Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.4% → 99.9%

Time: 8.1s

Alternatives: 11

Speedup: 5.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% 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, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 - 1.0;
	} else {
		tmp = ((fma((4.0 + a), a, fma((b * b), 2.0, 4.0)) * a) * a) - 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(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0)) * a) * a) - 1.0);
	end
	return tmp
end

Wolfram

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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+51)
   (- (* (* (fma (+ 4.0 a) a (fma (* b b) 2.0 4.0)) a) a) 1.0)
   (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* (fma a a a) a))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+51) {
		tmp = ((fma((4.0 + a), a, fma((b * b), 2.0, 4.0)) * a) * a) - 1.0;
	} else {
		tmp = (pow(((a * a) + (b * b)), 2.0) + (4.0 * (fma(a, a, a) * a))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+51)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0)) * a) * a) - 1.0);
	else
		tmp = Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(fma(a, a, a) * a))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+51], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1e51

   1. Initial program 17.7%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -1e51 < a

   1. Initial program 87.6%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lft-mult-inverse N/A

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

      7. *-lft-identity N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 3: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b \cdot b, 2, 4\right)\\ \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-43}:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, t\_0\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right), \left(t\_0 \cdot a\right) \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* b b) 2.0 4.0)))
   (if (<= (* b b) 4e-43)
     (- (* (* (fma (+ 4.0 a) a t_0) a) a) 1.0)
     (- (fma (* b b) (fma b b (fma -12.0 a 4.0)) (* (* t_0 a) a)) 1.0))))

C

double code(double a, double b) {
	double t_0 = fma((b * b), 2.0, 4.0);
	double tmp;
	if ((b * b) <= 4e-43) {
		tmp = ((fma((4.0 + a), a, t_0) * a) * a) - 1.0;
	} else {
		tmp = fma((b * b), fma(b, b, fma(-12.0, a, 4.0)), ((t_0 * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(b * b), 2.0, 4.0)
	tmp = 0.0
	if (Float64(b * b) <= 4e-43)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, t_0) * a) * a) - 1.0);
	else
		tmp = Float64(fma(Float64(b * b), fma(b, b, fma(-12.0, a, 4.0)), Float64(Float64(t_0 * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 4e-43], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + t$95$0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000031e-43

   1. Initial program 88.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 4.00000000000000031e-43 < (*.f64 b b)

   1. Initial program 64.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right), \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0098 \lor \neg \left(a \leq 1.8 \cdot 10^{-26}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.0098) (not (<= a 1.8e-26)))
   (- (* (* (fma (+ 4.0 a) a (fma (* b b) 2.0 4.0)) a) a) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.0098) || !(a <= 1.8e-26)) {
		tmp = ((fma((4.0 + a), a, fma((b * b), 2.0, 4.0)) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.0098) || !(a <= 1.8e-26))
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0)) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.0098], N[Not[LessEqual[a, 1.8e-26]], $MachinePrecision]], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0097999999999999997 or 1.8000000000000001e-26 < a

   1. Initial program 49.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.7%

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

   if -0.0097999999999999997 < a < 1.8000000000000001e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0098 \lor \neg \left(a \leq 1.8 \cdot 10^{-26}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+39}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(4 + a, a, 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+39)
   (- (* (* a a) (fma (+ 4.0 a) a 4.0)) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+39) {
		tmp = ((a * a) * fma((4.0 + a), a, 4.0)) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+39)
		tmp = Float64(Float64(Float64(a * a) * fma(Float64(4.0 + a), a, 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+39], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.9999999999999994e38

   1. Initial program 87.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-out N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 97.5

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

   8. Applied rewrites 97.5%

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

   if 9.9999999999999994e38 < (*.f64 b b)

   1. Initial program 62.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.6%

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

   9. Applied rewrites 94.7%

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

Alternative 6: 93.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+39}:\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+39)
   (- (* (* (* (- a -4.0) a) a) a) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+39) {
		tmp = ((((a - -4.0) * a) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+39)
		tmp = Float64(Float64(Float64(Float64(Float64(a - -4.0) * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+39], N[(N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.9999999999999994e38

   1. Initial program 87.6%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 96.1%

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

   if 9.9999999999999994e38 < (*.f64 b b)

   1. Initial program 62.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.6%

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

   9. Applied rewrites 94.7%

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

Alternative 7: 94.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -650000 \lor \neg \left(a \leq 3.15 \cdot 10^{+35}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -650000.0) (not (<= a 3.15e+35)))
   (* (* a a) (* a a))
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -650000.0) || !(a <= 3.15e+35)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -650000.0) || !(a <= 3.15e+35))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -650000.0], N[Not[LessEqual[a, 3.15e+35]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5e5 or 3.14999999999999985e35 < a

   1. Initial program 41.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.9

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

   8. Applied rewrites 93.9%

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

   10. Applied rewrites 93.8%

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

   if -6.5e5 < a < 3.14999999999999985e35

   1. Initial program 98.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 97.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -650000 \lor \neg \left(a \leq 3.15 \cdot 10^{+35}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -650000 \lor \neg \left(a \leq 3.15 \cdot 10^{+35}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -650000.0) (not (<= a 3.15e+35)))
   (* (* a a) (* a a))
   (- (* (* b b) (fma b b 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -650000.0) || !(a <= 3.15e+35)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * fma(b, b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -650000.0) || !(a <= 3.15e+35))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -650000.0], N[Not[LessEqual[a, 3.15e+35]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5e5 or 3.14999999999999985e35 < a

   1. Initial program 41.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.9

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

   8. Applied rewrites 93.9%

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

   10. Applied rewrites 93.8%

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

   if -6.5e5 < a < 3.14999999999999985e35

   1. Initial program 98.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -650000 \lor \neg \left(a \leq 3.15 \cdot 10^{+35}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -320 \lor \neg \left(a \leq 2.8 \cdot 10^{+26}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -320.0) (not (<= a 2.8e+26)))
   (* (* a a) (* a a))
   (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -320.0) || !(a <= 2.8e+26)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * 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 ((a <= (-320.0d0)) .or. (.not. (a <= 2.8d+26))) then
        tmp = (a * a) * (a * a)
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -320.0) || !(a <= 2.8e+26)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -320.0) or not (a <= 2.8e+26):
		tmp = (a * a) * (a * a)
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -320.0) || !(a <= 2.8e+26))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -320.0) || ~((a <= 2.8e+26)))
		tmp = (a * a) * (a * a);
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -320.0], N[Not[LessEqual[a, 2.8e+26]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -320 or 2.8e26 < a

   1. Initial program 42.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 91.5

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

   8. Applied rewrites 91.5%

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

   10. Applied rewrites 91.4%

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

   if -320 < a < 2.8e26

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -320 \lor \neg \left(a \leq 2.8 \cdot 10^{+26}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 9.5 \cdot 10^{+303}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 9.5e+303) (- (* (* a a) 4.0) 1.0) (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 9.5e+303) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * 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) <= 9.5d+303) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((b * 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) <= 9.5e+303) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 9.5e+303:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 9.5e+303)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 9.5e+303)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 9.5e+303], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.50000000000000015e303

   1. Initial program 82.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 60.8%

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

   if 9.50000000000000015e303 < (*.f64 b b)

   1. Initial program 53.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 98.6%

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

Alternative 11: 50.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 76.6

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

5. Applied rewrites 76.6%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 54.4%

   \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{4} - 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(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))