7. Simplified 44.8%

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

   8. Taylor expanded in a around 0 98.7%

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

      1. unpow2 98.7%

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

      2. *-commutative 98.7%

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

      3. associate-*r* 98.7%

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

   10. Simplified 98.7%

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

   if -4.40000000000000017e145 < a < -3.90000000000000016e-196 or 4.7e-7 < a < 6.59999999999999989e153

   1. Initial program 81.7%

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

      1. associate--l+ 81.7%

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

      2. fma-def 81.7%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in a around inf 66.0%

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

   5. Taylor expanded in b around inf 37.6%

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

      1. unpow2 37.6%

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

      2. unpow2 37.6%

         \[\leadsto 2 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

      3. *-commutative 37.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)} \]

      4. associate-*l* 37.6%

         \[\leadsto \color{blue}{\left(2 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)} \]

      5. *-commutative 37.6%

         \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)} \]

      6. associate-*l* 41.4%

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(2 \cdot \left(b \cdot b\right)\right)\right)} \]

      7. associate-*r* 41.4%

         \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(2 \cdot b\right) \cdot b\right)}\right) \]

   7. Simplified 41.4%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\left(2 \cdot b\right) \cdot b\right)\right)} \]

   if -3.90000000000000016e-196 < a < 4.7e-7

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 61.1%

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

      1. associate--l+ 61.1%

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

      2. associate-*r* 61.1%

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

      3. unpow2 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in a around 0 61.1%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{+145}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -3.9 \cdot 10^{-196}:\\ \;\;\;\;a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 10: 50.2% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.36 \cdot 10^{-14} \lor \neg \left(a \leq 0.00082\right):\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.36e-14) (not (<= a 0.00082))) (* a (* a 4.0)) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.36e-14) || !(a <= 0.00082)) {
		tmp = a * (a * 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.36d-14)) .or. (.not. (a <= 0.00082d0))) then
        tmp = a * (a * 4.0d0)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.36e-14) || !(a <= 0.00082)) {
		tmp = a * (a * 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.36e-14) or not (a <= 0.00082):
		tmp = a * (a * 4.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.36e-14) || !(a <= 0.00082))
		tmp = Float64(a * Float64(a * 4.0));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.36e-14) || ~((a <= 0.00082)))
		tmp = a * (a * 4.0);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.36e-14], N[Not[LessEqual[a, 0.00082]], $MachinePrecision]], N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -1.36e-14 or 8.1999999999999998e-4 < a

   1. Initial program 55.4%

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

      1. associate--l+ 55.4%

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

      2. fma-def 55.4%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in a around inf 63.1%

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

   5. Taylor expanded in b around 0 53.7%

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

      1. unpow2 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in a around 0 49.1%

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

      1. unpow2 49.1%

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

      2. *-commutative 49.1%

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

      3. associate-*r* 49.1%

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

   10. Simplified 49.1%

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

   if -1.36e-14 < a < 8.1999999999999998e-4

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 53.1%

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

      1. associate--l+ 53.1%

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

      2. associate-*r* 53.1%

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

      3. unpow2 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in a around 0 53.1%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.36 \cdot 10^{-14} \lor \neg \left(a \leq 0.00082\right):\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11: 24.5% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.9%

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

   1. associate--l+ 74.9%

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

   2. fma-def 74.9%

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

3. Simplified 76.1%

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

4. Taylor expanded in b around 0 53.7%

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

   1. associate--l+ 53.7%

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

   2. associate-*r* 53.7%

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

   3. unpow2 53.7%

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

6. Simplified 53.7%

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

7. Taylor expanded in a around 0 23.7%

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

8. Final simplification 23.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))