Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.6s

Alternatives: 8

Speedup: 1.0×

• 61.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. sqr-pow 99.9%

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

   3. sqr-pow 99.9%

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

   4. unpow2 99.9%

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

   5. unpow1 99.9%

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

   6. sqr-pow 99.9%

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

   7. associate-*r* 99.9%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. fma-def 99.9%

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

   3. sqr-neg 99.9%

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

   4. fma-def 99.9%

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

   5. sqr-neg 99.9%

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

   6. fma-def 99.9%

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

   7. sqr-neg 99.9%

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

   8. sqr-neg 99.9%

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

   9. *-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 50000:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + -1\right) + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(a \cdot \left(b \cdot \left(a \cdot b\right)\right)\right) + {b}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 50000.0)
   (+ (+ (* 4.0 (* b b)) -1.0) (pow a 4.0))
   (+ (+ (* 2.0 (* a (* b (* a b)))) (pow b 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 50000.0) {
		tmp = ((4.0 * (b * b)) + -1.0) + pow(a, 4.0);
	} else {
		tmp = ((2.0 * (a * (b * (a * b)))) + pow(b, 4.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 50000.0d0) then
        tmp = ((4.0d0 * (b * b)) + (-1.0d0)) + (a ** 4.0d0)
    else
        tmp = ((2.0d0 * (a * (b * (a * 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) <= 50000.0) {
		tmp = ((4.0 * (b * b)) + -1.0) + Math.pow(a, 4.0);
	} else {
		tmp = ((2.0 * (a * (b * (a * b)))) + Math.pow(b, 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 50000.0:
		tmp = ((4.0 * (b * b)) + -1.0) + math.pow(a, 4.0)
	else:
		tmp = ((2.0 * (a * (b * (a * b)))) + math.pow(b, 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 50000.0)
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + -1.0) + (a ^ 4.0));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(a * Float64(b * Float64(a * b)))) + (b ^ 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 50000.0)
		tmp = ((4.0 * (b * b)) + -1.0) + (a ^ 4.0);
	else
		tmp = ((2.0 * (a * (b * (a * b)))) + (b ^ 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 50000.0], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(a * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e4

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

      3. sqr-neg 99.9%

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

      4. fma-def 99.9%

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

      5. sqr-neg 99.9%

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

      6. fma-def 99.9%

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

      7. sqr-neg 99.9%

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

      8. sqr-neg 99.9%

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

      9. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. fma-udef 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in b around 0 98.3%

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

   if 5e4 < (*.f64 b b)

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf 96.5%

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

   3. Taylor expanded in a around inf 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-*l* 86.1%

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

   5. Simplified 86.1%

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

   6. Taylor expanded in a around 0 86.9%

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

      1. unpow2 86.9%

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

      2. unpow2 86.9%

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

      3. swap-sqr 96.1%

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

      4. unpow2 96.1%

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

   8. Simplified 96.1%

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

      1. unpow2 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-*r* 96.1%

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

   10. Applied egg-rr 96.1%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 50000:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + -1\right) + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(a \cdot \left(b \cdot \left(a \cdot b\right)\right)\right) + {b}^{4}\right) + -1\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 5: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{+48}:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + -1\right) + {a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 9.2e+48)
   (+ (pow b 4.0) -1.0)
   (+ (+ (* 4.0 (* b b)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 9.2e+48) {
		tmp = pow(b, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + -1.0) + 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 <= 9.2d+48) then
        tmp = (b ** 4.0d0) + (-1.0d0)
    else
        tmp = ((4.0d0 * (b * b)) + (-1.0d0)) + (a ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 9.2e+48) {
		tmp = Math.pow(b, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + -1.0) + Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 9.2e+48:
		tmp = math.pow(b, 4.0) + -1.0
	else:
		tmp = ((4.0 * (b * b)) + -1.0) + math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 9.2e+48)
		tmp = Float64((b ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + -1.0) + (a ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 9.2e+48)
		tmp = (b ^ 4.0) + -1.0;
	else
		tmp = ((4.0 * (b * b)) + -1.0) + (a ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 9.2e+48], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.2000000000000001e48

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf 87.2%

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

   3. Taylor expanded in a around inf 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*l* 78.9%

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

   5. Simplified 78.9%

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

   6. Taylor expanded in a around 0 76.7%

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

   if 9.2000000000000001e48 < a

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

      3. sqr-neg 99.9%

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

      4. fma-def 99.9%

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

      5. sqr-neg 99.9%

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

      6. fma-def 99.9%

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

      7. sqr-neg 99.9%

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

      8. sqr-neg 99.9%

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

      9. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. fma-udef 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in b around 0 100.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{+48}:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + -1\right) + {a}^{4}\\ \end{array} \]

Alternative 6: 47.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.5) -1.0 (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.5) {
		tmp = -1.0;
	} 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 <= 0.5d0) then
        tmp = -1.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.5:
		tmp = -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.5)
		tmp = -1.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.5)
		tmp = -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.5], -1.0, N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.5

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sqr-pow 99.9%

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

      3. sqr-pow 99.9%

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

      4. unpow2 99.9%

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

      5. unpow1 99.9%

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

      6. sqr-pow 99.9%

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

      7. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 64.0%

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

   5. Taylor expanded in b around 0 34.1%

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

   if 0.5 < b

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sqr-pow 99.8%

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

      3. sqr-pow 99.8%

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

      4. unpow2 99.8%

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

      5. unpow1 99.8%

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

      6. sqr-pow 99.8%

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

      7. associate-*r* 99.7%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 88.1%

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

   5. Taylor expanded in b around inf 85.2%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 7: 69.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in b around inf 81.4%

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

3. Taylor expanded in a around inf 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-*l* 75.0%

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

5. Simplified 75.0%

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

6. Taylor expanded in a around 0 69.1%

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

7. Final simplification 69.1%

   \[\leadsto {b}^{4} + -1 \]

Alternative 8: 24.6% accurate, 116.0× 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 99.9%

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

   1. associate--l+ 99.9%

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

   2. sqr-pow 99.9%

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

   3. sqr-pow 99.9%

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

   4. unpow2 99.9%

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

   5. unpow1 99.9%

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

   6. sqr-pow 99.9%

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

   7. associate-*r* 99.9%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 70.4%

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

5. Taylor expanded in b around 0 25.2%

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

6. Final simplification 25.2%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))