Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 4.2s

Alternatives: 6

Speedup: 1.0×

• 61.3% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + ((b * b) * 4.0)) + -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[(N[(b * b), $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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 2: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-266}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-204}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-161}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-85}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 16000:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.4e-266)
   (pow b 4.0)
   (if (<= a 3.8e-204)
     -1.0
     (if (<= a 5.2e-161)
       (pow b 4.0)
       (if (<= a 9.8e-85) -1.0 (if (<= a 16000.0) (pow b 4.0) (pow a 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.4e-266) {
		tmp = pow(b, 4.0);
	} else if (a <= 3.8e-204) {
		tmp = -1.0;
	} else if (a <= 5.2e-161) {
		tmp = pow(b, 4.0);
	} else if (a <= 9.8e-85) {
		tmp = -1.0;
	} else if (a <= 16000.0) {
		tmp = pow(b, 4.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 1.4d-266) then
        tmp = b ** 4.0d0
    else if (a <= 3.8d-204) then
        tmp = -1.0d0
    else if (a <= 5.2d-161) then
        tmp = b ** 4.0d0
    else if (a <= 9.8d-85) then
        tmp = -1.0d0
    else if (a <= 16000.0d0) then
        tmp = b ** 4.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.4e-266) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 3.8e-204) {
		tmp = -1.0;
	} else if (a <= 5.2e-161) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 9.8e-85) {
		tmp = -1.0;
	} else if (a <= 16000.0) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.4e-266:
		tmp = math.pow(b, 4.0)
	elif a <= 3.8e-204:
		tmp = -1.0
	elif a <= 5.2e-161:
		tmp = math.pow(b, 4.0)
	elif a <= 9.8e-85:
		tmp = -1.0
	elif a <= 16000.0:
		tmp = math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.4e-266)
		tmp = b ^ 4.0;
	elseif (a <= 3.8e-204)
		tmp = -1.0;
	elseif (a <= 5.2e-161)
		tmp = b ^ 4.0;
	elseif (a <= 9.8e-85)
		tmp = -1.0;
	elseif (a <= 16000.0)
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.4e-266)
		tmp = b ^ 4.0;
	elseif (a <= 3.8e-204)
		tmp = -1.0;
	elseif (a <= 5.2e-161)
		tmp = b ^ 4.0;
	elseif (a <= 9.8e-85)
		tmp = -1.0;
	elseif (a <= 16000.0)
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.4e-266], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 3.8e-204], -1.0, If[LessEqual[a, 5.2e-161], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 9.8e-85], -1.0, If[LessEqual[a, 16000.0], N[Power[b, 4.0], $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.4e-266 or 3.79999999999999983e-204 < a < 5.19999999999999991e-161 or 9.80000000000000029e-85 < a < 16000

   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. Step-by-step derivation

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\right)\right)} \]

      2. add-exp-log 99.1%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}} - 1\right)\right) \]

      3. expm1-define 99.1%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)\right)}\right)\right) \]

      4. log1p-expm1-u 99.1%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}\right) \]

      5. add-sqr-sqrt 99.1%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}\right) \]

      6. pow2 99.1%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left({\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}^{2}\right)}\right) \]

   4. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(\mathsf{hypot}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}, b \cdot 2\right)\right)}^{2}\right)\right)} \]

   5. Taylor expanded in b around inf 45.1%

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

   if 1.4e-266 < a < 3.79999999999999983e-204 or 5.19999999999999991e-161 < a < 9.80000000000000029e-85

   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. Add Preprocessing

   3. Taylor expanded in b around 0 56.8%

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

   4. Taylor expanded in a around 0 56.8%

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

   if 16000 < a

   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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 98.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\right)\right)} \]

      2. add-exp-log 98.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}} - 1\right)\right) \]

      3. expm1-define 98.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)\right)}\right)\right) \]

      4. log1p-expm1-u 98.7%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}\right) \]

      5. add-sqr-sqrt 98.7%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}\right) \]

      6. pow2 98.7%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left({\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}^{2}\right)}\right) \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(\mathsf{hypot}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}, b \cdot 2\right)\right)}^{2}\right)\right)} \]

   5. Taylor expanded in a around inf 92.5%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-266}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-204}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-161}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-85}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 16000:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+34} \lor \neg \left(b \leq 4.1 \cdot 10^{+64}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8000.0)
   (+ (pow a 4.0) -1.0)
   (if (or (<= b 2.05e+34) (not (<= b 4.1e+64))) (pow b 4.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else if ((b <= 2.05e+34) || !(b <= 4.1e+64)) {
		tmp = pow(b, 4.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 8000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else if ((b <= 2.05d+34) .or. (.not. (b <= 4.1d+64))) then
        tmp = b ** 4.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else if ((b <= 2.05e+34) || !(b <= 4.1e+64)) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8000.0:
		tmp = math.pow(a, 4.0) + -1.0
	elif (b <= 2.05e+34) or not (b <= 4.1e+64):
		tmp = math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	elseif ((b <= 2.05e+34) || !(b <= 4.1e+64))
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8000.0)
		tmp = (a ^ 4.0) + -1.0;
	elseif ((b <= 2.05e+34) || ~((b <= 4.1e+64)))
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], If[Or[LessEqual[b, 2.05e+34], N[Not[LessEqual[b, 4.1e+64]], $MachinePrecision]], N[Power[b, 4.0], $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 8e3

   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. Taylor expanded in b around 0 81.1%

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

   if 8e3 < b < 2.0499999999999999e34 or 4.09999999999999978e64 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\right)\right)} \]

      2. add-exp-log 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}} - 1\right)\right) \]

      3. expm1-define 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)\right)}\right)\right) \]

      4. log1p-expm1-u 99.2%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}\right) \]

      5. add-sqr-sqrt 99.2%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}\right) \]

      6. pow2 99.2%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left({\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}^{2}\right)}\right) \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(\mathsf{hypot}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}, b \cdot 2\right)\right)}^{2}\right)\right)} \]

   5. Taylor expanded in b around inf 99.3%

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

   if 2.0499999999999999e34 < b < 4.09999999999999978e64

   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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\right)\right)} \]

      2. add-exp-log 98.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}} - 1\right)\right) \]

      3. expm1-define 98.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)\right)}\right)\right) \]

      4. log1p-expm1-u 98.5%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}\right) \]

      5. add-sqr-sqrt 98.5%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}\right) \]

      6. pow2 98.5%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left({\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}^{2}\right)}\right) \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(\mathsf{hypot}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}, b \cdot 2\right)\right)}^{2}\right)\right)} \]

   5. Taylor expanded in a around inf 75.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+34} \lor \neg \left(b \leq 4.1 \cdot 10^{+64}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 230000:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 230000.0)
   (+ (+ (* (* b b) 4.0) (pow b 4.0)) -1.0)
   (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 230000.0) {
		tmp = (((b * b) * 4.0) + pow(b, 4.0)) + -1.0;
	} else {
		tmp = pow(a, 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 (a <= 230000.0d0) then
        tmp = (((b * b) * 4.0d0) + (b ** 4.0d0)) + (-1.0d0)
    else
        tmp = (a ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 230000.0) {
		tmp = (((b * b) * 4.0) + Math.pow(b, 4.0)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 230000.0:
		tmp = (((b * b) * 4.0) + math.pow(b, 4.0)) + -1.0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 230000.0)
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + (b ^ 4.0)) + -1.0);
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 230000.0)
		tmp = (((b * b) * 4.0) + (b ^ 4.0)) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 230000.0], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.3e5

   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. Taylor expanded in a around 0 76.6%

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

   if 2.3e5 < a

   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. Add Preprocessing

   3. Taylor expanded in b around 0 92.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 230000:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.5% 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. Add Preprocessing

   3. Taylor expanded in b around 0 65.7%

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

   4. Taylor expanded in a around 0 33.0%

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

   if 1 < a

   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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 98.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\right)\right)} \]

      2. add-exp-log 98.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}} - 1\right)\right) \]

      3. expm1-define 98.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)\right)}\right)\right) \]

      4. log1p-expm1-u 98.7%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right)}\right) \]

      5. add-sqr-sqrt 98.7%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}\right) \]

      6. pow2 98.7%

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left({\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right)}^{2}\right)}\right) \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(\mathsf{hypot}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}, b \cdot 2\right)\right)}^{2}\right)\right)} \]

   5. Taylor expanded in a around inf 91.1%

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

4. Final simplification 47.7%

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

Alternative 6: 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. Add Preprocessing

3. Taylor expanded in b around 0 72.2%

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

4. Taylor expanded in a around 0 24.8%

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

5. Final simplification 24.8%

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

Reproduce

herbie shell --seed 2024083 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))