Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.6% → 98.3%

Time: 9.4s

Alternatives: 6

Speedup: 1.2×

• 61.2% 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: 73.6% 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, 0.5× 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 7.9%

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

   4. Taylor expanded in a around inf 94.0%

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

4. Final simplification 98.4%

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

Alternative 2: 80.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -1450:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+68}:\\ \;\;\;\;4 \cdot {b}^{2} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+65)
   (pow a 4.0)
   (if (<= a -1450.0)
     (pow b 4.0)
     (if (<= a 1.45e+68) (+ (* 4.0 (pow b 2.0)) -1.0) (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+65) {
		tmp = pow(a, 4.0);
	} else if (a <= -1450.0) {
		tmp = pow(b, 4.0);
	} else if (a <= 1.45e+68) {
		tmp = (4.0 * pow(b, 2.0)) + -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 <= (-1d+65)) then
        tmp = a ** 4.0d0
    else if (a <= (-1450.0d0)) then
        tmp = b ** 4.0d0
    else if (a <= 1.45d+68) then
        tmp = (4.0d0 * (b ** 2.0d0)) + (-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 <= -1e+65) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -1450.0) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 1.45e+68) {
		tmp = (4.0 * Math.pow(b, 2.0)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e+65:
		tmp = math.pow(a, 4.0)
	elif a <= -1450.0:
		tmp = math.pow(b, 4.0)
	elif a <= 1.45e+68:
		tmp = (4.0 * math.pow(b, 2.0)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+65)
		tmp = a ^ 4.0;
	elseif (a <= -1450.0)
		tmp = b ^ 4.0;
	elseif (a <= 1.45e+68)
		tmp = Float64(Float64(4.0 * (b ^ 2.0)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+65)
		tmp = a ^ 4.0;
	elseif (a <= -1450.0)
		tmp = b ^ 4.0;
	elseif (a <= 1.45e+68)
		tmp = (4.0 * (b ^ 2.0)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+65], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -1450.0], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 1.45e+68], N[(N[(4.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.9999999999999999e64 or 1.45000000000000006e68 < a

   1. Initial program 43.2%

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

      1. associate--l+ 43.2%

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

      2. fma-def 43.2%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -9.9999999999999999e64 < a < -1450

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in b around inf 69.6%

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

   if -1450 < a < 1.45000000000000006e68

   1. Initial program 97.0%

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

      1. associate--l+ 97.0%

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

      2. fma-def 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in a around 0 99.2%

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

   5. Taylor expanded in b around 0 80.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -1450:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+68}:\\ \;\;\;\;4 \cdot {b}^{2} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 3: 80.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -1450:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.2e+64)
   (pow a 4.0)
   (if (<= a -1450.0)
     (pow b 4.0)
     (if (<= a 8.5e+69)
       (* (+ 1.0 (* b 2.0)) (+ (* b 2.0) -1.0))
       (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.2e+64) {
		tmp = pow(a, 4.0);
	} else if (a <= -1450.0) {
		tmp = pow(b, 4.0);
	} else if (a <= 8.5e+69) {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -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 <= (-5.2d+64)) then
        tmp = a ** 4.0d0
    else if (a <= (-1450.0d0)) then
        tmp = b ** 4.0d0
    else if (a <= 8.5d+69) then
        tmp = (1.0d0 + (b * 2.0d0)) * ((b * 2.0d0) + (-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 <= -5.2e+64) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -1450.0) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 8.5e+69) {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.2e+64:
		tmp = math.pow(a, 4.0)
	elif a <= -1450.0:
		tmp = math.pow(b, 4.0)
	elif a <= 8.5e+69:
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.2e+64)
		tmp = a ^ 4.0;
	elseif (a <= -1450.0)
		tmp = b ^ 4.0;
	elseif (a <= 8.5e+69)
		tmp = Float64(Float64(1.0 + Float64(b * 2.0)) * Float64(Float64(b * 2.0) + -1.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.2e+64)
		tmp = a ^ 4.0;
	elseif (a <= -1450.0)
		tmp = b ^ 4.0;
	elseif (a <= 8.5e+69)
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.2e+64], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -1450.0], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 8.5e+69], N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.19999999999999994e64 or 8.5000000000000002e69 < a

   1. Initial program 43.2%

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

      1. associate--l+ 43.2%

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

      2. fma-def 43.2%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -5.19999999999999994e64 < a < -1450

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in b around inf 69.6%

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

   if -1450 < a < 8.5000000000000002e69

   1. Initial program 97.0%

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

      1. associate--l+ 97.0%

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

      2. fma-def 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in a around 0 99.2%

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

   5. Taylor expanded in b around 0 80.2%

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

      1. add-sqr-sqrt 80.2%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2}} \cdot \sqrt{4 \cdot {b}^{2}}} - 1 \]

      2. difference-of-sqr-1 80.2%

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

      3. sqrt-prod 80.2%

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

      4. metadata-eval 80.2%

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

      5. unpow2 80.2%

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

      6. sqrt-prod 32.1%

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

      7. add-sqr-sqrt 58.4%

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

      8. sqrt-prod 58.4%

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

      9. metadata-eval 58.4%

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

      10. unpow2 58.4%

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

      11. sqrt-prod 32.1%

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

      12. add-sqr-sqrt 80.2%

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

   7. Applied egg-rr 80.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -1450:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+34} \lor \neg \left(a \leq 6.6 \cdot 10^{+67}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -6e+34) (not (<= a 6.6e+67)))
   (pow a 4.0)
   (* (+ 1.0 (* b 2.0)) (+ (* b 2.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -6e+34) || !(a <= 6.6e+67)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -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 <= (-6d+34)) .or. (.not. (a <= 6.6d+67))) then
        tmp = a ** 4.0d0
    else
        tmp = (1.0d0 + (b * 2.0d0)) * ((b * 2.0d0) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -6e+34) || !(a <= 6.6e+67)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -6e+34) or not (a <= 6.6e+67):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -6e+34) || !(a <= 6.6e+67))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(1.0 + Float64(b * 2.0)) * Float64(Float64(b * 2.0) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -6e+34) || ~((a <= 6.6e+67)))
		tmp = a ^ 4.0;
	else
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -6e+34], N[Not[LessEqual[a, 6.6e+67]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.00000000000000037e34 or 6.6000000000000006e67 < a

   1. Initial program 46.8%

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

      1. associate--l+ 46.8%

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

      2. fma-def 46.8%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in a around inf 96.6%

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

   if -6.00000000000000037e34 < a < 6.6000000000000006e67

   1. Initial program 97.1%

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

      1. associate--l+ 97.1%

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

      2. fma-def 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in a around 0 97.9%

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

   5. Taylor expanded in b around 0 76.8%

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

      1. add-sqr-sqrt 76.8%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2}} \cdot \sqrt{4 \cdot {b}^{2}}} - 1 \]

      2. difference-of-sqr-1 76.8%

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

      3. sqrt-prod 76.8%

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

      4. metadata-eval 76.8%

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

      5. unpow2 76.8%

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

      6. sqrt-prod 30.2%

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

      7. add-sqr-sqrt 54.8%

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

      8. sqrt-prod 54.8%

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

      9. metadata-eval 54.8%

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

      10. unpow2 54.8%

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

      11. sqrt-prod 30.2%

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

      12. add-sqr-sqrt 76.8%

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

   7. Applied egg-rr 76.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+34} \lor \neg \left(a \leq 6.6 \cdot 10^{+67}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \end{array} \]

Alternative 5: 50.5% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
}

Python

def code(a, b):
	return (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. associate--l+ 75.3%

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

   2. fma-def 75.3%

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

3. Simplified 77.3%

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

4. Taylor expanded in a around 0 73.9%

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

5. Taylor expanded in b around 0 56.0%

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

   1. add-sqr-sqrt 56.0%

      \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2}} \cdot \sqrt{4 \cdot {b}^{2}}} - 1 \]

   2. difference-of-sqr-1 56.0%

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

   3. sqrt-prod 56.0%

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

   4. metadata-eval 56.0%

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

   5. unpow2 56.0%

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

   6. sqrt-prod 24.6%

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

   7. add-sqr-sqrt 38.6%

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

   8. sqrt-prod 38.6%

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

   9. metadata-eval 38.6%

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

   10. unpow2 38.6%

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

   11. sqrt-prod 24.6%

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

   12. add-sqr-sqrt 56.0%

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

7. Applied egg-rr 56.0%

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

8. Final simplification 56.0%

   \[\leadsto \left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right) \]

Alternative 6: 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 75.3%

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

   1. associate--l+ 75.3%

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

   2. fma-def 75.3%

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

3. Simplified 77.3%

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

4. Taylor expanded in a around 0 73.9%

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

5. Taylor expanded in b around 0 25.0%

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

6. Final simplification 25.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023301 
(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))