Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.4% → 98.3%

Time: 7.5s

Alternatives: 10

Speedup: 1.2×

• 60.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 75.3%

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

   1. associate--l+ 75.3%

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

3. Simplified 78.5%

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

4. Taylor expanded in b around 0 85.1%

   \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \color{blue}{{a}^{2}} - {a}^{3}, -1\right) \]
5. Step-by-step derivation

   1. unpow2 85.1%

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

6. Simplified 85.1%

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

7. Taylor expanded in a around 0 98.5%

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

8. Final simplification 98.5%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

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) * (a + 3.0))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = -1.0 + t_0;
	} else {
		tmp = -1.0 + ((a * a) * fma(a, a, 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(-1.0 + t_0);
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * fma(a, a, 4.0)));
	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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(-1.0 + t$95$0), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

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

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 7.9%

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

      4. +-commutative 7.9%

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

      5. metadata-eval 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in b around 0 32.1%

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

      1. associate-*r* 32.1%

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

      2. unpow2 32.1%

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

   6. Simplified 32.1%

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

      1. sqr-pow 32.1%

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

      2. metadata-eval 32.1%

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

      3. pow2 32.1%

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

      4. metadata-eval 32.1%

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

      5. pow2 32.1%

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

      6. fma-def 32.1%

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

      7. associate-*l* 32.1%

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

   8. Applied egg-rr 32.1%

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

   9. Taylor expanded in a around 0 92.4%

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

       1. unpow2 92.4%

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

   11. Simplified 92.4%

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

   12. Taylor expanded in a around 0 92.4%

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

       1. +-commutative 92.4%

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

       2. metadata-eval 92.4%

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

       3. pow-plus 92.4%

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

       4. unpow3 92.4%

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

       5. associate-*r* 92.4%

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

       6. unpow2 92.4%

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

       7. distribute-rgt-in 92.4%

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

       8. fma-udef 92.4%

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

   14. Simplified 92.4%

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

4. Final simplification 98.0%

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

Alternative 3: 93.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+40}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+40}:\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.5e+40)
   (pow a 4.0)
   (if (<= a 6.5e+40) (+ -1.0 (pow b 4.0)) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.5e+40) {
		tmp = pow(a, 4.0);
	} else if (a <= 6.5e+40) {
		tmp = -1.0 + pow(b, 4.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-9.5d+40)) then
        tmp = a ** 4.0d0
    else if (a <= 6.5d+40) then
        tmp = (-1.0d0) + (b ** 4.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9.5e+40) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 6.5e+40) {
		tmp = -1.0 + Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9.5e+40:
		tmp = math.pow(a, 4.0)
	elif a <= 6.5e+40:
		tmp = -1.0 + math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.5e+40)
		tmp = a ^ 4.0;
	elseif (a <= 6.5e+40)
		tmp = Float64(-1.0 + (b ^ 4.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9.5e+40)
		tmp = a ^ 4.0;
	elseif (a <= 6.5e+40)
		tmp = -1.0 + (b ^ 4.0);
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.5e+40], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 6.5e+40], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.5000000000000003e40 or 6.5000000000000001e40 < a

   1. Initial program 45.3%

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

      1. associate--l+ 45.3%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in b around 0 64.8%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \color{blue}{{a}^{2}} - {a}^{3}, -1\right) \]
   5. Step-by-step derivation

      1. unpow2 64.8%

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

   6. Simplified 64.8%

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

   7. Taylor expanded in a around inf 95.3%

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

   if -9.5000000000000003e40 < a < 6.5000000000000001e40

   1. Initial program 97.2%

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

      1. sub-neg 97.2%

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

      2. fma-def 97.2%

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

      3. fma-def 97.2%

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

      4. +-commutative 97.2%

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

      5. metadata-eval 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in b around inf 95.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+40}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+40}:\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 4: 93.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+46}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+46) (+ -1.0 (pow a 4.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+46) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+46) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+46) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+46:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+46)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+46)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+46], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e46

   1. Initial program 83.8%

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

      1. sub-neg 83.8%

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

      2. fma-def 83.8%

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

      3. fma-def 83.8%

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

      4. +-commutative 83.8%

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

      5. metadata-eval 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in a around inf 94.7%

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

   if 2e46 < (*.f64 b b)

   1. Initial program 66.6%

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

      1. associate--l+ 66.6%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in b around 0 85.7%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \color{blue}{{a}^{2}} - {a}^{3}, -1\right) \]
   5. Step-by-step derivation

      1. unpow2 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in b around inf 92.5%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+46}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 74.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.005:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+37}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.005)
   (pow a 4.0)
   (if (<= a 1.75e+37) (+ -1.0 (* 4.0 (* a (* b b)))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.005) {
		tmp = pow(a, 4.0);
	} else if (a <= 1.75e+37) {
		tmp = -1.0 + (4.0 * (a * (b * b)));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.005d0)) then
        tmp = a ** 4.0d0
    else if (a <= 1.75d+37) then
        tmp = (-1.0d0) + (4.0d0 * (a * (b * b)))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.005) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 1.75e+37) {
		tmp = -1.0 + (4.0 * (a * (b * b)));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.005:
		tmp = math.pow(a, 4.0)
	elif a <= 1.75e+37:
		tmp = -1.0 + (4.0 * (a * (b * b)))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.005)
		tmp = a ^ 4.0;
	elseif (a <= 1.75e+37)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * Float64(b * b))));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.005)
		tmp = a ^ 4.0;
	elseif (a <= 1.75e+37)
		tmp = -1.0 + (4.0 * (a * (b * b)));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.005], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 1.75e+37], N[(-1.0 + N[(4.0 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0050000000000000001 or 1.75e37 < a

   1. Initial program 47.0%

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

      1. associate--l+ 47.0%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in b around 0 68.1%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \color{blue}{{a}^{2}} - {a}^{3}, -1\right) \]
   5. Step-by-step derivation

      1. unpow2 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in a around inf 88.0%

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

   if -0.0050000000000000001 < a < 1.75e37

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. fma-def 99.9%

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

      3. fma-def 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 84.0%

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

      1. associate-+r+ 84.0%

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

      2. associate-*r* 84.0%

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

      3. distribute-rgt-out 97.1%

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

      4. metadata-eval 97.1%

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

      5. distribute-lft-in 97.1%

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

      6. unpow2 97.1%

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

      7. distribute-rgt-in 97.1%

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

      8. metadata-eval 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in a around inf 58.1%

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

      1. unpow2 58.1%

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

   9. Simplified 58.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.005:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+37}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+46}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+46) (+ -1.0 (* 4.0 (* a a))) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+46) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+46) then
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+46) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+46:
		tmp = -1.0 + (4.0 * (a * a))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+46)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * a)));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+46)
		tmp = -1.0 + (4.0 * (a * a));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+46], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e46

   1. Initial program 83.8%

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

      1. sub-neg 83.8%

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

      2. fma-def 83.8%

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

      3. fma-def 83.8%

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

      4. +-commutative 83.8%

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

      5. metadata-eval 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in b around 0 81.9%

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

      1. associate-*r* 81.9%

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

      2. unpow2 81.9%

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

   6. Simplified 81.9%

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

   7. Taylor expanded in a around 0 76.4%

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

      1. unpow2 76.4%

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

   9. Simplified 76.4%

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

   if 2e46 < (*.f64 b b)

   1. Initial program 66.6%

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

      1. associate--l+ 66.6%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in b around 0 85.7%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \color{blue}{{a}^{2}} - {a}^{3}, -1\right) \]
   5. Step-by-step derivation

      1. unpow2 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in b around inf 92.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+46}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 7: 66.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-13) (+ -1.0 (* 4.0 (* a a))) (* 2.0 (* (* a a) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-13) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = 2.0 * ((a * a) * (b * b));
	}
	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) <= 2d-13) then
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    else
        tmp = 2.0d0 * ((a * a) * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-13) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = 2.0 * ((a * a) * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-13:
		tmp = -1.0 + (4.0 * (a * a))
	else:
		tmp = 2.0 * ((a * a) * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-13)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * a)));
	else
		tmp = Float64(2.0 * Float64(Float64(a * a) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-13)
		tmp = -1.0 + (4.0 * (a * a));
	else
		tmp = 2.0 * ((a * a) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-13], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(a * a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000001e-13

   1. Initial program 85.4%

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

      1. sub-neg 85.4%

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

      2. fma-def 85.4%

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

      3. fma-def 85.4%

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

      4. +-commutative 85.4%

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

      5. metadata-eval 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in b around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. unpow2 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in a around 0 78.7%

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

      1. unpow2 78.7%

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

   9. Simplified 78.7%

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

   if 2.0000000000000001e-13 < (*.f64 b b)

   1. Initial program 66.8%

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

      1. associate--l+ 66.8%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in b around 0 84.9%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \color{blue}{{a}^{2}} - {a}^{3}, -1\right) \]
   5. Step-by-step derivation

      1. unpow2 84.9%

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

   6. Simplified 84.9%

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

   7. Taylor expanded in b around inf 86.0%

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

      1. fma-def 86.0%

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

      2. unpow2 86.0%

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

      3. unpow2 86.0%

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

      4. unswap-sqr 96.1%

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

   9. Simplified 96.1%

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

   10. Taylor expanded in a around inf 58.3%

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

       1. unpow2 58.3%

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

       2. *-commutative 58.3%

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

       3. unpow2 58.3%

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

   12. Simplified 58.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 8: 50.8% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.005 \lor \neg \left(a \leq 0.7\right):\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.005) (not (<= a 0.7))) (* 4.0 (* a a)) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.005) || !(a <= 0.7)) {
		tmp = 4.0 * (a * a);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-0.005d0)) .or. (.not. (a <= 0.7d0))) then
        tmp = 4.0d0 * (a * a)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -0.005) || !(a <= 0.7)) {
		tmp = 4.0 * (a * a);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -0.005) or not (a <= 0.7):
		tmp = 4.0 * (a * a)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.005) || !(a <= 0.7))
		tmp = Float64(4.0 * Float64(a * a));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -0.005) || ~((a <= 0.7)))
		tmp = 4.0 * (a * a);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.005], N[Not[LessEqual[a, 0.7]], $MachinePrecision]], N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -0.0050000000000000001 or 0.69999999999999996 < a

   1. Initial program 49.5%

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

      1. sub-neg 49.5%

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

      2. fma-def 49.5%

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

      3. fma-def 53.5%

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

      4. +-commutative 53.5%

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

      5. metadata-eval 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in b around 0 55.2%

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

      1. associate-*r* 55.2%

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

      2. unpow2 55.2%

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

   6. Simplified 55.2%

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

   7. Taylor expanded in a around 0 48.9%

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

      1. unpow2 48.9%

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

   9. Simplified 48.9%

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

   10. Taylor expanded in a around inf 49.0%

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

       1. unpow2 49.0%

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

   12. Simplified 49.0%

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

   if -0.0050000000000000001 < a < 0.69999999999999996

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. fma-def 99.9%

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

      3. fma-def 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 51.3%

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

      1. associate-*r* 51.3%

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

      2. unpow2 51.3%

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

   6. Simplified 51.3%

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

   7. Taylor expanded in a around 0 50.3%

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

      1. unpow2 50.3%

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

   9. Simplified 50.3%

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

   10. Taylor expanded in a around 0 49.8%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.005 \lor \neg \left(a \leq 0.7\right):\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9: 50.9% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. sub-neg 75.3%

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

   2. fma-def 75.3%

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

   3. fma-def 77.2%

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

   4. +-commutative 77.2%

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

   5. metadata-eval 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in b around 0 53.2%

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

   1. associate-*r* 53.2%

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

   2. unpow2 53.2%

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

6. Simplified 53.2%

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

7. Taylor expanded in a around 0 49.6%

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

   1. unpow2 49.6%

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

9. Simplified 49.6%

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

10. Final simplification 49.6%

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

Alternative 10: 24.9% accurate, 128.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 75.3%

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

   1. sub-neg 75.3%

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

   2. fma-def 75.3%

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

   3. fma-def 77.2%

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

   4. +-commutative 77.2%

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

   5. metadata-eval 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in b around 0 53.2%

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

   1. associate-*r* 53.2%

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

   2. unpow2 53.2%

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

6. Simplified 53.2%

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

7. Taylor expanded in a around 0 49.6%

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

   1. unpow2 49.6%

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

9. Simplified 49.6%

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

10. Taylor expanded in a around 0 25.8%

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

11. Final simplification 25.8%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))