Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 99.5%

Time: 7.5s

Alternatives: 9

Speedup: 1.2×

• 61.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := 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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[N[Power[N[(4.0 * N[(a * a), $MachinePrecision] + N[(-1.0 + N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

   3. Simplified 100.0%

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

   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 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 9.9%

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

   4. Taylor expanded in b around 0 31.5%

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

      1. associate--l+ 31.5%

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

      2. associate-*r* 31.5%

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

      3. unpow2 31.5%

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

   6. Simplified 31.5%

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

      1. add-cbrt-cube 38.1%

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

   8. Applied egg-rr 38.1%

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

      1. unpow3 38.1%

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

   10. Simplified 98.6%

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

       1. fma-udef 98.6%

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

       2. +-commutative 98.6%

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

   12. Applied egg-rr 98.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := 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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(-1.0 + N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(4.0 * N[(a * a), $MachinePrecision] + N[(-1.0 + N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   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 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 9.9%

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

   4. Taylor expanded in b around 0 31.5%

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

      1. associate--l+ 31.5%

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

      2. associate-*r* 31.5%

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

      3. unpow2 31.5%

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

   6. Simplified 31.5%

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

      1. add-cbrt-cube 38.1%

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

   8. Applied egg-rr 38.1%

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

      1. unpow3 38.1%

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

   10. Simplified 98.6%

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

       1. fma-udef 98.6%

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

       2. +-commutative 98.6%

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

   12. Applied egg-rr 98.6%

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

4. Final simplification 99.5%

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

Alternative 3: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \left(-1 + 4 \cdot {a}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+103)
   (* (* a a) (* a a))
   (+ (pow (hypot a b) 4.0) (+ -1.0 (* 4.0 (pow a 3.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = pow(hypot(a, b), 4.0) + (-1.0 + (4.0 * pow(a, 3.0)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = Math.pow(Math.hypot(a, b), 4.0) + (-1.0 + (4.0 * Math.pow(a, 3.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e+103:
		tmp = (a * a) * (a * a)
	else:
		tmp = math.pow(math.hypot(a, b), 4.0) + (-1.0 + (4.0 * math.pow(a, 3.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+103)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64((hypot(a, b) ^ 4.0) + Float64(-1.0 + Float64(4.0 * (a ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+103)
		tmp = (a * a) * (a * a);
	else
		tmp = (hypot(a, b) ^ 4.0) + (-1.0 + (4.0 * (a ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+103], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(-1.0 + N[(4.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1e103

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in a around inf 100.0%

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

      1. sqr-pow 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -1e103 < a

   1. Initial program 86.7%

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

      1. associate--l+ 86.7%

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

      2. fma-def 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in a around inf 98.0%

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

      1. fma-def 98.0%

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

      2. add-sqr-sqrt 98.0%

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

      3. hypot-udef 98.0%

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

      4. hypot-udef 98.0%

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

      5. pow-prod-down 98.0%

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

      6. pow-prod-up 98.2%

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

      7. metadata-eval 98.2%

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

      8. expm1-log1p-u 96.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + \left(4 \cdot {a}^{3} - 1\right) \]

      9. expm1-udef 96.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot {a}^{3} - 1\right) \]

   6. Applied egg-rr 96.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot {a}^{3} - 1\right) \]
   7. Step-by-step derivation

      1. expm1-def 96.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + \left(4 \cdot {a}^{3} - 1\right) \]

      2. expm1-log1p 98.2%

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

   8. Simplified 98.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \left(-1 + 4 \cdot {a}^{3}\right)\\ \end{array} \]

Alternative 4: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = (a * a) * (a * a)
	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(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = (a * a) * (a * a);
	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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.8%

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

   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 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 9.9%

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

   4. Taylor expanded in a around inf 92.0%

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

      1. sqr-pow 92.0%

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

      2. metadata-eval 92.0%

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

      3. pow2 92.0%

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

      4. metadata-eval 92.0%

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

      5. pow2 92.0%

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

   6. Applied egg-rr 92.0%

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

4. Final simplification 97.7%

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

Alternative 5: 94.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+20)
   (+ (* (* a a) (* a a)) (+ -1.0 (* (* a a) 4.0)))
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+20) {
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * 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 * b) <= 2d+20) then
        tmp = ((a * a) * (a * a)) + ((-1.0d0) + ((a * 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 * b) <= 2e+20) {
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+20:
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+20)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) + Float64(-1.0 + Float64(Float64(a * a) * 4.0)));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+20)
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+20], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e20

   1. Initial program 85.3%

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

      1. associate--l+ 85.3%

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

      2. fma-def 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in b around 0 84.1%

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

      1. associate--l+ 84.1%

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

      2. associate-*r* 84.1%

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

      3. unpow2 84.1%

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

   6. Simplified 84.1%

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

      1. sqr-pow 47.3%

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

      2. metadata-eval 47.3%

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

      3. pow2 47.3%

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

      4. metadata-eval 47.3%

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

      5. pow2 47.3%

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

   8. Applied egg-rr 83.9%

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

   9. Taylor expanded in a around 0 96.1%

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

       1. unpow2 96.1%

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

   11. Simplified 96.1%

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

   if 2e20 < (*.f64 b b)

   1. Initial program 58.4%

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

      1. associate--l+ 58.4%

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

      2. fma-def 58.4%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in b around inf 91.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6: 71.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+289}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+289)
   (+ (* (* a a) (* a a)) (+ -1.0 (* (* a a) 4.0)))
   (* a (* (* b b) -12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+289) {
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0));
	} else {
		tmp = a * ((b * b) * -12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+289) then
        tmp = ((a * a) * (a * a)) + ((-1.0d0) + ((a * a) * 4.0d0))
    else
        tmp = a * ((b * b) * (-12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+289) {
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0));
	} else {
		tmp = a * ((b * b) * -12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+289:
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0))
	else:
		tmp = a * ((b * b) * -12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+289)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) + Float64(-1.0 + Float64(Float64(a * a) * 4.0)));
	else
		tmp = Float64(a * Float64(Float64(b * b) * -12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+289)
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0));
	else
		tmp = a * ((b * b) * -12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+289], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.0000000000000001e289

   1. Initial program 79.7%

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

      1. associate--l+ 79.7%

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

      2. fma-def 79.7%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in b around 0 66.7%

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

      1. associate--l+ 66.7%

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

      2. associate-*r* 66.7%

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

      3. unpow2 66.7%

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

   6. Simplified 66.7%

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

      1. sqr-pow 46.1%

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

      2. metadata-eval 46.1%

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

      3. pow2 46.1%

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

      4. metadata-eval 46.1%

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

      5. pow2 46.1%

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

   8. Applied egg-rr 66.6%

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

   9. Taylor expanded in a around 0 80.7%

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

       1. unpow2 80.7%

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

   11. Simplified 80.7%

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

   if 1.0000000000000001e289 < (*.f64 b b)

   1. Initial program 52.8%

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

      1. associate--l+ 52.8%

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

      2. fma-def 52.8%

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

   3. Simplified 61.1%

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

      1. fma-def 61.1%

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

      2. add-cbrt-cube 61.1%

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

      3. cbrt-prod 61.1%

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

      4. pow-prod-up 61.1%

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

      5. add-sqr-sqrt 61.1%

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

      6. hypot-udef 61.1%

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

      7. hypot-udef 61.1%

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

      8. metadata-eval 61.1%

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

      9. pow-prod-down 61.1%

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

      10. pow-prod-up 61.1%

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

      11. metadata-eval 61.1%

          \[\leadsto \sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{\color{blue}{8}}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

      12. fma-def 61.1%

          \[\leadsto \sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{8}} \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{8}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 51.4%

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

      1. sub-neg 51.4%

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

      2. associate-+r+ 51.4%

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

      3. metadata-eval 51.4%

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

      4. associate-+l+ 51.4%

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

      5. associate-*r* 51.4%

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

      6. distribute-rgt-out 72.2%

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

      7. unpow2 72.2%

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

      8. +-commutative 72.2%

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

      9. *-commutative 72.2%

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

   8. Simplified 72.2%

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

   9. Taylor expanded in a around inf 50.1%

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

       1. *-commutative 50.1%

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

       2. unpow2 50.1%

          \[\leadsto \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -12 \]

       3. *-commutative 50.1%

          \[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot a\right)} \cdot -12 \]

       4. associate-*r* 50.1%

          \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(a \cdot -12\right)} \]

       5. associate-*l* 39.9%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(a \cdot -12\right)\right)} \]

   11. Simplified 39.9%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(a \cdot -12\right)\right)} \]

   12. Taylor expanded in b around 0 50.1%

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

       1. *-commutative 50.1%

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

       2. unpow2 50.1%

          \[\leadsto \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -12 \]

       3. associate-*l* 50.1%

          \[\leadsto \color{blue}{a \cdot \left(\left(b \cdot b\right) \cdot -12\right)} \]

   14. Simplified 50.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+289}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \end{array} \]

Alternative 7: 68.7% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.4) || !(a <= 0.41))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if a < -2.39999999999999991 or 0.409999999999999976 < a

   1. Initial program 45.6%

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

      1. associate--l+ 45.6%

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

      2. fma-def 45.6%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in a around inf 85.6%

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

      1. sqr-pow 85.5%

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

      2. metadata-eval 85.5%

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

      3. pow2 85.5%

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

      4. metadata-eval 85.5%

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

      5. pow2 85.5%

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

   6. Applied egg-rr 85.5%

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

   if -2.39999999999999991 < a < 0.409999999999999976

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 53.5%

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

      1. associate--l+ 53.5%

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

      2. associate-*r* 53.5%

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

      3. unpow2 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in a around 0 53.0%

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

4. Final simplification 69.6%

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

Alternative 8: 40.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.004:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.004) -1.0 (* a (* (* b b) -12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.004) {
		tmp = -1.0;
	} else {
		tmp = a * ((b * b) * -12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.004d0) then
        tmp = -1.0d0
    else
        tmp = a * ((b * b) * (-12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.004) {
		tmp = -1.0;
	} else {
		tmp = a * ((b * b) * -12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.004:
		tmp = -1.0
	else:
		tmp = a * ((b * b) * -12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.004)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(Float64(b * b) * -12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.004)
		tmp = -1.0;
	else
		tmp = a * ((b * b) * -12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.004], -1.0, N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0040000000000000001

   1. Initial program 86.3%

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

      1. associate--l+ 86.3%

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

      2. fma-def 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in b around 0 85.3%

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

      1. associate--l+ 85.3%

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

      2. associate-*r* 85.3%

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

      3. unpow2 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in a around 0 52.6%

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

   if 0.0040000000000000001 < (*.f64 b b)

   1. Initial program 58.4%

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

      1. associate--l+ 58.4%

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

      2. fma-def 58.4%

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

   3. Simplified 63.8%

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

      1. fma-def 63.8%

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

      2. add-cbrt-cube 57.4%

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

      3. cbrt-prod 58.8%

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

      4. pow-prod-up 58.8%

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

      5. add-sqr-sqrt 58.8%

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

      6. hypot-udef 58.8%

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

      7. hypot-udef 58.8%

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

      8. metadata-eval 58.8%

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

      9. pow-prod-down 58.8%

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

      10. pow-prod-up 58.8%

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

      11. metadata-eval 58.8%

          \[\leadsto \sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{\color{blue}{8}}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

      12. fma-def 58.8%

          \[\leadsto \sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{8}} \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 58.8%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{8}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 58.8%

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

      1. sub-neg 58.8%

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

      2. associate-+r+ 58.8%

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

      3. metadata-eval 58.8%

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

      4. associate-+l+ 58.8%

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

      5. associate-*r* 58.8%

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

      6. distribute-rgt-out 70.3%

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

      7. unpow2 70.3%

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

      8. +-commutative 70.3%

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

      9. *-commutative 70.3%

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

   8. Simplified 70.3%

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

   9. Taylor expanded in a around inf 34.0%

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

       1. *-commutative 34.0%

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

       2. unpow2 34.0%

          \[\leadsto \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -12 \]

       3. *-commutative 34.0%

          \[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot a\right)} \cdot -12 \]

       4. associate-*r* 34.0%

          \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(a \cdot -12\right)} \]

       5. associate-*l* 28.4%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(a \cdot -12\right)\right)} \]

   11. Simplified 28.4%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(a \cdot -12\right)\right)} \]

   12. Taylor expanded in b around 0 34.0%

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

       1. *-commutative 34.0%

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

       2. unpow2 34.0%

          \[\leadsto \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -12 \]

       3. associate-*l* 34.0%

          \[\leadsto \color{blue}{a \cdot \left(\left(b \cdot b\right) \cdot -12\right)} \]

   14. Simplified 34.0%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.004:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \end{array} \]

Alternative 9: 24.7% accurate, 130.0× 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 72.1%

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

   1. associate--l+ 72.1%

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

   2. fma-def 72.1%

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

3. Simplified 74.9%

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

4. Taylor expanded in b around 0 54.1%

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

   1. associate--l+ 54.1%

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

   2. associate-*r* 54.1%

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

   3. unpow2 54.1%

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

6. Simplified 54.1%

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

7. Taylor expanded in a around 0 26.2%

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

8. Final simplification 26.2%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023192 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))