Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.9% → 98.2%

Time: 7.8s

Alternatives: 7

Speedup: 1.2×

• 61.4% 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: 73.9% 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: 98.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision] * N[(N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision] * N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a))))) < +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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-rgt-in 99.8%

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

      6. fma-def 99.8%

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

      7. sqr-neg 99.8%

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

      8. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. fma-def 99.8%

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

      2. unpow2 99.8%

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

      3. fma-def 99.8%

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

      4. add-sqr-sqrt 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

      7. hypot-def 99.9%

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

      8. fma-def 99.9%

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

      9. hypot-def 99.9%

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

      10. fma-def 99.9%

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

      11. add-sqr-sqrt 99.9%

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

      12. pow2 99.9%

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

      13. fma-def 99.9%

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

      14. hypot-def 99.9%

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

   6. Applied egg-rr 99.9%

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

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

   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. Step-by-step derivation

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. fma-def 6.6%

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

      7. sqr-neg 6.6%

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

      8. +-commutative 6.6%

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

   3. Simplified 6.6%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 95.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq \infty:\\ \;\;\;\;\mathsf{hypot}\left(a, b\right) \cdot \left(\mathsf{hypot}\left(a, b\right) \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right) + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a))))) < +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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-rgt-in 99.8%

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

      6. fma-def 99.8%

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

      7. sqr-neg 99.8%

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

      8. +-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

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

   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. Step-by-step derivation

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. fma-def 6.6%

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

      7. sqr-neg 6.6%

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

      8. +-commutative 6.6%

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

   3. Simplified 6.6%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 95.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq \infty:\\ \;\;\;\;\left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + -1\right) + {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.5× speedup

Math

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

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) (+ a 3.0)))))))
   (if (<= t_0 INFINITY) (+ t_0 -1.0) (pow a 4.0))))

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

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a))))) < +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 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a)))))

   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. Step-by-step derivation

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. fma-def 6.6%

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

      7. sqr-neg 6.6%

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

      8. +-commutative 6.6%

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

   3. Simplified 6.6%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 95.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq \infty:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-263}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-204}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-145}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 2800:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.26e-263)
   (pow a 4.0)
   (if (<= b 8.5e-204)
     -1.0
     (if (<= b 8e-145)
       (pow a 4.0)
       (if (<= b 6.5e-71) -1.0 (if (<= b 2800.0) (pow a 4.0) (pow b 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.26e-263) {
		tmp = pow(a, 4.0);
	} else if (b <= 8.5e-204) {
		tmp = -1.0;
	} else if (b <= 8e-145) {
		tmp = pow(a, 4.0);
	} else if (b <= 6.5e-71) {
		tmp = -1.0;
	} else if (b <= 2800.0) {
		tmp = pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.26d-263) then
        tmp = a ** 4.0d0
    else if (b <= 8.5d-204) then
        tmp = -1.0d0
    else if (b <= 8d-145) then
        tmp = a ** 4.0d0
    else if (b <= 6.5d-71) then
        tmp = -1.0d0
    else if (b <= 2800.0d0) then
        tmp = a ** 4.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.26e-263) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 8.5e-204) {
		tmp = -1.0;
	} else if (b <= 8e-145) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 6.5e-71) {
		tmp = -1.0;
	} else if (b <= 2800.0) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.26e-263:
		tmp = math.pow(a, 4.0)
	elif b <= 8.5e-204:
		tmp = -1.0
	elif b <= 8e-145:
		tmp = math.pow(a, 4.0)
	elif b <= 6.5e-71:
		tmp = -1.0
	elif b <= 2800.0:
		tmp = math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.26e-263)
		tmp = a ^ 4.0;
	elseif (b <= 8.5e-204)
		tmp = -1.0;
	elseif (b <= 8e-145)
		tmp = a ^ 4.0;
	elseif (b <= 6.5e-71)
		tmp = -1.0;
	elseif (b <= 2800.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.26e-263)
		tmp = a ^ 4.0;
	elseif (b <= 8.5e-204)
		tmp = -1.0;
	elseif (b <= 8e-145)
		tmp = a ^ 4.0;
	elseif (b <= 6.5e-71)
		tmp = -1.0;
	elseif (b <= 2800.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.26e-263], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 8.5e-204], -1.0, If[LessEqual[b, 8e-145], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 6.5e-71], -1.0, If[LessEqual[b, 2800.0], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.2599999999999999e-263 or 8.4999999999999997e-204 < b < 7.99999999999999932e-145 or 6.50000000000000005e-71 < b < 2800

   1. Initial program 76.1%

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

      1. associate--l+ 76.1%

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

      2. fma-def 76.1%

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

      3. distribute-rgt-in 76.1%

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

      4. sqr-neg 76.1%

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

      5. distribute-rgt-in 76.1%

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

      6. fma-def 77.3%

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

      7. sqr-neg 77.3%

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

      8. +-commutative 77.3%

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

   3. Simplified 77.3%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 52.0%

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

   if 1.2599999999999999e-263 < b < 8.4999999999999997e-204 or 7.99999999999999932e-145 < b < 6.50000000000000005e-71

   1. Initial program 93.0%

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

      1. associate--l+ 93.0%

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

      2. fma-def 93.0%

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

      3. distribute-rgt-in 93.0%

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

      4. sqr-neg 93.0%

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

      5. distribute-rgt-in 93.0%

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

      6. fma-def 93.0%

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

      7. sqr-neg 93.0%

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

      8. +-commutative 93.0%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 64.5%

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

   6. Taylor expanded in b around 0 64.5%

      \[\leadsto \color{blue}{12 \cdot {b}^{2}} - 1 \]

   7. Taylor expanded in b around 0 64.5%

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

   if 2800 < b

   1. Initial program 68.2%

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

      1. associate--l+ 68.2%

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

      2. fma-def 68.2%

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

      3. distribute-rgt-in 68.2%

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

      4. sqr-neg 68.2%

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

      5. distribute-rgt-in 68.2%

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

      6. fma-def 71.3%

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

      7. sqr-neg 71.3%

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

      8. +-commutative 71.3%

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

   3. Simplified 71.3%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 96.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-263}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-204}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-145}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 2800:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.41 \lor \neg \left(a \leq 2.4\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.41) (not (<= a 2.4))) (pow a 4.0) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.41) || !(a <= 2.4)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -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 <= (-0.41d0)) .or. (.not. (a <= 2.4d0))) then
        tmp = a ** 4.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -0.41) || !(a <= 2.4)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -0.41) or not (a <= 2.4):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.41) || !(a <= 2.4))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -0.41) || ~((a <= 2.4)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.41], N[Not[LessEqual[a, 2.4]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -0.409999999999999976 or 2.39999999999999991 < a

   1. Initial program 52.5%

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

      1. associate--l+ 52.5%

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

      2. fma-def 52.5%

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

      3. distribute-rgt-in 52.5%

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

      4. sqr-neg 52.5%

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

      5. distribute-rgt-in 52.5%

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

      6. fma-def 55.6%

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

      7. sqr-neg 55.6%

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

      8. +-commutative 55.6%

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

   3. Simplified 55.6%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 90.7%

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

   if -0.409999999999999976 < a < 2.39999999999999991

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. sqr-neg 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. fma-def 99.9%

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

      7. sqr-neg 99.9%

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

      8. +-commutative 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 99.6%

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

   6. Taylor expanded in b around 0 78.6%

      \[\leadsto \color{blue}{12 \cdot {b}^{2}} - 1 \]

   7. Taylor expanded in b around 0 51.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.41 \lor \neg \left(a \leq 2.4\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2800:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2800.0) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2800.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2800.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2800.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2800.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2800.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2800.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2800.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2800

   1. Initial program 78.6%

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

      1. +-commutative 78.6%

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

      2. +-commutative 78.6%

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

      3. fma-udef 79.7%

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

      4. add-cube-cbrt 79.7%

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

      5. fma-def 79.7%

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

   4. Applied egg-rr 80.2%

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

   5. Taylor expanded in a around inf 82.0%

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

   if 2800 < b

   1. Initial program 68.2%

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

      1. associate--l+ 68.2%

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

      2. fma-def 68.2%

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

      3. distribute-rgt-in 68.2%

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

      4. sqr-neg 68.2%

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

      5. distribute-rgt-in 68.2%

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

      6. fma-def 71.3%

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

      7. sqr-neg 71.3%

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

      8. +-commutative 71.3%

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

   3. Simplified 71.3%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 96.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2800:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 24.2% accurate, 128.0× 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 76.0%

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

   1. associate--l+ 76.0%

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

   2. fma-def 76.0%

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

   3. distribute-rgt-in 76.0%

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

   4. sqr-neg 76.0%

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

   5. distribute-rgt-in 76.0%

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

   6. fma-def 77.6%

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

   7. sqr-neg 77.6%

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

   8. +-commutative 77.6%

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

3. Simplified 77.6%

   \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right) - 1\right)} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 69.3%

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

6. Taylor expanded in b around 0 53.4%

   \[\leadsto \color{blue}{12 \cdot {b}^{2}} - 1 \]

7. Taylor expanded in b around 0 26.1%

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

8. Final simplification 26.1%

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

Reproduce

herbie shell --seed 2024026 
(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))