Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.1% → 98.3%

Time: 6.4s

Alternatives: 11

Speedup: 1.2×

• 61.1% 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: 73.1% accurate, 1.0× speedup

Math

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

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: 98.3% accurate, 0.2× speedup

Math

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

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 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, 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 = 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[Power[a, 4.0], $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 5.3%

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

   4. Taylor expanded in a around inf 98.3%

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

4. Final simplification 99.6%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

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

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 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, 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, 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, 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 = 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 ^ 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[Power[a, 4.0], $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 5.3%

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

   4. Taylor expanded in a around inf 98.3%

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

4. Final simplification 99.6%

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

Alternative 3: 93.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot 4\\ \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;{a}^{4} + \left(-1 + t_0\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-18}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) 4.0)))
   (if (<= a -0.019)
     (+ (pow a 4.0) (+ -1.0 t_0))
     (if (<= a 3.6e-18)
       (+ -1.0 (+ (pow b 4.0) (* (* b b) 4.0)))
       (+ (pow a 4.0) (+ -1.0 (* (+ a 1.0) t_0)))))))

C

double code(double a, double b) {
	double t_0 = (a * a) * 4.0;
	double tmp;
	if (a <= -0.019) {
		tmp = pow(a, 4.0) + (-1.0 + t_0);
	} else if (a <= 3.6e-18) {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 4.0));
	} else {
		tmp = pow(a, 4.0) + (-1.0 + ((a + 1.0) * 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) :: tmp
    t_0 = (a * a) * 4.0d0
    if (a <= (-0.019d0)) then
        tmp = (a ** 4.0d0) + ((-1.0d0) + t_0)
    else if (a <= 3.6d-18) then
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * 4.0d0))
    else
        tmp = (a ** 4.0d0) + ((-1.0d0) + ((a + 1.0d0) * 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 tmp;
	if (a <= -0.019) {
		tmp = Math.pow(a, 4.0) + (-1.0 + t_0);
	} else if (a <= 3.6e-18) {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * 4.0));
	} else {
		tmp = Math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * t_0));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * 4.0
	tmp = 0
	if a <= -0.019:
		tmp = math.pow(a, 4.0) + (-1.0 + t_0)
	elif a <= 3.6e-18:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * 4.0))
	else:
		tmp = math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * t_0))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * 4.0)
	tmp = 0.0
	if (a <= -0.019)
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + t_0));
	elseif (a <= 3.6e-18)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0)));
	else
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a + 1.0) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * 4.0;
	tmp = 0.0;
	if (a <= -0.019)
		tmp = (a ^ 4.0) + (-1.0 + t_0);
	elseif (a <= 3.6e-18)
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * 4.0));
	else
		tmp = (a ^ 4.0) + (-1.0 + ((a + 1.0) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[a, -0.019], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.6e-18], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0189999999999999995

   1. Initial program 27.0%

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

      1. associate--l+ 27.0%

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

      2. fma-def 27.0%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in b around 0 23.1%

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

      1. associate--l+ 23.1%

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

      2. associate-*r* 23.1%

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

      3. unpow2 23.1%

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

   6. Simplified 23.1%

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

   7. Taylor expanded in a around 0 94.4%

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

      1. unpow2 94.4%

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

   9. Simplified 94.4%

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

   if -0.0189999999999999995 < a < 3.6000000000000001e-18

   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 0 99.8%

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

      1. sub-neg 99.8%

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

      2. fma-def 99.8%

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

      3. unpow2 99.8%

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

      4. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   if 3.6000000000000001e-18 < a

   1. Initial program 68.0%

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

      1. associate--l+ 68.0%

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

      2. fma-def 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in b around 0 92.5%

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

      1. associate--l+ 92.5%

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

      2. associate-*r* 92.5%

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

      3. unpow2 92.5%

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

   6. Simplified 92.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-18}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \end{array} \]

Alternative 4: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4\right)\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.019)
   (+ -1.0 (* (* a a) (fma a a 4.0)))
   (if (<= a 2.95e+54) (+ -1.0 (+ (pow b 4.0) (* (* b b) 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.019) {
		tmp = -1.0 + ((a * a) * fma(a, a, 4.0));
	} else if (a <= 2.95e+54) {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 4.0));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.019)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * fma(a, a, 4.0)));
	elseif (a <= 2.95e+54)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.019], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.95e+54], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0189999999999999995

   1. Initial program 27.0%

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

      1. associate--l+ 27.0%

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

      2. fma-def 27.0%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in b around 0 23.1%

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

      1. associate--l+ 23.1%

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

      2. associate-*r* 23.1%

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

      3. unpow2 23.1%

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

   6. Simplified 23.1%

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

   7. Taylor expanded in a around 0 94.4%

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

      1. unpow2 94.4%

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

   9. Simplified 94.4%

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

       1. associate-+r- 94.4%

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

       2. sqr-pow 94.3%

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

       3. metadata-eval 94.3%

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

       4. pow2 94.3%

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

       5. metadata-eval 94.3%

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

       6. pow2 94.3%

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

       7. distribute-rgt-out 94.3%

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

       8. fma-def 94.3%

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

   11. Applied egg-rr 94.3%

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

   if -0.0189999999999999995 < a < 2.9499999999999999e54

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around 0 95.6%

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

      1. sub-neg 95.6%

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

      2. fma-def 95.6%

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

      3. unpow2 95.6%

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

      4. metadata-eval 95.6%

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

   6. Simplified 95.6%

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

      1. fma-udef 95.6%

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

      2. +-commutative 95.6%

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

   8. Applied egg-rr 95.6%

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

   if 2.9499999999999999e54 < a

   1. Initial program 61.1%

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

      1. associate--l+ 61.1%

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

      2. fma-def 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in a around inf 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4\right)\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.019)
   (+ (pow a 4.0) (+ -1.0 (* (* a a) 4.0)))
   (if (<= a 4.1e+54) (+ -1.0 (+ (pow b 4.0) (* (* b b) 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.019) {
		tmp = pow(a, 4.0) + (-1.0 + ((a * a) * 4.0));
	} else if (a <= 4.1e+54) {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 4.0));
	} 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 <= (-0.019d0)) then
        tmp = (a ** 4.0d0) + ((-1.0d0) + ((a * a) * 4.0d0))
    else if (a <= 4.1d+54) then
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * 4.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.019) {
		tmp = Math.pow(a, 4.0) + (-1.0 + ((a * a) * 4.0));
	} else if (a <= 4.1e+54) {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * 4.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.019:
		tmp = math.pow(a, 4.0) + (-1.0 + ((a * a) * 4.0))
	elif a <= 4.1e+54:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * 4.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.019)
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a * a) * 4.0)));
	elseif (a <= 4.1e+54)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.019)
		tmp = (a ^ 4.0) + (-1.0 + ((a * a) * 4.0));
	elseif (a <= 4.1e+54)
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * 4.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.019], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.1e+54], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0189999999999999995

   1. Initial program 27.0%

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

      1. associate--l+ 27.0%

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

      2. fma-def 27.0%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in b around 0 23.1%

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

      1. associate--l+ 23.1%

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

      2. associate-*r* 23.1%

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

      3. unpow2 23.1%

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

   6. Simplified 23.1%

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

   7. Taylor expanded in a around 0 94.4%

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

      1. unpow2 94.4%

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

   9. Simplified 94.4%

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

   if -0.0189999999999999995 < a < 4.09999999999999967e54

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around 0 95.6%

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

      1. sub-neg 95.6%

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

      2. fma-def 95.6%

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

      3. unpow2 95.6%

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

      4. metadata-eval 95.6%

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

   6. Simplified 95.6%

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

      1. fma-udef 95.6%

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

      2. +-commutative 95.6%

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

   8. Applied egg-rr 95.6%

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

   if 4.09999999999999967e54 < a

   1. Initial program 61.1%

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

      1. associate--l+ 61.1%

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

      2. fma-def 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in a around inf 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6: 93.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.019)
   (+ -1.0 (* (* a a) (fma a a 4.0)))
   (if (<= a 1.8e+54) (+ -1.0 (* (* b b) (+ (* b b) 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.019) {
		tmp = -1.0 + ((a * a) * fma(a, a, 4.0));
	} else if (a <= 1.8e+54) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.019)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * fma(a, a, 4.0)));
	elseif (a <= 1.8e+54)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.019], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e+54], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0189999999999999995

   1. Initial program 27.0%

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

      1. associate--l+ 27.0%

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

      2. fma-def 27.0%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in b around 0 23.1%

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

      1. associate--l+ 23.1%

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

      2. associate-*r* 23.1%

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

      3. unpow2 23.1%

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

   6. Simplified 23.1%

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

   7. Taylor expanded in a around 0 94.4%

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

      1. unpow2 94.4%

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

   9. Simplified 94.4%

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

       1. associate-+r- 94.4%

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

       2. sqr-pow 94.3%

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

       3. metadata-eval 94.3%

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

       4. pow2 94.3%

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

       5. metadata-eval 94.3%

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

       6. pow2 94.3%

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

       7. distribute-rgt-out 94.3%

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

       8. fma-def 94.3%

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

   11. Applied egg-rr 94.3%

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

   if -0.0189999999999999995 < a < 1.8000000000000001e54

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around 0 95.6%

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

      1. sub-neg 95.6%

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

      2. fma-def 95.6%

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

      3. unpow2 95.6%

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

      4. metadata-eval 95.6%

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

   6. Simplified 95.6%

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

      1. fma-udef 95.6%

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

      2. sqr-pow 95.5%

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

      3. metadata-eval 95.5%

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

      4. pow2 95.5%

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

      5. metadata-eval 95.5%

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

      6. pow2 95.5%

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

      7. distribute-rgt-out 95.5%

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

   8. Applied egg-rr 95.5%

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

   if 1.8000000000000001e54 < a

   1. Initial program 61.1%

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

      1. associate--l+ 61.1%

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

      2. fma-def 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in a around inf 100.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7: 94.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -460000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+59}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -460000000.0)
   (pow a 4.0)
   (if (<= a 6.8e+59) (+ -1.0 (* (* b b) (+ (* b b) 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -460000000.0) {
		tmp = pow(a, 4.0);
	} else if (a <= 6.8e+59) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} 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 <= (-460000000.0d0)) then
        tmp = a ** 4.0d0
    else if (a <= 6.8d+59) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -460000000.0) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 6.8e+59) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -460000000.0:
		tmp = math.pow(a, 4.0)
	elif a <= 6.8e+59:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -460000000.0)
		tmp = a ^ 4.0;
	elseif (a <= 6.8e+59)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -460000000.0)
		tmp = a ^ 4.0;
	elseif (a <= 6.8e+59)
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -460000000.0], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 6.8e+59], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.6e8 or 6.80000000000000012e59 < a

   1. Initial program 44.5%

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

      1. associate--l+ 44.5%

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

      2. fma-def 44.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in a around inf 97.9%

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

   if -4.6e8 < a < 6.80000000000000012e59

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around 0 95.2%

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

      1. sub-neg 95.2%

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

      2. fma-def 95.2%

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

      3. unpow2 95.2%

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

      4. metadata-eval 95.2%

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

   6. Simplified 95.2%

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

      1. fma-udef 95.2%

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

      2. sqr-pow 95.1%

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

      3. metadata-eval 95.1%

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

      4. pow2 95.1%

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

      5. metadata-eval 95.1%

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

      6. pow2 95.1%

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

      7. distribute-rgt-out 95.1%

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

   8. Applied egg-rr 95.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -460000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+59}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 8: 69.5% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. associate--l+ 77.7%

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

   2. fma-def 77.7%

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

3. Simplified 78.8%

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

4. Taylor expanded in a around 0 72.9%

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

   1. sub-neg 72.9%

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

   2. fma-def 72.9%

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

   3. unpow2 72.9%

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

   4. metadata-eval 72.9%

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

6. Simplified 72.9%

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

   1. fma-udef 72.9%

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

   2. sqr-pow 72.9%

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

   3. metadata-eval 72.9%

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

   4. pow2 72.9%

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

   5. metadata-eval 72.9%

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

   6. pow2 72.9%

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

   7. distribute-rgt-out 72.9%

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

8. Applied egg-rr 72.9%

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

9. Final simplification 72.9%

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

Alternative 9: 68.9% accurate, 14.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. associate--l+ 77.7%

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

   2. fma-def 77.7%

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

3. Simplified 78.8%

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

4. Taylor expanded in a around 0 72.9%

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

   1. sub-neg 72.9%

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

   2. fma-def 72.9%

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

   3. unpow2 72.9%

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

   4. metadata-eval 72.9%

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

6. Simplified 72.9%

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

   1. fma-udef 72.9%

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

   2. sqr-pow 72.9%

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

   3. metadata-eval 72.9%

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

   4. pow2 72.9%

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

   5. metadata-eval 72.9%

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

   6. pow2 72.9%

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

   7. distribute-rgt-out 72.9%

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

8. Applied egg-rr 72.9%

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

9. Taylor expanded in b around inf 71.2%

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

    1. unpow2 71.2%

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

11. Simplified 71.2%

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

12. Final simplification 71.2%

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

Alternative 10: 50.7% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. associate--l+ 77.7%

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

   2. fma-def 77.7%

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

3. Simplified 78.8%

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

4. Taylor expanded in a around 0 72.9%

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

   1. sub-neg 72.9%

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

   2. fma-def 72.9%

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

   3. unpow2 72.9%

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

   4. metadata-eval 72.9%

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

6. Simplified 72.9%

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

7. Taylor expanded in b around 0 49.9%

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

   1. unpow2 49.9%

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

9. Simplified 49.9%

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

10. Final simplification 49.9%

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

Alternative 11: 24.8% 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 77.7%

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

   1. associate--l+ 77.7%

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

   2. fma-def 77.7%

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

3. Simplified 78.8%

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

4. Taylor expanded in b around 0 55.0%

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

   1. associate--l+ 55.0%

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

   2. associate-*r* 55.0%

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

   3. unpow2 55.0%

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

6. Simplified 55.0%

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

7. Taylor expanded in a around 0 27.3%

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

8. Final simplification 27.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023261 
(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))