Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.1% → 99.9%

Time: 9.5s

Alternatives: 9

Speedup: 1.2×

• 60.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% 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: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{2} \cdot \left(2 \cdot {b}^{2} + a \cdot \left(a + 4\right)\right) + {b}^{2} \cdot \left(4 + {b}^{2}\right)\right) + -1\\ \end{array} \end{array} \]

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 2.0) (+ (* 2.0 (pow b 2.0)) (* a (+ a 4.0))))
       (* (pow b 2.0) (+ 4.0 (pow b 2.0))))
      -1.0))))

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, 2.0) * ((2.0 * pow(b, 2.0)) + (a * (a + 4.0)))) + (pow(b, 2.0) * (4.0 + pow(b, 2.0)))) + -1.0;
	}
	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, 2.0) * ((2.0 * Math.pow(b, 2.0)) + (a * (a + 4.0)))) + (Math.pow(b, 2.0) * (4.0 + Math.pow(b, 2.0)))) + -1.0;
	}
	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, 2.0) * ((2.0 * math.pow(b, 2.0)) + (a * (a + 4.0)))) + (math.pow(b, 2.0) * (4.0 + math.pow(b, 2.0)))) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(Float64(Float64((a ^ 2.0) * Float64(Float64(2.0 * (b ^ 2.0)) + Float64(a * Float64(a + 4.0)))) + Float64((b ^ 2.0) * Float64(4.0 + (b ^ 2.0)))) + -1.0);
	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 ^ 2.0) * ((2.0 * (b ^ 2.0)) + (a * (a + 4.0)))) + ((b ^ 2.0) * (4.0 + (b ^ 2.0)))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.8%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) 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. Add Preprocessing

   3. Taylor expanded in b around 0 35.1%

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

   4. Taylor expanded in a around inf 35.1%

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

   5. Taylor expanded in a around 0 35.1%

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

   6. Taylor expanded in a around 0 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 95.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-54}:\\ \;\;\;\;{a}^{2} \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-54)
   (+ (* (pow a 2.0) (+ 4.0 (* a (+ a 4.0)))) -1.0)
   (if (<= (* b b) 2e+129)
     (+
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
      -1.0)
     (+ (pow b 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-54) {
		tmp = (pow(a, 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else if ((b * b) <= 2e+129) {
		tmp = (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) + -1.0;
	} else {
		tmp = pow(b, 4.0) + -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 ((b * b) <= 5d-54) then
        tmp = ((a ** 2.0d0) * (4.0d0 + (a * (a + 4.0d0)))) + (-1.0d0)
    else if ((b * b) <= 2d+129) then
        tmp = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (a + 1.0d0)) + ((b * b) * (1.0d0 - (a * 3.0d0)))))) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-54) {
		tmp = (Math.pow(a, 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else if ((b * b) <= 2e+129) {
		tmp = (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-54:
		tmp = (math.pow(a, 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0
	elif (b * b) <= 2e+129:
		tmp = (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-54)
		tmp = Float64(Float64((a ^ 2.0) * Float64(4.0 + Float64(a * Float64(a + 4.0)))) + -1.0);
	elseif (Float64(b * b) <= 2e+129)
		tmp = Float64(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)))))) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-54)
		tmp = ((a ^ 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0;
	elseif ((b * b) <= 2e+129)
		tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-54], N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 2e+129], 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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 5.00000000000000015e-54

   1. Initial program 84.2%

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

   3. Taylor expanded in b around 0 80.2%

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

   4. Taylor expanded in b around 0 84.3%

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

   5. Taylor expanded in a around 0 99.9%

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

   if 5.00000000000000015e-54 < (*.f64 b b) < 2e129

   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. Add Preprocessing

   if 2e129 < (*.f64 b b)

   1. Initial program 60.6%

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

      1. sub-neg 60.6%

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

      2. +-commutative 60.6%

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

      3. fma-define 68.1%

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

      4. +-commutative 68.1%

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

      5. associate-*l* 68.1%

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

      6. cancel-sign-sub-inv 68.1%

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

      7. metadata-eval 68.1%

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

      8. fma-define 68.1%

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

      9. metadata-eval 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in b around inf 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-54}:\\ \;\;\;\;{a}^{2} \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;{a}^{2} \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.2e+15)
   (+ (* (pow a 2.0) (+ 4.0 (* a (+ a 4.0)))) -1.0)
   (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.2e+15) {
		tmp = (pow(a, 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = pow(b, 4.0) + -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 (b <= 3.2d+15) then
        tmp = ((a ** 2.0d0) * (4.0d0 + (a * (a + 4.0d0)))) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.2e+15) {
		tmp = (Math.pow(a, 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.2e+15:
		tmp = (math.pow(a, 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.2e+15)
		tmp = Float64(Float64((a ^ 2.0) * Float64(4.0 + Float64(a * Float64(a + 4.0)))) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.2e+15)
		tmp = ((a ^ 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.2e+15], N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.2e15

   1. Initial program 82.5%

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

   3. Taylor expanded in b around 0 84.6%

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

   4. Taylor expanded in b around 0 64.6%

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

   5. Taylor expanded in a around 0 77.3%

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

   if 3.2e15 < b

   1. Initial program 61.5%

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

      1. sub-neg 61.5%

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

      2. +-commutative 61.5%

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

      3. fma-define 73.2%

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

      4. +-commutative 73.2%

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

      5. associate-*l* 73.2%

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

      6. cancel-sign-sub-inv 73.2%

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

      7. metadata-eval 73.2%

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

      8. fma-define 73.2%

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

      9. metadata-eval 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in b around inf 86.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;{a}^{2} \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.2e+15)
   (+ -1.0 (* (pow a 4.0) (+ 1.0 (/ 4.0 a))))
   (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.2e+15) {
		tmp = -1.0 + (pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else {
		tmp = pow(b, 4.0) + -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 (b <= 3.2d+15) then
        tmp = (-1.0d0) + ((a ** 4.0d0) * (1.0d0 + (4.0d0 / a)))
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.2e+15) {
		tmp = -1.0 + (Math.pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.2e+15:
		tmp = -1.0 + (math.pow(a, 4.0) * (1.0 + (4.0 / a)))
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.2e+15)
		tmp = Float64(-1.0 + Float64((a ^ 4.0) * Float64(1.0 + Float64(4.0 / a))));
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.2e+15)
		tmp = -1.0 + ((a ^ 4.0) * (1.0 + (4.0 / a)));
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.2e+15], N[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.2e15

   1. Initial program 82.5%

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

      1. sub-neg 82.5%

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

      2. +-commutative 82.5%

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

      3. fma-define 82.5%

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

      4. +-commutative 82.5%

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

      5. associate-*l* 82.5%

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

      6. cancel-sign-sub-inv 82.5%

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

      7. metadata-eval 82.5%

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

      8. fma-define 82.5%

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

      9. metadata-eval 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in a around inf 76.6%

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

      1. associate-*r/ 76.6%

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

      2. metadata-eval 76.6%

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

   7. Simplified 76.6%

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

   if 3.2e15 < b

   1. Initial program 61.5%

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

      1. sub-neg 61.5%

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

      2. +-commutative 61.5%

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

      3. fma-define 73.2%

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

      4. +-commutative 73.2%

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

      5. associate-*l* 73.2%

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

      6. cancel-sign-sub-inv 73.2%

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

      7. metadata-eval 73.2%

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

      8. fma-define 73.2%

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

      9. metadata-eval 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in b around inf 86.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;-1 + \left(a + 4\right) \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.2e+15) (+ -1.0 (* (+ a 4.0) (pow a 3.0))) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.2e+15) {
		tmp = -1.0 + ((a + 4.0) * pow(a, 3.0));
	} else {
		tmp = pow(b, 4.0) + -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 (b <= 3.2d+15) then
        tmp = (-1.0d0) + ((a + 4.0d0) * (a ** 3.0d0))
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.2e+15) {
		tmp = -1.0 + ((a + 4.0) * Math.pow(a, 3.0));
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.2e+15:
		tmp = -1.0 + ((a + 4.0) * math.pow(a, 3.0))
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.2e+15)
		tmp = Float64(-1.0 + Float64(Float64(a + 4.0) * (a ^ 3.0)));
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.2e+15)
		tmp = -1.0 + ((a + 4.0) * (a ^ 3.0));
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.2e+15], N[(-1.0 + N[(N[(a + 4.0), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.2e15

   1. Initial program 82.5%

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

      1. sub-neg 82.5%

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

      2. +-commutative 82.5%

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

      3. fma-define 82.5%

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

      4. +-commutative 82.5%

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

      5. associate-*l* 82.5%

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

      6. cancel-sign-sub-inv 82.5%

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

      7. metadata-eval 82.5%

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

      8. fma-define 82.5%

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

      9. metadata-eval 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in a around inf 76.6%

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

      1. associate-*r/ 76.6%

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

      2. metadata-eval 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in a around 0 76.5%

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

   if 3.2e15 < b

   1. Initial program 61.5%

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

      1. sub-neg 61.5%

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

      2. +-commutative 61.5%

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

      3. fma-define 73.2%

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

      4. +-commutative 73.2%

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

      5. associate-*l* 73.2%

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

      6. cancel-sign-sub-inv 73.2%

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

      7. metadata-eval 73.2%

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

      8. fma-define 73.2%

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

      9. metadata-eval 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in b around inf 86.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;-1 + \left(a + 4\right) \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.4e+15) (+ -1.0 (pow a 4.0)) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.4e+15) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0) + -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 (b <= 3.4d+15) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.4e+15) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.4e+15:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.4e+15)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.4e+15)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.4e+15], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.4e15

   1. Initial program 82.5%

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

      1. sub-neg 82.5%

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

      2. +-commutative 82.5%

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

      3. fma-define 82.5%

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

      4. +-commutative 82.5%

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

      5. associate-*l* 82.5%

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

      6. cancel-sign-sub-inv 82.5%

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

      7. metadata-eval 82.5%

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

      8. fma-define 82.5%

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

      9. metadata-eval 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in a around inf 76.4%

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

   if 3.4e15 < b

   1. Initial program 61.5%

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

      1. sub-neg 61.5%

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

      2. +-commutative 61.5%

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

      3. fma-define 73.2%

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

      4. +-commutative 73.2%

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

      5. associate-*l* 73.2%

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

      6. cancel-sign-sub-inv 73.2%

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

      7. metadata-eval 73.2%

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

      8. fma-define 73.2%

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

      9. metadata-eval 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in b around inf 86.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ -1 + {a}^{4} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. sub-neg 77.6%

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

   2. +-commutative 77.6%

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

   3. fma-define 80.3%

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

   4. +-commutative 80.3%

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

   5. associate-*l* 80.3%

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

   6. cancel-sign-sub-inv 80.3%

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

   7. metadata-eval 80.3%

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

   8. fma-define 80.3%

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

   9. metadata-eval 80.3%

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

3. Simplified 80.3%

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

5. Taylor expanded in a around inf 68.6%

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

6. Final simplification 68.6%

   \[\leadsto -1 + {a}^{4} \]
7. Add Preprocessing

Alternative 8: 50.6% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in b around 0 83.5%

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

4. Taylor expanded in b around 0 54.8%

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

5. Taylor expanded in a around 0 48.4%

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

   1. add-sqr-sqrt 48.4%

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

   2. difference-of-sqr-1 48.4%

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

   3. *-commutative 48.4%

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

   4. sqrt-prod 48.4%

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

   5. sqrt-pow1 38.7%

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

   6. metadata-eval 38.7%

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

   7. pow1 38.7%

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

   8. metadata-eval 38.7%

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

   9. *-commutative 38.7%

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

   10. sqrt-prod 38.7%

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

   11. sqrt-pow1 48.4%

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

   12. metadata-eval 48.4%

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

   13. pow1 48.4%

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

   14. metadata-eval 48.4%

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

7. Applied egg-rr 48.4%

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

8. Final simplification 48.4%

   \[\leadsto \left(1 + a \cdot 2\right) \cdot \left(-1 + a \cdot 2\right) \]
9. Add Preprocessing

Alternative 9: 24.9% 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 77.6%

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

3. Taylor expanded in b around 0 83.5%

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

4. Taylor expanded in b around 0 54.8%

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

5. Taylor expanded in a around 0 48.4%

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

6. Taylor expanded in a around 0 27.3%

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

7. Final simplification 27.3%

   \[\leadsto -1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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))