Bouland and Aaronson, Equation (25)

Percentage Accurate: 72.7% → 98.3%

Time: 8.5s

Alternatives: 7

Speedup: 1.1×

• 62.4% 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: 72.7% 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: 98.3% accurate, 0.2× speedup

Math

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

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 (hypot a b) 4.0)
    (fma 4.0 (fma a (* a (+ a 1.0)) (* (* b b) (+ 1.0 (* a -3.0)))) -1.0))
   (* (pow a 3.0) (+ a 4.0))))

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(hypot(a, b), 4.0) + fma(4.0, fma(a, (a * (a + 1.0)), ((b * b) * (1.0 + (a * -3.0)))), -1.0);
	} else {
		tmp = pow(a, 3.0) * (a + 4.0);
	}
	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((hypot(a, b) ^ 4.0) + fma(4.0, fma(a, Float64(a * Float64(a + 1.0)), Float64(Float64(b * b) * Float64(1.0 + Float64(a * -3.0)))), -1.0));
	else
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	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[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(a * N[(a * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $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 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

      3. fma-neg 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

      7. sub-neg 99.9%

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

      8. *-commutative 99.9%

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

      9. distribute-rgt-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-udef 98.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      3. fma-def 98.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(a \cdot a + b \cdot b\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      4. fma-def 98.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      5. add-sqr-sqrt 98.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      6. pow2 98.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}^{2}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      7. fma-def 98.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\sqrt{\color{blue}{a \cdot a + b \cdot b}}\right)}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      8. hypot-def 98.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

   6. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]
   7. Step-by-step derivation

      1. expm1-def 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-log1p 99.9%

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

      3. unpow2 99.9%

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

      4. pow-sqr 100.0%

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

      5. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   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 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. fma-neg 0.0%

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

      4. associate-*l* 0.0%

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

      5. fma-def 2.9%

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

      6. +-commutative 2.9%

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

      7. sub-neg 2.9%

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

      8. *-commutative 2.9%

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

      9. distribute-rgt-neg-in 2.9%

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

      10. metadata-eval 2.9%

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

      11. metadata-eval 2.9%

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

   3. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-udef 2.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      3. fma-def 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(a \cdot a + b \cdot b\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      4. fma-def 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      5. add-sqr-sqrt 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      6. pow2 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}^{2}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      7. fma-def 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\sqrt{\color{blue}{a \cdot a + b \cdot b}}\right)}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      8. hypot-def 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

   6. Applied egg-rr 2.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]
   7. Step-by-step derivation

      1. expm1-def 2.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-log1p 2.9%

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

      3. unpow2 2.9%

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

      4. pow-sqr 2.9%

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

      5. metadata-eval 2.9%

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

   8. Simplified 2.9%

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

   9. Taylor expanded in b around 0 26.5%

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

       1. fma-def 26.5%

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

       2. distribute-rgt-in 26.5%

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

       3. *-lft-identity 26.5%

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

       4. unpow2 26.5%

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

       5. unpow2 26.5%

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

       6. cube-mult 26.5%

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

       7. fma-udef 26.5%

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

       8. fma-udef 26.5%

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

       9. +-commutative 26.5%

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

       10. associate--l+ 26.5%

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

       11. fma-neg 26.5%

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

       12. metadata-eval 26.5%

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

   11. Simplified 26.5%

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

   12. Taylor expanded in a around inf 26.5%

       \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   13. Step-by-step derivation

       1. +-commutative 26.5%

          \[\leadsto \color{blue}{{a}^{4} + 4 \cdot {a}^{3}} \]

       2. metadata-eval 26.5%

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

       3. pow-plus 26.5%

          \[\leadsto \color{blue}{{a}^{3} \cdot a} + 4 \cdot {a}^{3} \]

       4. *-commutative 26.5%

          \[\leadsto {a}^{3} \cdot a + \color{blue}{{a}^{3} \cdot 4} \]

       5. distribute-lft-out 90.2%

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

   14. Simplified 90.2%

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

4. Final simplification 97.4%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

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

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) (* (pow a 3.0) (+ a 4.0)))))

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 = pow(a, 3.0) * (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) * (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 = Math.pow(a, 3.0) * (a + 4.0);
	}
	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 = math.pow(a, 3.0) * (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(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((a ^ 3.0) * Float64(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) * (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 ^ 3.0) * (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[(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[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $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 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 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. fma-neg 0.0%

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

      4. associate-*l* 0.0%

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

      5. fma-def 2.9%

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

      6. +-commutative 2.9%

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

      7. sub-neg 2.9%

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

      8. *-commutative 2.9%

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

      9. distribute-rgt-neg-in 2.9%

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

      10. metadata-eval 2.9%

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

      11. metadata-eval 2.9%

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

   3. Simplified 2.9%

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

      1. expm1-log1p-u 2.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-udef 2.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      3. fma-def 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(a \cdot a + b \cdot b\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      4. fma-def 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      5. add-sqr-sqrt 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      6. pow2 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}^{2}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      7. fma-def 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\sqrt{\color{blue}{a \cdot a + b \cdot b}}\right)}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      8. hypot-def 2.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

   6. Applied egg-rr 2.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]
   7. Step-by-step derivation

      1. expm1-def 2.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-log1p 2.9%

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

      3. unpow2 2.9%

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

      4. pow-sqr 2.9%

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

      5. metadata-eval 2.9%

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

   8. Simplified 2.9%

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

   9. Taylor expanded in b around 0 26.5%

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

       1. fma-def 26.5%

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

       2. distribute-rgt-in 26.5%

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

       3. *-lft-identity 26.5%

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

       4. unpow2 26.5%

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

       5. unpow2 26.5%

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

       6. cube-mult 26.5%

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

       7. fma-udef 26.5%

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

       8. fma-udef 26.5%

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

       9. +-commutative 26.5%

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

       10. associate--l+ 26.5%

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

       11. fma-neg 26.5%

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

       12. metadata-eval 26.5%

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

   11. Simplified 26.5%

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

   12. Taylor expanded in a around inf 26.5%

       \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   13. Step-by-step derivation

       1. +-commutative 26.5%

          \[\leadsto \color{blue}{{a}^{4} + 4 \cdot {a}^{3}} \]

       2. metadata-eval 26.5%

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

       3. pow-plus 26.5%

          \[\leadsto \color{blue}{{a}^{3} \cdot a} + 4 \cdot {a}^{3} \]

       4. *-commutative 26.5%

          \[\leadsto {a}^{3} \cdot a + \color{blue}{{a}^{3} \cdot 4} \]

       5. distribute-lft-out 90.2%

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

   14. Simplified 90.2%

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

4. Final simplification 97.3%

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

Alternative 3: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 580:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+40} \lor \neg \left(b \leq 2.8 \cdot 10^{+49}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 580.0)
   (+ -1.0 (pow a 4.0))
   (if (or (<= b 7.2e+40) (not (<= b 2.8e+49))) (pow b 4.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 580.0) {
		tmp = -1.0 + pow(a, 4.0);
	} else if ((b <= 7.2e+40) || !(b <= 2.8e+49)) {
		tmp = pow(b, 4.0);
	} else {
		tmp = pow(a, 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 <= 580.0d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else if ((b <= 7.2d+40) .or. (.not. (b <= 2.8d+49))) then
        tmp = b ** 4.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 580.0) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else if ((b <= 7.2e+40) || !(b <= 2.8e+49)) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 580.0:
		tmp = -1.0 + math.pow(a, 4.0)
	elif (b <= 7.2e+40) or not (b <= 2.8e+49):
		tmp = math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 580.0)
		tmp = Float64(-1.0 + (a ^ 4.0));
	elseif ((b <= 7.2e+40) || !(b <= 2.8e+49))
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 580.0)
		tmp = -1.0 + (a ^ 4.0);
	elseif ((b <= 7.2e+40) || ~((b <= 2.8e+49)))
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 580.0], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 7.2e+40], N[Not[LessEqual[b, 2.8e+49]], $MachinePrecision]], N[Power[b, 4.0], $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 580

   1. Initial program 77.1%

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

      1. sub-neg 77.1%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in a around inf 77.6%

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

   if 580 < b < 7.19999999999999993e40 or 2.7999999999999998e49 < b

   1. Initial program 66.1%

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

      1. associate--l+ 66.1%

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

      2. fma-def 66.1%

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

      3. fma-neg 66.1%

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

      4. associate-*l* 66.1%

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

      5. fma-def 67.5%

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

      6. +-commutative 67.5%

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

      7. sub-neg 67.5%

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

      8. *-commutative 67.5%

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

      9. distribute-rgt-neg-in 67.5%

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

      10. metadata-eval 67.5%

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

      11. metadata-eval 67.5%

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

   3. Simplified 67.5%

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

      1. expm1-log1p-u 66.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-udef 66.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      3. fma-def 66.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(a \cdot a + b \cdot b\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      4. fma-def 66.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      5. add-sqr-sqrt 66.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      6. pow2 66.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}^{2}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      7. fma-def 66.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\sqrt{\color{blue}{a \cdot a + b \cdot b}}\right)}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      8. hypot-def 66.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

   6. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]
   7. Step-by-step derivation

      1. expm1-def 66.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-log1p 67.5%

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

      3. unpow2 67.5%

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

      4. pow-sqr 67.6%

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

      5. metadata-eval 67.6%

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

   8. Simplified 67.6%

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

   9. Taylor expanded in b around inf 95.3%

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

   if 7.19999999999999993e40 < b < 2.7999999999999998e49

   1. Initial program 20.0%

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

      1. associate--l+ 20.0%

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

      2. fma-def 20.0%

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

      3. fma-neg 20.0%

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

      4. associate-*l* 20.0%

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

      5. fma-def 20.0%

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

      6. +-commutative 20.0%

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

      7. sub-neg 20.0%

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

      8. *-commutative 20.0%

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

      9. distribute-rgt-neg-in 20.0%

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

      10. metadata-eval 20.0%

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

      11. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

      1. expm1-log1p-u 20.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-udef 20.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      3. fma-def 20.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(a \cdot a + b \cdot b\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      4. fma-def 20.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      5. add-sqr-sqrt 20.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      6. pow2 20.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}^{2}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      7. fma-def 20.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\sqrt{\color{blue}{a \cdot a + b \cdot b}}\right)}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      8. hypot-def 20.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

   6. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]
   7. Step-by-step derivation

      1. expm1-def 20.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-log1p 20.0%

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

      3. unpow2 20.0%

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

      4. pow-sqr 20.0%

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

      5. metadata-eval 20.0%

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

   8. Simplified 20.0%

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

   9. Taylor expanded in a around inf 100.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 580:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+40} \lor \neg \left(b \leq 2.8 \cdot 10^{+49}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 3.9 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 475:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* a (* a 4.0)))))
   (if (<= b 3.9e-128)
     t_0
     (if (<= b 1.15e-120) (pow a 4.0) (if (<= b 475.0) t_0 (pow b 4.0))))))

C

double code(double a, double b) {
	double t_0 = -1.0 + (a * (a * 4.0));
	double tmp;
	if (b <= 3.9e-128) {
		tmp = t_0;
	} else if (b <= 1.15e-120) {
		tmp = pow(a, 4.0);
	} else if (b <= 475.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + (a * (a * 4.0d0))
    if (b <= 3.9d-128) then
        tmp = t_0
    else if (b <= 1.15d-120) then
        tmp = a ** 4.0d0
    else if (b <= 475.0d0) then
        tmp = t_0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = -1.0 + (a * (a * 4.0));
	double tmp;
	if (b <= 3.9e-128) {
		tmp = t_0;
	} else if (b <= 1.15e-120) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 475.0) {
		tmp = t_0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = -1.0 + (a * (a * 4.0))
	tmp = 0
	if b <= 3.9e-128:
		tmp = t_0
	elif b <= 1.15e-120:
		tmp = math.pow(a, 4.0)
	elif b <= 475.0:
		tmp = t_0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(-1.0 + Float64(a * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 3.9e-128)
		tmp = t_0;
	elseif (b <= 1.15e-120)
		tmp = a ^ 4.0;
	elseif (b <= 475.0)
		tmp = t_0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = -1.0 + (a * (a * 4.0));
	tmp = 0.0;
	if (b <= 3.9e-128)
		tmp = t_0;
	elseif (b <= 1.15e-120)
		tmp = a ^ 4.0;
	elseif (b <= 475.0)
		tmp = t_0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.9e-128], t$95$0, If[LessEqual[b, 1.15e-120], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 475.0], t$95$0, N[Power[b, 4.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.89999999999999997e-128 or 1.14999999999999993e-120 < b < 475

   1. Initial program 77.4%

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

      1. associate--l+ 77.5%

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

      2. fma-def 77.5%

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

      3. fma-neg 77.5%

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

      4. associate-*l* 77.5%

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

      5. fma-def 78.0%

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

      6. +-commutative 78.0%

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

      7. sub-neg 78.0%

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

      8. *-commutative 78.0%

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

      9. distribute-rgt-neg-in 78.0%

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

      10. metadata-eval 78.0%

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

      11. metadata-eval 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in b around 0 62.2%

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

   6. Taylor expanded in a around 0 60.7%

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

      1. expm1-log1p-u 60.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(4 \cdot {a}^{2}\right)\right)} - 1 \]

      2. expm1-udef 60.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(4 \cdot {a}^{2}\right)} - 1\right)} - 1 \]

      3. log1p-udef 60.7%

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

      4. rem-exp-log 60.7%

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

      5. +-commutative 60.7%

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

   8. Applied egg-rr 60.7%

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

      1. associate--l+ 60.7%

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

      2. metadata-eval 60.7%

         \[\leadsto \left(4 \cdot {a}^{2} + \color{blue}{0}\right) - 1 \]

      3. +-rgt-identity 60.7%

         \[\leadsto \color{blue}{4 \cdot {a}^{2}} - 1 \]

      4. unpow2 60.7%

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

      5. associate-*r* 60.7%

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

   10. Applied egg-rr 60.7%

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

   if 3.89999999999999997e-128 < b < 1.14999999999999993e-120

   1. Initial program 49.2%

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

      1. associate--l+ 49.2%

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

      2. fma-def 49.2%

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

      3. fma-neg 49.2%

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

      4. associate-*l* 49.2%

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

      5. fma-def 49.2%

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

      6. +-commutative 49.2%

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

      7. sub-neg 49.2%

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

      8. *-commutative 49.2%

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

      9. distribute-rgt-neg-in 49.2%

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

      10. metadata-eval 49.2%

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

      11. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

      1. expm1-log1p-u 44.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-udef 44.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      3. fma-def 44.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(a \cdot a + b \cdot b\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      4. fma-def 44.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      5. add-sqr-sqrt 44.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      6. pow2 44.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}^{2}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      7. fma-def 44.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\sqrt{\color{blue}{a \cdot a + b \cdot b}}\right)}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      8. hypot-def 44.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

   6. Applied egg-rr 44.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]
   7. Step-by-step derivation

      1. expm1-def 44.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-log1p 49.2%

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

      3. unpow2 49.2%

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

      4. pow-sqr 50.0%

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

      5. metadata-eval 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in a around inf 100.0%

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

   if 475 < b

   1. Initial program 63.1%

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

      1. associate--l+ 63.1%

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

      2. fma-def 63.1%

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

      3. fma-neg 63.1%

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

      4. associate-*l* 63.1%

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

      5. fma-def 64.4%

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

      6. +-commutative 64.4%

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

      7. sub-neg 64.4%

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

      8. *-commutative 64.4%

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

      9. distribute-rgt-neg-in 64.4%

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

      10. metadata-eval 64.4%

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

      11. metadata-eval 64.4%

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

   3. Simplified 64.4%

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

      1. expm1-log1p-u 63.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-udef 63.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      3. fma-def 63.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(a \cdot a + b \cdot b\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      4. fma-def 63.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      5. add-sqr-sqrt 63.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      6. pow2 63.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}^{2}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      7. fma-def 63.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\sqrt{\color{blue}{a \cdot a + b \cdot b}}\right)}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      8. hypot-def 63.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

   6. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]
   7. Step-by-step derivation

      1. expm1-def 63.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-log1p 64.4%

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

      3. unpow2 64.4%

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

      4. pow-sqr 64.5%

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

      5. metadata-eval 64.5%

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

   8. Simplified 64.5%

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

   9. Taylor expanded in b around inf 89.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{-128}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 475:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0065 \lor \neg \left(a \leq 3.9 \cdot 10^{-7}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.0065) (not (<= a 3.9e-7)))
   (pow a 4.0)
   (+ -1.0 (* a (* a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.0065) || !(a <= 3.9e-7)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + (a * (a * 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 ((a <= (-0.0065d0)) .or. (.not. (a <= 3.9d-7))) then
        tmp = a ** 4.0d0
    else
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -0.0065) || !(a <= 3.9e-7)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (a * (a * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -0.0065) or not (a <= 3.9e-7):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0 + (a * (a * 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.0065) || !(a <= 3.9e-7))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + Float64(a * Float64(a * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -0.0065) || ~((a <= 3.9e-7)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0 + (a * (a * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.0065], N[Not[LessEqual[a, 3.9e-7]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0064999999999999997 or 3.90000000000000025e-7 < a

   1. Initial program 48.4%

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

      1. associate--l+ 48.4%

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

      2. fma-def 48.4%

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

      3. fma-neg 48.4%

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

      4. associate-*l* 48.4%

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

      5. fma-def 49.9%

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

      6. +-commutative 49.9%

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

      7. sub-neg 49.9%

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

      8. *-commutative 49.9%

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

      9. distribute-rgt-neg-in 49.9%

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

      10. metadata-eval 49.9%

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

      11. metadata-eval 49.9%

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

   3. Simplified 49.9%

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

      1. expm1-log1p-u 48.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-udef 48.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      3. fma-def 48.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(a \cdot a + b \cdot b\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      4. fma-def 48.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      5. add-sqr-sqrt 48.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      6. pow2 48.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}^{2}\right)}}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      7. fma-def 48.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\sqrt{\color{blue}{a \cdot a + b \cdot b}}\right)}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      8. hypot-def 48.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right) + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

   6. Applied egg-rr 48.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]
   7. Step-by-step derivation

      1. expm1-def 48.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(a + 1\right), \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right), -1\right) \]

      2. expm1-log1p 49.9%

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

      3. unpow2 49.9%

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

      4. pow-sqr 50.0%

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

      5. metadata-eval 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in a around inf 83.1%

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

   if -0.0064999999999999997 < a < 3.90000000000000025e-7

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

      3. fma-neg 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

      7. sub-neg 99.9%

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

      8. *-commutative 99.9%

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

      9. distribute-rgt-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 48.0%

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

   6. Taylor expanded in a around 0 48.0%

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

      1. expm1-log1p-u 48.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(4 \cdot {a}^{2}\right)\right)} - 1 \]

      2. expm1-udef 48.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(4 \cdot {a}^{2}\right)} - 1\right)} - 1 \]

      3. log1p-udef 48.0%

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

      4. rem-exp-log 48.0%

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

      5. +-commutative 48.0%

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

   8. Applied egg-rr 48.0%

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

      1. associate--l+ 48.0%

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

      2. metadata-eval 48.0%

         \[\leadsto \left(4 \cdot {a}^{2} + \color{blue}{0}\right) - 1 \]

      3. +-rgt-identity 48.0%

         \[\leadsto \color{blue}{4 \cdot {a}^{2}} - 1 \]

      4. unpow2 48.0%

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

      5. associate-*r* 48.0%

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

   10. Applied egg-rr 48.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0065 \lor \neg \left(a \leq 3.9 \cdot 10^{-7}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.8% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

   1. associate--l+ 73.0%

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

   2. fma-def 73.0%

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

   3. fma-neg 73.0%

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

   4. associate-*l* 73.0%

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

   5. fma-def 73.7%

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

   6. +-commutative 73.7%

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

   7. sub-neg 73.7%

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

   8. *-commutative 73.7%

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

   9. distribute-rgt-neg-in 73.7%

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

   10. metadata-eval 73.7%

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

   11. metadata-eval 73.7%

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

3. Simplified 73.7%

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

5. Taylor expanded in b around 0 49.8%

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

6. Taylor expanded in a around 0 50.2%

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

   1. expm1-log1p-u 50.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(4 \cdot {a}^{2}\right)\right)} - 1 \]

   2. expm1-udef 50.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(4 \cdot {a}^{2}\right)} - 1\right)} - 1 \]

   3. log1p-udef 50.2%

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

   4. rem-exp-log 50.2%

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

   5. +-commutative 50.2%

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

8. Applied egg-rr 50.2%

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

   1. associate--l+ 50.2%

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

   2. metadata-eval 50.2%

      \[\leadsto \left(4 \cdot {a}^{2} + \color{blue}{0}\right) - 1 \]

   3. +-rgt-identity 50.2%

      \[\leadsto \color{blue}{4 \cdot {a}^{2}} - 1 \]

   4. unpow2 50.2%

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

   5. associate-*r* 50.2%

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

10. Applied egg-rr 50.2%

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

11. Final simplification 50.2%

    \[\leadsto -1 + a \cdot \left(a \cdot 4\right) \]
12. Add Preprocessing

Alternative 7: 24.1% 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 73.0%

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

   1. sub-neg 73.0%

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

3. Simplified 73.7%

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

5. Taylor expanded in a around inf 66.2%

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

6. Taylor expanded in a around 0 23.1%

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

7. Final simplification 23.1%

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

Reproduce

herbie shell --seed 2024018 
(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))