Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 3.9s

Alternatives: 7

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 100.0%

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

   1. associate--l+ 100.0%

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

   2. unpow2 100.0%

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

   3. unpow1 100.0%

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

   4. sqr-pow 100.0%

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

   5. associate-*r* 100.0%

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

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, 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]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 92.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+154}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+154)
   (+ (pow a 4.0) -1.0)
   (+ (* (* b b) (+ 4.0 (* b b))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+154) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = ((b * b) * (4.0 + (b * b))) + -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) <= 1d+154) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = ((b * b) * (4.0d0 + (b * b))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+154) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+154:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+154)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+154)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+154], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.00000000000000004e154

   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. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. 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 b around 0 96.0%

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

   if 1.00000000000000004e154 < (*.f64 b b)

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

      2. sqrt-pow2 100.0%

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

      3. hypot-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. log-pow 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. sqr-pow 100.0%

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

      3. metadata-eval 100.0%

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

      4. pow2 100.0%

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

      5. metadata-eval 100.0%

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

      6. pow2 100.0%

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

      7. distribute-rgt-out 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+154}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \end{array} \]

Alternative 4: 82.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 3.3e+35) (+ (* (* b b) (+ 4.0 (* b b))) -1.0) (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 3.3e+35) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} 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 <= 3.3d+35) then
        tmp = ((b * b) * (4.0d0 + (b * b))) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 3.3e+35) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 3.3e+35:
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 3.3e+35)
		tmp = Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 3.3e+35)
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 3.3e+35], N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.3000000000000002e35

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

      2. sqrt-pow2 100.0%

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

      3. hypot-udef 100.0%

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

      4. add-exp-log 99.5%

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

      5. log-pow 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in a around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. sqr-pow 82.3%

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

      3. metadata-eval 82.3%

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

      4. pow2 82.3%

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

      5. metadata-eval 82.3%

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

      6. pow2 82.3%

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

      7. distribute-rgt-out 82.3%

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

   6. Applied egg-rr 82.3%

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

   if 3.3000000000000002e35 < a

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow2 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. associate-*r* 100.0%

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

   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 inf 95.0%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5: 70.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. metadata-eval 100.0%

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

   2. sqrt-pow2 100.0%

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

   3. hypot-udef 100.0%

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

   4. add-exp-log 99.4%

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

   5. log-pow 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in a around 0 72.4%

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

   1. +-commutative 72.4%

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

   2. sqr-pow 72.4%

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

   3. metadata-eval 72.4%

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

   4. pow2 72.4%

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

   5. metadata-eval 72.4%

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

   6. pow2 72.4%

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

   7. distribute-rgt-out 72.4%

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

6. Applied egg-rr 72.4%

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

7. Final simplification 72.4%

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

Alternative 6: 52.1% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. metadata-eval 100.0%

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

   2. sqrt-pow2 100.0%

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

   3. hypot-udef 100.0%

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

   4. add-exp-log 99.4%

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

   5. log-pow 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in a around 0 72.4%

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

5. Taylor expanded in b around 0 57.2%

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

   1. unpow2 57.2%

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

7. Simplified 57.2%

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

8. Final simplification 57.2%

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

Alternative 7: 25.1% 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 100.0%

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

   1. associate--l+ 100.0%

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

   2. unpow2 100.0%

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

   3. unpow1 100.0%

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

   4. sqr-pow 100.0%

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

   5. associate-*r* 100.0%

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

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 b around 0 73.9%

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

5. Taylor expanded in a around 0 29.7%

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

6. Final simplification 29.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023250 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))