Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 7.9s

Alternatives: 12

Speedup: 1.0×

• 61.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. 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. unpow1 99.9%

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

   4. sqr-pow 99.9%

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

   5. associate-*r* 99.9%

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

   6. unpow1 99.9%

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

   7. sqr-pow 99.9%

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

   8. unpow3 99.9%

      \[\leadsto \color{blue}{{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)}^{3}} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   9. pow-plus 100.0%

      \[\leadsto \color{blue}{{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)}^{\left(3 + 1\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   10. metadata-eval 100.0%

       \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   11. unpow1/2 100.0%

       \[\leadsto {\color{blue}{\left(\sqrt{a \cdot a + b \cdot b}\right)}}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   12. hypot-def 100.0%

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

   13. metadata-eval 100.0%

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

   14. associate-*r* 100.0%

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

   15. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.024:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+74}:\\ \;\;\;\;\left({a}^{4} + \left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 0.024)
   (+ (+ (pow b 4.0) (* 4.0 (* b b))) -1.0)
   (if (<= a 6.2e+74)
     (+ (+ (pow a 4.0) (* (* a a) (* (* b b) 2.0))) -1.0)
     (+ (pow a 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.024) {
		tmp = (pow(b, 4.0) + (4.0 * (b * b))) + -1.0;
	} else if (a <= 6.2e+74) {
		tmp = (pow(a, 4.0) + ((a * a) * ((b * b) * 2.0))) + -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 <= 0.024d0) then
        tmp = ((b ** 4.0d0) + (4.0d0 * (b * b))) + (-1.0d0)
    else if (a <= 6.2d+74) then
        tmp = ((a ** 4.0d0) + ((a * a) * ((b * b) * 2.0d0))) + (-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 <= 0.024) {
		tmp = (Math.pow(b, 4.0) + (4.0 * (b * b))) + -1.0;
	} else if (a <= 6.2e+74) {
		tmp = (Math.pow(a, 4.0) + ((a * a) * ((b * b) * 2.0))) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.024:
		tmp = (math.pow(b, 4.0) + (4.0 * (b * b))) + -1.0
	elif a <= 6.2e+74:
		tmp = (math.pow(a, 4.0) + ((a * a) * ((b * b) * 2.0))) + -1.0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.024)
		tmp = Float64(Float64((b ^ 4.0) + Float64(4.0 * Float64(b * b))) + -1.0);
	elseif (a <= 6.2e+74)
		tmp = Float64(Float64((a ^ 4.0) + Float64(Float64(a * a) * Float64(Float64(b * b) * 2.0))) + -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 <= 0.024)
		tmp = ((b ^ 4.0) + (4.0 * (b * b))) + -1.0;
	elseif (a <= 6.2e+74)
		tmp = ((a ^ 4.0) + ((a * a) * ((b * b) * 2.0))) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.024], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 6.2e+74], N[(N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 0.024

   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. Taylor expanded in a around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-*r* 85.9%

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

      5. fma-def 85.9%

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

   4. Simplified 85.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot 4, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. fma-udef 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-*r* 85.9%

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

   6. Applied egg-rr 85.9%

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

   if 0.024 < a < 6.20000000000000043e74

   1. Initial program 99.5%

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

   2. Taylor expanded in b around 0 89.9%

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

      1. unpow2 89.9%

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

      2. unpow2 89.9%

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

   4. Simplified 89.9%

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

   5. Taylor expanded in a around inf 89.9%

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

      1. unpow2 89.9%

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

      2. unpow2 89.9%

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

      3. associate-*r* 89.9%

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

      4. *-commutative 89.9%

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

      5. associate-*r* 89.9%

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

   7. Simplified 89.9%

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

   if 6.20000000000000043e74 < a

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 88.0%

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

      1. unpow2 88.0%

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

      2. unpow2 88.0%

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

   4. Simplified 88.0%

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

   5. Taylor expanded in a around inf 100.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 0.024:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+74}:\\ \;\;\;\;\left({a}^{4} + \left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

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. Final simplification 99.9%

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

Alternative 4: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.48:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 0.48)
   (+ (+ (pow b 4.0) (* 4.0 (* b b))) -1.0)
   (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.48) {
		tmp = (pow(b, 4.0) + (4.0 * (b * b))) + -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 <= 0.48d0) then
        tmp = ((b ** 4.0d0) + (4.0d0 * (b * b))) + (-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 <= 0.48) {
		tmp = (Math.pow(b, 4.0) + (4.0 * (b * b))) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.48:
		tmp = (math.pow(b, 4.0) + (4.0 * (b * b))) + -1.0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.48)
		tmp = Float64(Float64((b ^ 4.0) + Float64(4.0 * Float64(b * b))) + -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 <= 0.48)
		tmp = ((b ^ 4.0) + (4.0 * (b * b))) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.48], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $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 < 0.47999999999999998

   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. Taylor expanded in a around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-*r* 85.9%

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

      5. fma-def 85.9%

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

   4. Simplified 85.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot 4, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. fma-udef 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-*r* 85.9%

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

   6. Applied egg-rr 85.9%

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

   if 0.47999999999999998 < 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. Taylor expanded in b around 0 88.5%

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

      1. unpow2 88.5%

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

      2. unpow2 88.5%

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

   4. Simplified 88.5%

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

   5. Taylor expanded in a around inf 90.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 0.48:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

Alternative 5: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.2) (+ (* b (* b (fma b b 4.0))) -1.0) (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.2) {
		tmp = (b * (b * fma(b, b, 4.0))) + -1.0;
	} else {
		tmp = pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.2)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) + -1.0);
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.2], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $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 < 1.19999999999999996

   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. Taylor expanded in a around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-*r* 85.9%

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

      5. fma-def 85.9%

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

   4. Simplified 85.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot 4, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. fma-udef 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-*r* 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in b around 0 85.9%

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

      1. metadata-eval 85.9%

         \[\leadsto \left({b}^{\color{blue}{\left(3 + 1\right)}} + 4 \cdot {b}^{2}\right) - 1 \]

      2. pow-plus 85.9%

         \[\leadsto \left(\color{blue}{{b}^{3} \cdot b} + 4 \cdot {b}^{2}\right) - 1 \]

      3. unpow3 85.9%

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

      4. associate-*r* 85.8%

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

      5. unpow2 85.8%

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

      6. distribute-rgt-in 85.8%

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

      7. fma-udef 85.8%

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

      8. associate-*l* 85.9%

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

   9. Simplified 85.9%

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

   if 1.19999999999999996 < 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. Taylor expanded in b around 0 88.5%

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

      1. unpow2 88.5%

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

      2. unpow2 88.5%

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

   4. Simplified 88.5%

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

   5. Taylor expanded in a around inf 90.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.2:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

Alternative 6: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6.2:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 6.2)
   (+ (+ (* 4.0 (* b b)) (* (* b b) (* b b))) -1.0)
   (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 6.2) {
		tmp = ((4.0 * (b * b)) + ((b * b) * (b * b))) + -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 <= 6.2d0) then
        tmp = ((4.0d0 * (b * b)) + ((b * b) * (b * b))) + (-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 <= 6.2) {
		tmp = ((4.0 * (b * b)) + ((b * b) * (b * b))) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 6.2:
		tmp = ((4.0 * (b * b)) + ((b * b) * (b * b))) + -1.0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 6.2)
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + Float64(Float64(b * b) * Float64(b * b))) + -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 <= 6.2)
		tmp = ((4.0 * (b * b)) + ((b * b) * (b * b))) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 6.2], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $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 < 6.20000000000000018

   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. Taylor expanded in a around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-*r* 85.9%

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

      5. fma-def 85.9%

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

   4. Simplified 85.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot 4, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. fma-udef 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-*r* 85.9%

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

   6. Applied egg-rr 85.9%

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

      1. metadata-eval 84.6%

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

      2. pow-prod-up 84.5%

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

      3. pow-prod-down 84.5%

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

      4. pow2 84.5%

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

   8. Applied egg-rr 85.8%

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

   if 6.20000000000000018 < 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. Taylor expanded in b around 0 88.5%

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

      1. unpow2 88.5%

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

      2. unpow2 88.5%

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

   4. Simplified 88.5%

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

   5. Taylor expanded in a around inf 90.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.2:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

Alternative 7: 69.3% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return ((4.0 * (b * b)) + ((b * b) * (b * b))) + -1.0

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 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. Taylor expanded in a around 0 70.4%

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

   1. +-commutative 70.4%

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

   2. unpow2 70.4%

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

   3. *-commutative 70.4%

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

   4. associate-*r* 70.4%

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

   5. fma-def 70.4%

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

4. Simplified 70.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot 4, {b}^{4}\right)} - 1 \]
5. Step-by-step derivation

   1. fma-udef 70.4%

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

   2. +-commutative 70.4%

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

   3. associate-*r* 70.4%

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

6. Applied egg-rr 70.4%

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

   1. metadata-eval 69.4%

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

   2. pow-prod-up 69.4%

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

   3. pow-prod-down 69.4%

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

   4. pow2 69.4%

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

8. Applied egg-rr 70.3%

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

9. Final simplification 70.3%

   \[\leadsto \left(4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) + -1 \]

Alternative 8: 69.3% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return ((b * b) * (4.0 + (b * b))) + -1.0

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $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. Taylor expanded in a around 0 70.4%

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

   1. +-commutative 70.4%

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

   2. unpow2 70.4%

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

   3. *-commutative 70.4%

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

   4. associate-*r* 70.4%

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

   5. fma-def 70.4%

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

4. Simplified 70.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot 4, {b}^{4}\right)} - 1 \]
5. Step-by-step derivation

   1. fma-udef 70.4%

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

   2. +-commutative 70.4%

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

   3. associate-*r* 70.4%

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

6. Applied egg-rr 70.4%

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

   1. +-commutative 70.4%

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

   2. metadata-eval 70.4%

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

   3. pow-prod-up 70.3%

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

   4. pow2 70.3%

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

   5. pow2 70.3%

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

   6. distribute-lft-out 70.3%

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

8. Applied egg-rr 70.3%

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

9. Final simplification 70.3%

   \[\leadsto \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1 \]

Alternative 9: 68.8% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return ((b * b) * (b * b)) + -1.0;
}

Python

def code(a, b):
	return ((b * b) * (b * b)) + -1.0

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $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. Taylor expanded in a around 0 70.4%

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

   1. +-commutative 70.4%

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

   2. unpow2 70.4%

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

   3. *-commutative 70.4%

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

   4. associate-*r* 70.4%

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

   5. fma-def 70.4%

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

4. Simplified 70.4%

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

5. Taylor expanded in b around inf 69.4%

   \[\leadsto \color{blue}{{b}^{4}} - 1 \]
6. Step-by-step derivation

   1. metadata-eval 69.4%

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

   2. pow-prod-up 69.4%

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

   3. pow-prod-down 69.4%

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

   4. pow2 69.4%

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

7. Applied egg-rr 69.4%

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

8. Final simplification 69.4%

   \[\leadsto \left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1 \]

Alternative 10: 38.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.48) -1.0 (* b (* b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		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 <= 0.48d0) 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 <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.48:
		tmp = -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.48)
		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 <= 0.48)
		tmp = -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.48], -1.0, N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.47999999999999998

   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. Taylor expanded in b around 0 83.8%

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

      1. unpow2 83.8%

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

      2. unpow2 83.8%

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

   4. Simplified 83.8%

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

   5. Taylor expanded in a around 0 51.0%

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

      1. fma-neg 51.0%

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

      2. unpow2 51.0%

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

      3. metadata-eval 51.0%

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

   7. Simplified 51.0%

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

   8. Taylor expanded in b around 0 33.9%

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

   if 0.47999999999999998 < 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. Taylor expanded in b around 0 73.2%

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

      1. unpow2 73.2%

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

      2. unpow2 73.2%

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

   4. Simplified 73.2%

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

   5. Taylor expanded in a around 0 45.6%

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

      1. fma-neg 45.6%

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

      2. unpow2 45.6%

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

      3. metadata-eval 45.6%

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

   7. Simplified 45.6%

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

   8. Taylor expanded in b around inf 45.7%

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

      1. unpow2 45.7%

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

      2. *-commutative 45.7%

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

      3. associate-*r* 45.7%

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

   10. Simplified 45.7%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 11: 51.4% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0 + (b * (b * 4.0))

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(-1.0 + N[(b * N[(b * 4.0), $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. Taylor expanded in b around 0 81.4%

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

   1. unpow2 81.4%

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

   2. unpow2 81.4%

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

4. Simplified 81.4%

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

5. Taylor expanded in a around 0 49.8%

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

   1. fma-neg 49.8%

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

   2. unpow2 49.8%

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

   3. metadata-eval 49.8%

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

7. Simplified 49.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, -1\right)} \]
8. Step-by-step derivation

   1. fma-udef 49.8%

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

   2. *-commutative 49.8%

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

   3. associate-*l* 49.8%

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

9. Applied egg-rr 49.8%

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

10. Final simplification 49.8%

    \[\leadsto -1 + b \cdot \left(b \cdot 4\right) \]

Alternative 12: 24.9% 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. Taylor expanded in b around 0 81.4%

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

   1. unpow2 81.4%

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

   2. unpow2 81.4%

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

4. Simplified 81.4%

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

5. Taylor expanded in a around 0 49.8%

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

   1. fma-neg 49.8%

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

   2. unpow2 49.8%

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

   3. metadata-eval 49.8%

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

7. Simplified 49.8%

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

8. Taylor expanded in b around 0 26.4%

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

9. Final simplification 26.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))