Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.3% → 98.3%

Time: 8.2s

Alternatives: 10

Speedup: 6.2×

• 60.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, a + -4, 4\right) \cdot a\right) \cdot a - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (-
          (+
           (pow (+ (* a a) (* b b)) 2.0)
           (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
          1.0)))
   (if (<= t_0 INFINITY) t_0 (- (* (* (fma a (+ a -4.0) 4.0) a) a) 1.0))))

C

double code(double a, double b) {
	double t_0 = (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = ((fma(a, (a + -4.0), 4.0) * a) * a) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(fma(a, Float64(a + -4.0), 4.0) * a) * a) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   if +inf.0 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 94.0%

      \[\leadsto \left(\mathsf{fma}\left(a, a + -4, 4\right) \cdot a\right) \cdot \color{blue}{a} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, a + -4, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.55e-8)
   (- (* (* (fma a (+ a -4.0) 4.0) a) a) 1.0)
   (- (fma (* b b) 12.0 (pow b 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.55e-8) {
		tmp = ((fma(a, (a + -4.0), 4.0) * a) * a) - 1.0;
	} else {
		tmp = fma((b * b), 12.0, pow(b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.55e-8)
		tmp = Float64(Float64(Float64(fma(a, Float64(a + -4.0), 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(fma(Float64(b * b), 12.0, (b ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.55e-8], N[(N[(N[(N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.55e-8

   1. Initial program 79.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 81.5%

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

   if 2.55e-8 < b

   1. Initial program 63.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot 12} + {b}^{4}\right) - 1 \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, 12, {b}^{4}\right)} - 1 \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 12, {b}^{4}\right) - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 12, {b}^{4}\right) - 1 \]

      5. lower-pow.f64 92.9

         \[\leadsto \mathsf{fma}\left(b \cdot b, 12, \color{blue}{{b}^{4}}\right) - 1 \]

   5. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right)} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, a + -4, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.55e-8)
   (- (* (* (fma a (+ a -4.0) 4.0) a) a) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.55e-8) {
		tmp = ((fma(a, (a + -4.0), 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.55e-8)
		tmp = Float64(Float64(Float64(fma(a, Float64(a + -4.0), 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.55e-8], N[(N[(N[(N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.55e-8

   1. Initial program 79.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 81.5%

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

   if 2.55e-8 < b

   1. Initial program 63.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left(a \cdot {b}^{2}\right) + \left(12 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(12 \cdot {b}^{2} + {b}^{4}\right) + 4 \cdot \left(a \cdot {b}^{2}\right)\right)} - 1 \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({b}^{4} + 12 \cdot {b}^{2}\right)} + 4 \cdot \left(a \cdot {b}^{2}\right)\right) - 1 \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left({b}^{4} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right)} - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right) - 1 \]

      5. pow-sqr N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot {b}^{2}} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right) - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \left(12 \cdot {b}^{2} + \color{blue}{\left(4 \cdot a\right) \cdot {b}^{2}}\right)\right) - 1 \]

      7. distribute-rgt-in N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot \left(12 + 4 \cdot a\right)}\right) - 1 \]

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(b \cdot \left(12 + {b}^{2}\right)\right) \cdot b - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 92.9%

      \[\leadsto \left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(\left(a - 4\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.55e-8)
   (- (* (* (* (- a 4.0) a) a) a) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.55e-8) {
		tmp = ((((a - 4.0) * a) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.55e-8)
		tmp = Float64(Float64(Float64(Float64(Float64(a - 4.0) * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.55e-8], N[(N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.55e-8

   1. Initial program 79.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 38.4%

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

   10. Taylor expanded in a around inf

       \[\leadsto \left({a}^{3} \cdot \left(1 - 4 \cdot \frac{1}{a}\right)\right) \cdot a - 1 \]

   12. Applied rewrites 81.1%

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

   if 2.55e-8 < b

   1. Initial program 63.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left(a \cdot {b}^{2}\right) + \left(12 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(12 \cdot {b}^{2} + {b}^{4}\right) + 4 \cdot \left(a \cdot {b}^{2}\right)\right)} - 1 \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({b}^{4} + 12 \cdot {b}^{2}\right)} + 4 \cdot \left(a \cdot {b}^{2}\right)\right) - 1 \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left({b}^{4} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right)} - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right) - 1 \]

      5. pow-sqr N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot {b}^{2}} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right) - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \left(12 \cdot {b}^{2} + \color{blue}{\left(4 \cdot a\right) \cdot {b}^{2}}\right)\right) - 1 \]

      7. distribute-rgt-in N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot \left(12 + 4 \cdot a\right)}\right) - 1 \]

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(b \cdot \left(12 + {b}^{2}\right)\right) \cdot b - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 92.9%

      \[\leadsto \left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.55e-8)
   (- (* (* (fma a a 4.0) a) a) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.55e-8) {
		tmp = ((fma(a, a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.55e-8)
		tmp = Float64(Float64(Float64(fma(a, a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.55e-8], N[(N[(N[(N[(a * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.55e-8

   1. Initial program 79.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 38.4%

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

   10. Taylor expanded in a around -inf

       \[\leadsto \left(\mathsf{fma}\left(a, -1 \cdot \left(a \cdot {\left(\sqrt{-1}\right)}^{2}\right), 4\right) \cdot a\right) \cdot a - 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 80.9%

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

   if 2.55e-8 < b

   1. Initial program 63.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left(a \cdot {b}^{2}\right) + \left(12 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(12 \cdot {b}^{2} + {b}^{4}\right) + 4 \cdot \left(a \cdot {b}^{2}\right)\right)} - 1 \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({b}^{4} + 12 \cdot {b}^{2}\right)} + 4 \cdot \left(a \cdot {b}^{2}\right)\right) - 1 \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left({b}^{4} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right)} - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right) - 1 \]

      5. pow-sqr N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot {b}^{2}} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right) - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \left(12 \cdot {b}^{2} + \color{blue}{\left(4 \cdot a\right) \cdot {b}^{2}}\right)\right) - 1 \]

      7. distribute-rgt-in N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot \left(12 + 4 \cdot a\right)}\right) - 1 \]

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(b \cdot \left(12 + {b}^{2}\right)\right) \cdot b - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 92.9%

      \[\leadsto \left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 80.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55 \cdot 10^{-8}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.55e-8)
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.55e-8) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.55e-8)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.55e-8], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.55e-8

   1. Initial program 79.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 80.8%

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

   if 2.55e-8 < b

   1. Initial program 63.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left(a \cdot {b}^{2}\right) + \left(12 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(12 \cdot {b}^{2} + {b}^{4}\right) + 4 \cdot \left(a \cdot {b}^{2}\right)\right)} - 1 \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({b}^{4} + 12 \cdot {b}^{2}\right)} + 4 \cdot \left(a \cdot {b}^{2}\right)\right) - 1 \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left({b}^{4} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right)} - 1 \]

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right) - 1 \]

      5. pow-sqr N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot {b}^{2}} + \left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right)\right) - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \left(12 \cdot {b}^{2} + \color{blue}{\left(4 \cdot a\right) \cdot {b}^{2}}\right)\right) - 1 \]

      7. distribute-rgt-in N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot \left(12 + 4 \cdot a\right)}\right) - 1 \]

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(b \cdot \left(12 + {b}^{2}\right)\right) \cdot b - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 92.9%

      \[\leadsto \left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e+142) (- (* (* a a) (* a a)) 1.0) (- (* (* b b) 12.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+142) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b * b) * 12.0) - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.5d+142) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = ((b * b) * 12.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+142) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b * b) * 12.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.5e+142:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = ((b * b) * 12.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e+142)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 12.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.5e+142)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = ((b * b) * 12.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e+142], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.49999999999999987e142

   1. Initial program 78.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 74.7%

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

   if 1.49999999999999987e142 < b

   1. Initial program 52.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot 12} + {b}^{4}\right) - 1 \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, 12, {b}^{4}\right)} - 1 \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 12, {b}^{4}\right) - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 12, {b}^{4}\right) - 1 \]

      5. lower-pow.f64 100.0

         \[\leadsto \mathsf{fma}\left(b \cdot b, 12, \color{blue}{{b}^{4}}\right) - 1 \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right)} - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 97.6%

      \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e+142) (- (* (* a a) 4.0) 1.0) (- (* (* b b) 12.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+142) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 12.0) - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.5d+142) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((b * b) * 12.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+142) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 12.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.5e+142:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * b) * 12.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e+142)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 12.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.5e+142)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * b) * 12.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e+142], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.49999999999999987e142

   1. Initial program 78.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 56.8%

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

   if 1.49999999999999987e142 < b

   1. Initial program 52.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot 12} + {b}^{4}\right) - 1 \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, 12, {b}^{4}\right)} - 1 \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 12, {b}^{4}\right) - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 12, {b}^{4}\right) - 1 \]

      5. lower-pow.f64 100.0

         \[\leadsto \mathsf{fma}\left(b \cdot b, 12, \color{blue}{{b}^{4}}\right) - 1 \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right)} - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 97.6%

      \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.0% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \left(b \cdot b\right) \cdot 12 - 1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* b b) 12.0) 1.0))

C

double code(double a, double b) {
	return ((b * b) * 12.0) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((b * b) * 12.0d0) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return ((b * b) * 12.0) - 1.0;
}

Python

def code(a, b):
	return ((b * b) * 12.0) - 1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(b * b) * 12.0) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((b * b) * 12.0) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 75.3%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{{b}^{2} \cdot 12} + {b}^{4}\right) - 1 \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, 12, {b}^{4}\right)} - 1 \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 12, {b}^{4}\right) - 1 \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 12, {b}^{4}\right) - 1 \]

   5. lower-pow.f64 67.7

      \[\leadsto \mathsf{fma}\left(b \cdot b, 12, \color{blue}{{b}^{4}}\right) - 1 \]

5. Applied rewrites 67.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right)} - 1 \]

6. Taylor expanded in b around 0

   \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
7. Step-by-step derivation

8. Applied rewrites 50.6%

   \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} - 1 \]
9. Add Preprocessing

Alternative 10: 25.0% accurate, 155.0× 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 75.3%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in b around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. mul-1-neg N/A

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

   8. distribute-lft-in N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. distribute-lft-neg-in N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower--.f64 N/A

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

   19. lower-pow.f64 57.6

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

5. Applied rewrites 57.6%

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

6. Taylor expanded in a around 0

   \[\leadsto -1 \]
7. Step-by-step derivation

8. Applied rewrites 23.7%

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

Reproduce

herbie shell --seed 2024344 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))