Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

Alternatives: 6

Speedup: 1.0×

• 60.6% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[(N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision] * N[(N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision] * N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

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. unpow2 99.8%

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

   2. add-sqr-sqrt 99.8%

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

   3. associate-*l* 99.9%

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

   4. fma-define 99.9%

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

   5. hypot-define 99.9%

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

   6. hypot-define 99.9%

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

   7. add-sqr-sqrt 99.9%

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

   8. pow2 99.9%

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

   9. hypot-define 99.9%

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

   10. *-commutative 99.9%

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

   11. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

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 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. Final simplification 99.8%

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

Alternative 3: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.395:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 0.395)
   (+ (pow b 4.0) -1.0)
   (+ (+ (* 4.0 (* b b)) (pow a 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.395) {
		tmp = pow(b, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + 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 <= 0.395d0) then
        tmp = (b ** 4.0d0) + (-1.0d0)
    else
        tmp = ((4.0d0 * (b * b)) + (a ** 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 0.395) {
		tmp = Math.pow(b, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + Math.pow(a, 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.395:
		tmp = math.pow(b, 4.0) + -1.0
	else:
		tmp = ((4.0 * (b * b)) + math.pow(a, 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.395)
		tmp = Float64((b ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + (a ^ 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 0.395)
		tmp = (b ^ 4.0) + -1.0;
	else
		tmp = ((4.0 * (b * b)) + (a ^ 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.395], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 0.39500000000000002

   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 a around 0 77.6%

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

   4. Taylor expanded in b around inf 77.1%

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

   if 0.39500000000000002 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf 97.4%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 0.395:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq 13.5:\\ \;\;\;\;\left(t\_0 + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= a 13.5) (+ (+ t_0 (pow b 4.0)) -1.0) (+ (+ t_0 (pow a 4.0)) -1.0))))

C

double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if (a <= 13.5) {
		tmp = (t_0 + pow(b, 4.0)) + -1.0;
	} else {
		tmp = (t_0 + 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) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (b * b)
    if (a <= 13.5d0) then
        tmp = (t_0 + (b ** 4.0d0)) + (-1.0d0)
    else
        tmp = (t_0 + (a ** 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if (a <= 13.5) {
		tmp = (t_0 + Math.pow(b, 4.0)) + -1.0;
	} else {
		tmp = (t_0 + Math.pow(a, 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 4.0 * (b * b)
	tmp = 0
	if a <= 13.5:
		tmp = (t_0 + math.pow(b, 4.0)) + -1.0
	else:
		tmp = (t_0 + math.pow(a, 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(4.0 * Float64(b * b))
	tmp = 0.0
	if (a <= 13.5)
		tmp = Float64(Float64(t_0 + (b ^ 4.0)) + -1.0);
	else
		tmp = Float64(Float64(t_0 + (a ^ 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 4.0 * (b * b);
	tmp = 0.0;
	if (a <= 13.5)
		tmp = (t_0 + (b ^ 4.0)) + -1.0;
	else
		tmp = (t_0 + (a ^ 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 13.5], N[(N[(t$95$0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(t$95$0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 13.5

   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 a around 0 77.6%

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

   if 13.5 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf 97.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 13.5:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 900:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 900.0) (+ (pow a 4.0) -1.0) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 900.0) {
		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 <= 900.0d0) 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 <= 900.0) {
		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 <= 900.0:
		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 <= 900.0)
		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 <= 900.0)
		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, 900.0], 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 < 900

   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. unpow2 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. associate-*l* 99.9%

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

      4. fma-define 99.9%

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

      5. hypot-define 99.9%

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

      6. hypot-define 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. pow2 99.9%

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

      9. hypot-define 99.9%

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

      10. *-commutative 99.9%

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

      11. pow2 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf 80.2%

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

   if 900 < b

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0 90.8%

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

   4. Taylor expanded in b around inf 90.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 900:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. unpow2 99.8%

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

   2. add-sqr-sqrt 99.8%

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

   3. associate-*l* 99.9%

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

   4. fma-define 99.9%

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

   5. hypot-define 99.9%

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

   6. hypot-define 99.9%

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

   7. add-sqr-sqrt 99.9%

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

   8. pow2 99.9%

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

   9. hypot-define 99.9%

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

   10. *-commutative 99.9%

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

   11. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around inf 68.4%

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

6. Final simplification 68.4%

   \[\leadsto {a}^{4} + -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))