Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 3.6s

Alternatives: 4

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[(b * N[(b * 4.0), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $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. 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-neg 99.9%

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

   3. sqr-neg 99.9%

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

   4. associate--l+ 99.9%

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

   5. sub-neg 99.9%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(-1.0 + 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]), $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%

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

Alternative 3: 69.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(-1.0 + N[Power[b, 4.0], $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-neg 99.9%

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

   3. sqr-neg 99.9%

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

   4. associate--l+ 99.9%

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

   5. sub-neg 99.9%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 71.8%

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

5. Taylor expanded in b around inf 71.1%

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

6. Final simplification 71.1%

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

Alternative 4: 51.6% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(-1.0 + N[(4.0 * N[(b * b), $MachinePrecision]), $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-neg 99.9%

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

   3. sqr-neg 99.9%

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

   4. associate--l+ 99.9%

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

   5. sub-neg 99.9%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 71.8%

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

5. Taylor expanded in b around 0 51.5%

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

   1. unpow2 51.5%

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

7. Applied egg-rr 51.5%

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

8. Final simplification 51.5%

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

Reproduce

herbie shell --seed 2023336 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))