Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.6% → 98.4%

Time: 7.7s

Alternatives: 12

Speedup: 1.2×

• 61.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.6% 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: 98.4% accurate, 0.2× speedup

Math

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

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)
   (+
    (fma
     4.0
     (fma a (fma a a a) (* b (* b (fma a -3.0 1.0))))
     (pow (hypot a b) 4.0))
    -1.0)
   (* (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 = fma(4.0, fma(a, fma(a, a, a), (b * (b * fma(a, -3.0, 1.0)))), pow(hypot(a, b), 4.0)) + -1.0;
	} else {
		tmp = 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(fma(4.0, fma(a, fma(a, a, a), Float64(b * Float64(b * fma(a, -3.0, 1.0)))), (hypot(a, b) ^ 4.0)) + -1.0);
	else
		tmp = 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[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(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 99.8%

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

      1. sub-neg 99.8%

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

   3. Simplified 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in a around inf 17.8%

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

      1. *-commutative 17.8%

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

      2. metadata-eval 17.8%

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

      3. pow-plus 17.8%

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

      4. distribute-lft-out 91.1%

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

   6. Simplified 91.1%

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

4. Final simplification 97.4%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

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 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in a around inf 17.8%

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

      1. *-commutative 17.8%

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

      2. metadata-eval 17.8%

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

      3. pow-plus 17.8%

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

      4. distribute-lft-out 91.1%

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

   6. Simplified 91.1%

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

4. Final simplification 97.3%

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

Alternative 3: 94.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -50000:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;a \leq 3000:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -50000.0)
   (* (pow a 3.0) (+ a 4.0))
   (if (<= a 3000.0)
     (+ -1.0 (+ (pow b 4.0) (* (* b b) 4.0)))
     (+ (pow a 4.0) (+ -1.0 (* (+ a 1.0) (* (* a a) 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -50000.0) {
		tmp = pow(a, 3.0) * (a + 4.0);
	} else if (a <= 3000.0) {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 4.0));
	} else {
		tmp = pow(a, 4.0) + (-1.0 + ((a + 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 <= (-50000.0d0)) then
        tmp = (a ** 3.0d0) * (a + 4.0d0)
    else if (a <= 3000.0d0) then
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * 4.0d0))
    else
        tmp = (a ** 4.0d0) + ((-1.0d0) + ((a + 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 <= -50000.0) {
		tmp = Math.pow(a, 3.0) * (a + 4.0);
	} else if (a <= 3000.0) {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * 4.0));
	} else {
		tmp = Math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -50000.0:
		tmp = math.pow(a, 3.0) * (a + 4.0)
	elif a <= 3000.0:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * 4.0))
	else:
		tmp = math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -50000.0)
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	elseif (a <= 3000.0)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0)));
	else
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a + 1.0) * Float64(Float64(a * a) * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -50000.0)
		tmp = (a ^ 3.0) * (a + 4.0);
	elseif (a <= 3000.0)
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * 4.0));
	else
		tmp = (a ^ 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -50000.0], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3000.0], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5e4

   1. Initial program 27.5%

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

      1. associate--l+ 27.5%

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

      2. fma-def 27.5%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in a around inf 20.3%

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

      1. *-commutative 20.3%

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

      2. metadata-eval 20.3%

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

      3. pow-plus 20.2%

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

      4. distribute-lft-out 92.6%

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

   6. Simplified 92.6%

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

   if -5e4 < a < 3e3

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-+l+ 91.9%

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

      3. *-commutative 91.9%

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

      4. associate-*r* 91.9%

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

      5. *-commutative 91.9%

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

      6. distribute-lft-out 99.2%

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

      7. unpow2 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in a around 0 100.0%

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

   if 3e3 < a

   1. Initial program 66.5%

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

      1. associate--l+ 66.5%

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

      2. fma-def 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in b around 0 86.6%

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

      1. associate--l+ 86.6%

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

      2. associate-*r* 86.6%

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

      3. unpow2 86.6%

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

   6. Simplified 86.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -50000:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;a \leq 3000:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \end{array} \]

Alternative 4: 94.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3100000 \lor \neg \left(a \leq 8500\right):\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3100000.0) (not (<= a 8500.0)))
   (* (pow a 3.0) (+ a 4.0))
   (+ -1.0 (+ (pow b 4.0) (* (* b b) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3100000.0) || !(a <= 8500.0)) {
		tmp = pow(a, 3.0) * (a + 4.0);
	} else {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-3100000.0d0)) .or. (.not. (a <= 8500.0d0))) then
        tmp = (a ** 3.0d0) * (a + 4.0d0)
    else
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -3100000.0) || !(a <= 8500.0)) {
		tmp = Math.pow(a, 3.0) * (a + 4.0);
	} else {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -3100000.0) or not (a <= 8500.0):
		tmp = math.pow(a, 3.0) * (a + 4.0)
	else:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3100000.0) || !(a <= 8500.0))
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	else
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -3100000.0) || ~((a <= 8500.0)))
		tmp = (a ^ 3.0) * (a + 4.0);
	else
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3100000.0], N[Not[LessEqual[a, 8500.0]], $MachinePrecision]], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.1e6 or 8500 < a

   1. Initial program 44.2%

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

      1. associate--l+ 44.2%

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

      2. fma-def 44.2%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in a around inf 47.8%

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

      1. *-commutative 47.8%

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

      2. metadata-eval 47.8%

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

      3. pow-plus 47.8%

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

      4. distribute-lft-out 89.1%

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

   6. Simplified 89.1%

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

   if -3.1e6 < a < 8500

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-+l+ 91.9%

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

      3. *-commutative 91.9%

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

      4. associate-*r* 91.9%

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

      5. *-commutative 91.9%

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

      6. distribute-lft-out 99.2%

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

      7. unpow2 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in a around 0 100.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3100000 \lor \neg \left(a \leq 8500\right):\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \end{array} \]

Alternative 5: 94.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -18000 \lor \neg \left(a \leq 14000\right):\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -18000.0) (not (<= a 14000.0)))
   (* (pow a 3.0) (+ a 4.0))
   (+ -1.0 (* b (* b (fma b b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -18000.0) || !(a <= 14000.0)) {
		tmp = pow(a, 3.0) * (a + 4.0);
	} else {
		tmp = -1.0 + (b * (b * fma(b, b, 4.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -18000.0) || !(a <= 14000.0))
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 4.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -18000.0], N[Not[LessEqual[a, 14000.0]], $MachinePrecision]], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -18000 or 14000 < a

   1. Initial program 44.2%

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

      1. associate--l+ 44.2%

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

      2. fma-def 44.2%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in a around inf 47.8%

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

      1. *-commutative 47.8%

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

      2. metadata-eval 47.8%

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

      3. pow-plus 47.8%

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

      4. distribute-lft-out 89.1%

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

   6. Simplified 89.1%

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

   if -18000 < a < 14000

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-+l+ 91.9%

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

      3. *-commutative 91.9%

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

      4. associate-*r* 91.9%

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

      5. *-commutative 91.9%

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

      6. distribute-lft-out 99.2%

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

      7. unpow2 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in a around 0 100.0%

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

   8. Taylor expanded in b around 0 100.0%

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

      1. metadata-eval 100.0%

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

      2. pow-sqr 99.9%

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

      3. unpow2 99.9%

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

      4. unpow2 99.9%

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

      5. unpow2 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. associate-*l* 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-def 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -18000 \lor \neg \left(a \leq 14000\right):\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \]

Alternative 6: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -210000 \lor \neg \left(a \leq 12500\right):\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -210000.0) (not (<= a 12500.0)))
   (* (pow a 3.0) (+ a 4.0))
   (+ -1.0 (* (* b b) (+ (* b b) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -210000.0) || !(a <= 12500.0)) {
		tmp = pow(a, 3.0) * (a + 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-210000.0d0)) .or. (.not. (a <= 12500.0d0))) then
        tmp = (a ** 3.0d0) * (a + 4.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -210000.0) || !(a <= 12500.0)) {
		tmp = Math.pow(a, 3.0) * (a + 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -210000.0) or not (a <= 12500.0):
		tmp = math.pow(a, 3.0) * (a + 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -210000.0) || !(a <= 12500.0))
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -210000.0) || ~((a <= 12500.0)))
		tmp = (a ^ 3.0) * (a + 4.0);
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -210000.0], N[Not[LessEqual[a, 12500.0]], $MachinePrecision]], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.1e5 or 12500 < a

   1. Initial program 44.2%

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

      1. associate--l+ 44.2%

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

      2. fma-def 44.2%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in a around inf 47.8%

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

      1. *-commutative 47.8%

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

      2. metadata-eval 47.8%

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

      3. pow-plus 47.8%

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

      4. distribute-lft-out 89.1%

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

   6. Simplified 89.1%

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

   if -2.1e5 < a < 12500

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-+l+ 91.9%

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

      3. *-commutative 91.9%

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

      4. associate-*r* 91.9%

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

      5. *-commutative 91.9%

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

      6. distribute-lft-out 99.2%

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

      7. unpow2 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. sqr-pow 99.9%

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

      2. metadata-eval 99.9%

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

      3. pow2 99.9%

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

      4. metadata-eval 99.9%

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

      5. pow2 99.9%

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

      6. distribute-lft-out 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -210000 \lor \neg \left(a \leq 12500\right):\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 7: 94.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -66000000 \lor \neg \left(a \leq 14000\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -66000000.0) (not (<= a 14000.0)))
   (pow a 4.0)
   (+ -1.0 (* (* b b) (+ (* b b) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -66000000.0) || !(a <= 14000.0)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-66000000.0d0)) .or. (.not. (a <= 14000.0d0))) then
        tmp = a ** 4.0d0
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -66000000.0) || !(a <= 14000.0)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -66000000.0) or not (a <= 14000.0):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -66000000.0) || !(a <= 14000.0))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -66000000.0) || ~((a <= 14000.0)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -66000000.0], N[Not[LessEqual[a, 14000.0]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.6e7 or 14000 < a

   1. Initial program 44.2%

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

      1. associate--l+ 44.2%

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

      2. fma-def 44.2%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in a around inf 87.5%

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

   if -6.6e7 < a < 14000

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-+l+ 91.9%

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

      3. *-commutative 91.9%

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

      4. associate-*r* 91.9%

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

      5. *-commutative 91.9%

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

      6. distribute-lft-out 99.2%

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

      7. unpow2 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. sqr-pow 99.9%

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

      2. metadata-eval 99.9%

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

      3. pow2 99.9%

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

      4. metadata-eval 99.9%

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

      5. pow2 99.9%

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

      6. distribute-lft-out 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -66000000 \lor \neg \left(a \leq 14000\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 8: 54.1% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;-1 + a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.33) (+ -1.0 (* a (* (* b b) -12.0))) (+ -1.0 (* b (* b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.33) {
		tmp = -1.0 + (a * ((b * b) * -12.0));
	} else {
		tmp = -1.0 + (b * (b * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.33d0)) then
        tmp = (-1.0d0) + (a * ((b * b) * (-12.0d0)))
    else
        tmp = (-1.0d0) + (b * (b * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.33) {
		tmp = -1.0 + (a * ((b * b) * -12.0));
	} else {
		tmp = -1.0 + (b * (b * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.33:
		tmp = -1.0 + (a * ((b * b) * -12.0))
	else:
		tmp = -1.0 + (b * (b * 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.33)
		tmp = Float64(-1.0 + Float64(a * Float64(Float64(b * b) * -12.0)));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.33)
		tmp = -1.0 + (a * ((b * b) * -12.0));
	else
		tmp = -1.0 + (b * (b * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.33], N[(-1.0 + N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.330000000000000016

   1. Initial program 28.4%

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

      1. sub-neg 28.4%

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

   3. Simplified 36.2%

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

   4. Taylor expanded in a around 0 52.9%

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

      1. +-commutative 52.9%

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

      2. associate-+l+ 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*r* 54.1%

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

      5. *-commutative 54.1%

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

      6. distribute-lft-out 54.1%

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

      7. unpow2 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in a around inf 48.2%

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

      1. unpow2 48.2%

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

      2. associate-*r* 49.4%

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

      3. *-commutative 49.4%

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

      4. associate-*r* 49.4%

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

      5. *-commutative 49.4%

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

      6. *-commutative 49.4%

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

      7. associate-*l* 48.2%

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

   9. Simplified 48.2%

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

   10. Taylor expanded in b around 0 48.2%

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

       1. *-commutative 48.2%

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

       2. unpow2 48.2%

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

       3. associate-*r* 48.2%

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

       4. *-commutative 48.2%

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

       5. associate-*r* 48.2%

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

       6. *-commutative 48.2%

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

       7. associate-*l* 48.2%

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

   12. Simplified 48.2%

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

   13. Taylor expanded in b around 0 48.2%

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

       1. unpow2 48.2%

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

   15. Simplified 48.2%

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

   if -0.330000000000000016 < a

   1. Initial program 88.7%

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

      1. sub-neg 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in a around 0 63.8%

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

      1. +-commutative 63.8%

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

      2. associate-+l+ 63.8%

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

      3. *-commutative 63.8%

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

      4. associate-*r* 63.8%

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

      5. *-commutative 63.8%

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

      6. distribute-lft-out 68.9%

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

      7. unpow2 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in a around 0 80.1%

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

   8. Taylor expanded in b around 0 80.1%

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

      1. metadata-eval 80.1%

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

      2. pow-sqr 80.0%

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

      3. unpow2 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

      6. distribute-rgt-in 80.0%

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

      7. associate-*l* 80.0%

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

      8. +-commutative 80.0%

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

      9. +-commutative 80.0%

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

      10. fma-def 80.0%

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

   10. Simplified 80.0%

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

   11. Taylor expanded in b around 0 63.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;-1 + a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 9: 54.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.4:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.4) (+ -1.0 (* b (* b (* a -12.0)))) (+ -1.0 (* b (* b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.4) {
		tmp = -1.0 + (b * (b * (a * -12.0)));
	} else {
		tmp = -1.0 + (b * (b * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.4d0)) then
        tmp = (-1.0d0) + (b * (b * (a * (-12.0d0))))
    else
        tmp = (-1.0d0) + (b * (b * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.4) {
		tmp = -1.0 + (b * (b * (a * -12.0)));
	} else {
		tmp = -1.0 + (b * (b * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.4:
		tmp = -1.0 + (b * (b * (a * -12.0)))
	else:
		tmp = -1.0 + (b * (b * 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.4)
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(a * -12.0))));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.4)
		tmp = -1.0 + (b * (b * (a * -12.0)));
	else
		tmp = -1.0 + (b * (b * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.4], N[(-1.0 + N[(b * N[(b * N[(a * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.40000000000000002

   1. Initial program 28.4%

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

      1. sub-neg 28.4%

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

   3. Simplified 36.2%

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

   4. Taylor expanded in a around 0 52.9%

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

      1. +-commutative 52.9%

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

      2. associate-+l+ 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*r* 54.1%

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

      5. *-commutative 54.1%

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

      6. distribute-lft-out 54.1%

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

      7. unpow2 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in a around inf 48.2%

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

      1. *-commutative 48.2%

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

      2. unpow2 48.2%

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

      3. *-commutative 48.2%

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

      4. associate-*r* 49.4%

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

      5. associate-*l* 49.4%

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

   9. Simplified 49.4%

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

   if -0.40000000000000002 < a

   1. Initial program 88.7%

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

      1. sub-neg 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in a around 0 63.8%

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

      1. +-commutative 63.8%

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

      2. associate-+l+ 63.8%

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

      3. *-commutative 63.8%

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

      4. associate-*r* 63.8%

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

      5. *-commutative 63.8%

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

      6. distribute-lft-out 68.9%

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

      7. unpow2 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in a around 0 80.1%

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

   8. Taylor expanded in b around 0 80.1%

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

      1. metadata-eval 80.1%

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

      2. pow-sqr 80.0%

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

      3. unpow2 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

      6. distribute-rgt-in 80.0%

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

      7. associate-*l* 80.0%

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

      8. +-commutative 80.0%

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

      9. +-commutative 80.0%

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

      10. fma-def 80.0%

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

   10. Simplified 80.0%

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

   11. Taylor expanded in b around 0 63.6%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.4:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 10: 69.4% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. sub-neg 70.6%

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

3. Simplified 72.9%

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

4. Taylor expanded in a around 0 60.6%

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

   1. +-commutative 60.6%

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

   2. associate-+l+ 60.6%

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

   3. *-commutative 60.6%

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

   4. associate-*r* 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-lft-out 64.4%

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

   7. unpow2 64.4%

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

6. Simplified 64.4%

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

7. Taylor expanded in a around 0 71.8%

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

   1. sqr-pow 71.7%

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

   2. metadata-eval 71.7%

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

   3. pow2 71.7%

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

   4. metadata-eval 71.7%

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

   5. pow2 71.7%

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

   6. distribute-lft-out 71.7%

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

9. Applied egg-rr 71.7%

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

10. Final simplification 71.7%

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

Alternative 11: 51.0% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. sub-neg 70.6%

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

3. Simplified 72.9%

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

4. Taylor expanded in a around 0 60.6%

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

   1. +-commutative 60.6%

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

   2. associate-+l+ 60.6%

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

   3. *-commutative 60.6%

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

   4. associate-*r* 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-lft-out 64.4%

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

   7. unpow2 64.4%

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

6. Simplified 64.4%

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

7. Taylor expanded in a around 0 71.8%

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

8. Taylor expanded in b around 0 71.8%

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

   1. metadata-eval 71.8%

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

   2. pow-sqr 71.7%

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

   3. unpow2 71.7%

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

   4. unpow2 71.7%

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

   5. unpow2 71.7%

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

   6. distribute-rgt-in 71.7%

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

   7. associate-*l* 71.7%

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

   8. +-commutative 71.7%

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

   9. +-commutative 71.7%

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

   10. fma-def 71.7%

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

10. Simplified 71.7%

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

11. Taylor expanded in b around 0 56.6%

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

12. Final simplification 56.6%

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

Alternative 12: 24.5% 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 70.6%

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

   1. associate--l+ 70.6%

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

   2. fma-def 70.6%

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

3. Simplified 72.5%

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

4. Taylor expanded in b around 0 53.1%

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

   1. associate--l+ 53.1%

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

   2. associate-*r* 53.1%

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

   3. unpow2 53.1%

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

6. Simplified 53.1%

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

7. Taylor expanded in a around 0 28.1%

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

8. Final simplification 28.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023238 
(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))