Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 6.2s

Alternatives: 6

Speedup: 1.0×

• 62.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[(b * N[(b * 4.0), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $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. 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-neg 99.9%

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

   3. sqr-neg 99.9%

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

   4. associate--l+ 99.9%

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

   5. sub-neg 99.9%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b)));
end

Wolfram

code[a_, b_] := N[(-1.0 + 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]), $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. Final simplification 99.9%

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

Alternative 3: 47.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(4 + {b}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.6e-5) -1.0 (* b (* b (+ 4.0 (pow b 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.6e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * (4.0 + pow(b, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 6.6d-5) then
        tmp = -1.0d0
    else
        tmp = b * (b * (4.0d0 + (b ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.6e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * (4.0 + Math.pow(b, 2.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.6e-5:
		tmp = -1.0
	else:
		tmp = b * (b * (4.0 + math.pow(b, 2.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.6e-5)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * Float64(4.0 + (b ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.6e-5)
		tmp = -1.0;
	else
		tmp = b * (b * (4.0 + (b ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.6e-5], -1.0, N[(b * N[(b * N[(4.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.6000000000000005e-5

   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-neg 99.9%

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

      3. sqr-neg 99.9%

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

      4. associate--l+ 99.9%

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

      5. sub-neg 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 61.8%

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

   5. Taylor expanded in b around 0 26.8%

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

   if 6.6000000000000005e-5 < 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-neg 99.9%

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

      3. sqr-neg 99.9%

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

      4. associate--l+ 99.9%

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

      5. sub-neg 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 90.0%

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

   5. Taylor expanded in b around inf 90.1%

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

      1. +-commutative 90.1%

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

      2. *-commutative 90.1%

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

      3. unpow2 90.1%

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

      4. associate-*r* 90.1%

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

      5. metadata-eval 90.1%

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

      6. pow-prod-up 90.1%

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

      7. unpow2 90.1%

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

      8. associate-*r* 90.1%

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

      9. unpow2 90.1%

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

      10. *-commutative 90.1%

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

      11. distribute-lft-out 90.1%

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

      12. pow3 90.1%

          \[\leadsto b \cdot \left(\color{blue}{{b}^{3}} + b \cdot 4\right) \]

      13. *-commutative 90.1%

          \[\leadsto b \cdot \left({b}^{3} + \color{blue}{4 \cdot b}\right) \]

   7. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{b \cdot \left({b}^{3} + 4 \cdot b\right)} \]
   8. Step-by-step derivation

      1. unpow3 90.1%

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

      2. distribute-rgt-out 90.1%

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

      3. +-commutative 90.1%

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

      4. unpow2 90.1%

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

   9. Applied egg-rr 90.1%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(4 + {b}^{2}\right)\right)\\ \end{array} \]

Alternative 4: 46.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 6.6e-5) -1.0 (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.6e-5) {
		tmp = -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 <= 6.6d-5) then
        tmp = -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 <= 6.6e-5) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.6e-5:
		tmp = -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.6e-5)
		tmp = -1.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.6e-5)
		tmp = -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.6e-5], -1.0, N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.6000000000000005e-5

   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-neg 99.9%

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

      3. sqr-neg 99.9%

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

      4. associate--l+ 99.9%

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

      5. sub-neg 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 61.8%

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

   5. Taylor expanded in b around 0 26.8%

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

   if 6.6000000000000005e-5 < 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-neg 99.9%

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

      3. sqr-neg 99.9%

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

      4. associate--l+ 99.9%

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

      5. sub-neg 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 90.0%

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

   5. Taylor expanded in b around inf 90.1%

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

   6. Taylor expanded in b around inf 89.7%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 38.3% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 6.6e-5) -1.0 (* b (* b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.6e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (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 <= 6.6d-5) then
        tmp = -1.0d0
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.6e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.6e-5:
		tmp = -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.6e-5)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.6e-5)
		tmp = -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.6e-5], -1.0, N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.6000000000000005e-5

   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-neg 99.9%

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

      3. sqr-neg 99.9%

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

      4. associate--l+ 99.9%

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

      5. sub-neg 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 61.8%

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

   5. Taylor expanded in b around 0 26.8%

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

   if 6.6000000000000005e-5 < 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-neg 99.9%

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

      3. sqr-neg 99.9%

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

      4. associate--l+ 99.9%

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

      5. sub-neg 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 90.0%

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

   5. Taylor expanded in b around inf 90.1%

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

      1. +-commutative 90.1%

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

      2. *-commutative 90.1%

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

      3. unpow2 90.1%

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

      4. associate-*r* 90.1%

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

      5. metadata-eval 90.1%

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

      6. pow-prod-up 90.1%

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

      7. unpow2 90.1%

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

      8. associate-*r* 90.1%

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

      9. unpow2 90.1%

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

      10. *-commutative 90.1%

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

      11. distribute-lft-out 90.1%

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

      12. pow3 90.1%

          \[\leadsto b \cdot \left(\color{blue}{{b}^{3}} + b \cdot 4\right) \]

      13. *-commutative 90.1%

          \[\leadsto b \cdot \left({b}^{3} + \color{blue}{4 \cdot b}\right) \]

   7. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{b \cdot \left({b}^{3} + 4 \cdot b\right)} \]

   8. Taylor expanded in b around 0 51.8%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 6: 24.4% 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-neg 99.9%

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

   3. sqr-neg 99.9%

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

   4. associate--l+ 99.9%

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

   5. sub-neg 99.9%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 69.3%

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

5. Taylor expanded in b around 0 19.8%

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

6. Final simplification 19.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023326 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))