Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

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: 99.9% accurate, 0.2× speedup

Math

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

FPCore

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

Julia

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

Wolfram

code[a_, b_] := 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 + -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. Add Preprocessing
3. 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. add-sqr-sqrt 99.9%

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

   4. associate-*l* 99.9%

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

   5. fma-define 99.9%

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

   6. hypot-define 99.9%

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

   7. hypot-define 99.9%

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

   8. add-sqr-sqrt 99.9%

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

   9. pow2 99.9%

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

   10. hypot-define 99.9%

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

   11. *-commutative 99.9%

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

   12. fma-neg 99.9%

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

   13. pow2 99.9%

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

   14. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{-281}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-231}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-115}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-47}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.16 \cdot 10^{+19}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.2e-281)
   (pow b 4.0)
   (if (<= a 5.2e-231)
     -1.0
     (if (<= a 9e-115)
       (pow b 4.0)
       (if (<= a 7.5e-47)
         -1.0
         (if (<= a 1.16e+19) (pow b 4.0) (pow a 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.2e-281) {
		tmp = pow(b, 4.0);
	} else if (a <= 5.2e-231) {
		tmp = -1.0;
	} else if (a <= 9e-115) {
		tmp = pow(b, 4.0);
	} else if (a <= 7.5e-47) {
		tmp = -1.0;
	} else if (a <= 1.16e+19) {
		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 <= 2.2d-281) then
        tmp = b ** 4.0d0
    else if (a <= 5.2d-231) then
        tmp = -1.0d0
    else if (a <= 9d-115) then
        tmp = b ** 4.0d0
    else if (a <= 7.5d-47) then
        tmp = -1.0d0
    else if (a <= 1.16d+19) 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 <= 2.2e-281) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 5.2e-231) {
		tmp = -1.0;
	} else if (a <= 9e-115) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 7.5e-47) {
		tmp = -1.0;
	} else if (a <= 1.16e+19) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2.2e-281:
		tmp = math.pow(b, 4.0)
	elif a <= 5.2e-231:
		tmp = -1.0
	elif a <= 9e-115:
		tmp = math.pow(b, 4.0)
	elif a <= 7.5e-47:
		tmp = -1.0
	elif a <= 1.16e+19:
		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 <= 2.2e-281)
		tmp = b ^ 4.0;
	elseif (a <= 5.2e-231)
		tmp = -1.0;
	elseif (a <= 9e-115)
		tmp = b ^ 4.0;
	elseif (a <= 7.5e-47)
		tmp = -1.0;
	elseif (a <= 1.16e+19)
		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 <= 2.2e-281)
		tmp = b ^ 4.0;
	elseif (a <= 5.2e-231)
		tmp = -1.0;
	elseif (a <= 9e-115)
		tmp = b ^ 4.0;
	elseif (a <= 7.5e-47)
		tmp = -1.0;
	elseif (a <= 1.16e+19)
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.2e-281], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 5.2e-231], -1.0, If[LessEqual[a, 9e-115], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 7.5e-47], -1.0, If[LessEqual[a, 1.16e+19], N[Power[b, 4.0], $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.20000000000000004e-281 or 5.20000000000000006e-231 < a < 9.00000000000000046e-115 or 7.49999999999999969e-47 < a < 1.16e19

   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. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate-*l* 99.9%

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

      5. fma-define 99.9%

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

      6. hypot-define 99.9%

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

      7. hypot-define 99.9%

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

      8. add-sqr-sqrt 99.9%

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

      9. pow2 99.9%

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

      10. hypot-define 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. pow2 99.9%

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

      14. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around inf 53.4%

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

   if 2.20000000000000004e-281 < a < 5.20000000000000006e-231 or 9.00000000000000046e-115 < a < 7.49999999999999969e-47

   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 75.5%

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

   4. Taylor expanded in a around 0 75.5%

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

   if 1.16e19 < 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. 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. add-sqr-sqrt 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-define 99.9%

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

      6. hypot-define 99.9%

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

      7. hypot-define 99.9%

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

      8. add-sqr-sqrt 99.9%

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

      9. pow2 99.9%

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

      10. hypot-define 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. pow2 99.9%

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

      14. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf 95.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{-281}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-231}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-115}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-47}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.16 \cdot 10^{+19}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
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: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.45 \cdot 10^{+18}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 3.45e+18)
   (+ (+ (* 4.0 (* b b)) (pow b 4.0)) -1.0)
   (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 3.45e+18) {
		tmp = ((4.0 * (b * b)) + 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 <= 3.45d+18) then
        tmp = ((4.0d0 * (b * b)) + (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 <= 3.45e+18) {
		tmp = ((4.0 * (b * b)) + 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 <= 3.45e+18:
		tmp = ((4.0 * (b * b)) + 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 <= 3.45e+18)
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + (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 <= 3.45e+18)
		tmp = ((4.0 * (b * b)) + (b ^ 4.0)) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 3.45e+18], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $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 < 3.45e18

   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 79.1%

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

   if 3.45e18 < 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 b around 0 95.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.45 \cdot 10^{+18}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 45000000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 45000000000.0) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 45000000000.0) {
		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 <= 45000000000.0d0) 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 <= 45000000000.0) {
		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 <= 45000000000.0:
		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 <= 45000000000.0)
		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 <= 45000000000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 45000000000.0], 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 < 4.5e10

   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 78.5%

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

   if 4.5e10 < 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. Add Preprocessing
   3. 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. add-sqr-sqrt 99.9%

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

      4. associate-*l* 100.0%

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

      5. fma-define 100.0%

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

      6. hypot-define 100.0%

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

      7. hypot-define 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. pow2 100.0%

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

      10. hypot-define 100.0%

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

      11. *-commutative 100.0%

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

      12. fma-neg 100.0%

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

      13. pow2 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in b around inf 93.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 45000000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.5% 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 61.6%

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

   4. Taylor expanded in a around 0 30.2%

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

   if 1 < 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. 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. add-sqr-sqrt 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-define 99.9%

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

      6. hypot-define 99.9%

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

      7. hypot-define 99.9%

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

      8. add-sqr-sqrt 99.9%

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

      9. pow2 99.9%

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

      10. hypot-define 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-neg 99.9%

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

      13. pow2 99.9%

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

      14. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf 90.7%

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

4. Final simplification 47.0%

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

3. Taylor expanded in b around 0 69.7%

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

4. Taylor expanded in a around 0 22.0%

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

5. Final simplification 22.0%

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

Reproduce

herbie shell --seed 2024077 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))