Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.5s

Alternatives: 7

Speedup: 1.0×

• 60.7% 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. 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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 47.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-284}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-221}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-78}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-33}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 620:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.25e-284)
   -1.0
   (if (<= b 1.65e-221)
     (pow a 4.0)
     (if (<= b 1.1e-78)
       -1.0
       (if (<= b 4.7e-33)
         (pow a 4.0)
         (if (<= b 1.1e-18)
           -1.0
           (if (<= b 620.0) (pow a 4.0) (pow b 4.0))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.25e-284) {
		tmp = -1.0;
	} else if (b <= 1.65e-221) {
		tmp = pow(a, 4.0);
	} else if (b <= 1.1e-78) {
		tmp = -1.0;
	} else if (b <= 4.7e-33) {
		tmp = pow(a, 4.0);
	} else if (b <= 1.1e-18) {
		tmp = -1.0;
	} else if (b <= 620.0) {
		tmp = 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 <= 1.25d-284) then
        tmp = -1.0d0
    else if (b <= 1.65d-221) then
        tmp = a ** 4.0d0
    else if (b <= 1.1d-78) then
        tmp = -1.0d0
    else if (b <= 4.7d-33) then
        tmp = a ** 4.0d0
    else if (b <= 1.1d-18) then
        tmp = -1.0d0
    else if (b <= 620.0d0) then
        tmp = 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 <= 1.25e-284) {
		tmp = -1.0;
	} else if (b <= 1.65e-221) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 1.1e-78) {
		tmp = -1.0;
	} else if (b <= 4.7e-33) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 1.1e-18) {
		tmp = -1.0;
	} else if (b <= 620.0) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.25e-284:
		tmp = -1.0
	elif b <= 1.65e-221:
		tmp = math.pow(a, 4.0)
	elif b <= 1.1e-78:
		tmp = -1.0
	elif b <= 4.7e-33:
		tmp = math.pow(a, 4.0)
	elif b <= 1.1e-18:
		tmp = -1.0
	elif b <= 620.0:
		tmp = math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.25e-284)
		tmp = -1.0;
	elseif (b <= 1.65e-221)
		tmp = a ^ 4.0;
	elseif (b <= 1.1e-78)
		tmp = -1.0;
	elseif (b <= 4.7e-33)
		tmp = a ^ 4.0;
	elseif (b <= 1.1e-18)
		tmp = -1.0;
	elseif (b <= 620.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.25e-284)
		tmp = -1.0;
	elseif (b <= 1.65e-221)
		tmp = a ^ 4.0;
	elseif (b <= 1.1e-78)
		tmp = -1.0;
	elseif (b <= 4.7e-33)
		tmp = a ^ 4.0;
	elseif (b <= 1.1e-18)
		tmp = -1.0;
	elseif (b <= 620.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.25e-284], -1.0, If[LessEqual[b, 1.65e-221], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 1.1e-78], -1.0, If[LessEqual[b, 4.7e-33], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 1.1e-18], -1.0, If[LessEqual[b, 620.0], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.24999999999999993e-284 or 1.64999999999999995e-221 < b < 1.0999999999999999e-78 or 4.7000000000000002e-33 < b < 1.0999999999999999e-18

   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%

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

      6. sqr-pow 99.9%

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

      7. associate-*l* 99.9%

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

      8. fma-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 37.5%

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

   6. Taylor expanded in b around 0 37.7%

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

   if 1.24999999999999993e-284 < b < 1.64999999999999995e-221 or 1.0999999999999999e-78 < b < 4.7000000000000002e-33 or 1.0999999999999999e-18 < b < 620

   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. 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. Add Preprocessing

   5. Taylor expanded in a around inf 64.5%

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

   if 620 < b

   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. 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. Add Preprocessing

   5. Taylor expanded in b around inf 95.0%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-284}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-221}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-78}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-33}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 620:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 82.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1250:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1250.0) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1250.0) {
		tmp = pow(a, 4.0) + -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 <= 1250.0d0) then
        tmp = (a ** 4.0d0) + (-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 <= 1250.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1250.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1250.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1250.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1250.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1250

   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. 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. Add Preprocessing

   5. Taylor expanded in b around 0 81.3%

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

   if 1250 < b

   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. 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. Add Preprocessing

   5. Taylor expanded in b around inf 95.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1250:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00062:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.00062) (+ (pow a 4.0) -1.0) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00062) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = 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 <= 0.00062d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.00062) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.00062:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00062)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.00062)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00062], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.2e-4

   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. 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. Add Preprocessing

   5. Taylor expanded in b around 0 81.7%

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

   if 6.2e-4 < b

   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. 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. Add Preprocessing

   5. Taylor expanded in a around 0 95.8%

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

      1. add-sqr-sqrt 95.7%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2} + {b}^{4}} \cdot \sqrt{4 \cdot {b}^{2} + {b}^{4}}} - 1 \]

      2. pow2 95.7%

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

      3. +-commutative 95.7%

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

      4. metadata-eval 95.7%

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

      5. pow-prod-up 95.7%

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

      6. add-sqr-sqrt 95.7%

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

      7. hypot-def 95.7%

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

      8. *-commutative 95.7%

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

      9. sqrt-prod 95.7%

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

      10. unpow2 95.7%

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

      11. sqrt-prod 95.7%

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

      12. add-sqr-sqrt 95.7%

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

      13. metadata-eval 95.7%

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

   7. Applied egg-rr 95.7%

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

   8. Taylor expanded in b around inf 93.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.00062:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.0) {
		tmp = -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 <= 1.0d0) then
        tmp = -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 <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if a < 1

   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%

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

      6. sqr-pow 99.9%

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

      7. associate-*l* 99.9%

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

      8. fma-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 62.0%

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

   6. Taylor expanded in b around 0 37.5%

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

   if 1 < 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. sqr-pow 100.0%

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

   3. 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. Add Preprocessing

   5. Taylor expanded in a around inf 89.4%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 25.5% 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%

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

   6. sqr-pow 99.9%

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

   7. associate-*l* 99.9%

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

   8. fma-def 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in a around 0 53.1%

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

6. Taylor expanded in b around 0 28.0%

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

7. Final simplification 28.0%

   \[\leadsto -1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))