Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.6% → 98.5%

Time: 5.6s

Alternatives: 9

Speedup: 1.2×

• 60.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= ((double) INFINITY)) {
		tmp = pow(hypot(a, b), 4.0) + fma(4.0, (fma((b * b), (a + 3.0), (a * a)) - pow(a, 3.0)), -1.0);
	} else {
		tmp = -1.0 + pow(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(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0))))) <= Inf)
		tmp = Float64((hypot(a, b) ^ 4.0) + fma(4.0, Float64(fma(Float64(b * b), Float64(a + 3.0), Float64(a * a)) - (a ^ 3.0)), -1.0));
	else
		tmp = Float64(-1.0 + (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[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[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 3 a))))) < +inf.0

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

   3. Simplified 100.0%

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

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

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 6.8%

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

      4. +-commutative 6.8%

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

      5. metadata-eval 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in a around inf 94.9%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

   1. Initial program 99.9%

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

   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 3 a)))))

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 6.8%

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

      4. +-commutative 6.8%

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

      5. metadata-eval 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in a around inf 94.9%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

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

Alternative 3: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+51}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.3e+51) (+ -1.0 (pow a 4.0)) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.3e+51) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.3d+51) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.3e+51) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.3e+51:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.3e+51)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.3e+51)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.3e+51], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3000000000000001e51

   1. Initial program 72.4%

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

      1. sub-neg 72.4%

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

      2. fma-def 72.4%

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

      3. fma-def 74.5%

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

      4. +-commutative 74.5%

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

      5. metadata-eval 74.5%

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

   3. Simplified 74.5%

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

   4. Taylor expanded in a around inf 79.3%

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

   if 1.3000000000000001e51 < b

   1. Initial program 68.7%

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

      1. sub-neg 68.7%

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

      2. fma-def 68.7%

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

      3. fma-def 70.1%

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

      4. +-commutative 70.1%

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

      5. metadata-eval 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in b around inf 98.6%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+51}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 4: 68.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ -1 + {a}^{4} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

   1. sub-neg 71.4%

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

   2. fma-def 71.4%

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

   3. fma-def 73.3%

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

   4. +-commutative 73.3%

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

   5. metadata-eval 73.3%

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

3. Simplified 73.3%

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

4. Taylor expanded in a around inf 67.4%

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

5. Final simplification 67.4%

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

Alternative 5: 61.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot 4\\ \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1 + t_0\\ \mathbf{elif}\;a \leq 2.4:\\ \;\;\;\;-1 + a \cdot \left(4 \cdot \frac{a \cdot a - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{a + a \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - t_0 \cdot \left(-1 - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) 4.0)))
   (if (<= a -2e+154)
     (+ -1.0 t_0)
     (if (<= a 2.4)
       (+ -1.0 (* a (* 4.0 (/ (- (* a a) (* (* a a) (* a a))) (+ a (* a a))))))
       (- -1.0 (* t_0 (- -1.0 a)))))))

C

double code(double a, double b) {
	double t_0 = (a * a) * 4.0;
	double tmp;
	if (a <= -2e+154) {
		tmp = -1.0 + t_0;
	} else if (a <= 2.4) {
		tmp = -1.0 + (a * (4.0 * (((a * a) - ((a * a) * (a * a))) / (a + (a * a)))));
	} else {
		tmp = -1.0 - (t_0 * (-1.0 - a));
	}
	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 = (a * a) * 4.0d0
    if (a <= (-2d+154)) then
        tmp = (-1.0d0) + t_0
    else if (a <= 2.4d0) then
        tmp = (-1.0d0) + (a * (4.0d0 * (((a * a) - ((a * a) * (a * a))) / (a + (a * a)))))
    else
        tmp = (-1.0d0) - (t_0 * ((-1.0d0) - a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * 4.0;
	double tmp;
	if (a <= -2e+154) {
		tmp = -1.0 + t_0;
	} else if (a <= 2.4) {
		tmp = -1.0 + (a * (4.0 * (((a * a) - ((a * a) * (a * a))) / (a + (a * a)))));
	} else {
		tmp = -1.0 - (t_0 * (-1.0 - a));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * 4.0
	tmp = 0
	if a <= -2e+154:
		tmp = -1.0 + t_0
	elif a <= 2.4:
		tmp = -1.0 + (a * (4.0 * (((a * a) - ((a * a) * (a * a))) / (a + (a * a)))))
	else:
		tmp = -1.0 - (t_0 * (-1.0 - a))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * 4.0)
	tmp = 0.0
	if (a <= -2e+154)
		tmp = Float64(-1.0 + t_0);
	elseif (a <= 2.4)
		tmp = Float64(-1.0 + Float64(a * Float64(4.0 * Float64(Float64(Float64(a * a) - Float64(Float64(a * a) * Float64(a * a))) / Float64(a + Float64(a * a))))));
	else
		tmp = Float64(-1.0 - Float64(t_0 * Float64(-1.0 - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * 4.0;
	tmp = 0.0;
	if (a <= -2e+154)
		tmp = -1.0 + t_0;
	elseif (a <= 2.4)
		tmp = -1.0 + (a * (4.0 * (((a * a) - ((a * a) * (a * a))) / (a + (a * a)))));
	else
		tmp = -1.0 - (t_0 * (-1.0 - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[a, -2e+154], N[(-1.0 + t$95$0), $MachinePrecision], If[LessEqual[a, 2.4], N[(-1.0 + N[(a * N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(t$95$0 * N[(-1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000007e154

   1. Initial program 40.7%

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

      1. sub-neg 40.7%

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

      2. fma-def 40.7%

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

      3. fma-def 40.7%

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

      4. +-commutative 40.7%

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

      5. metadata-eval 40.7%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in b around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if -2.00000000000000007e154 < a < 2.39999999999999991

   1. Initial program 93.5%

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

      1. sub-neg 93.5%

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

      2. fma-def 93.5%

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

      3. fma-def 93.5%

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

      4. +-commutative 93.5%

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

      5. metadata-eval 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in b around 0 51.3%

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

      1. associate-*r* 51.3%

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

      2. unpow2 51.3%

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

   6. Simplified 51.3%

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

   7. Taylor expanded in a around 0 43.5%

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

      1. unpow2 43.5%

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

      2. *-commutative 43.5%

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

      3. associate-*r* 43.5%

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

      4. *-commutative 43.5%

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

      5. cube-mult 43.5%

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

      6. associate-*l* 43.5%

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

      7. metadata-eval 43.5%

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

      8. distribute-rgt-neg-in 43.5%

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

      9. metadata-eval 43.5%

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

      10. swap-sqr 43.5%

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

      11. unpow2 43.5%

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

      12. distribute-lft-out 43.5%

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

      13. unpow2 43.5%

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

      14. swap-sqr 43.5%

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

      15. metadata-eval 43.5%

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

      16. associate-*r* 43.5%

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

      17. distribute-lft-neg-in 43.5%

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

      18. distribute-rgt1-in 43.5%

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

      19. +-commutative 43.5%

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

      20. sub-neg 43.5%

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

      21. *-commutative 43.5%

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

      22. *-commutative 43.5%

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

   9. Simplified 43.5%

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

       1. sub-neg 43.5%

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

       2. flip-+ 47.0%

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

       3. distribute-rgt-neg-in 47.0%

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

       4. distribute-rgt-neg-in 47.0%

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

       5. distribute-rgt-neg-in 47.0%

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

   11. Applied egg-rr 47.0%

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

   if 2.39999999999999991 < a

   1. Initial program 35.5%

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

      1. sub-neg 35.5%

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

      2. fma-def 35.5%

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

      3. fma-def 42.3%

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

      4. +-commutative 42.3%

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

      5. metadata-eval 42.3%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in b around 0 26.6%

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

      1. associate-*r* 26.6%

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

      2. unpow2 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in a around 0 0.1%

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

      1. unpow2 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-*r* 0.1%

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

      4. *-commutative 0.1%

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

      5. cube-mult 0.1%

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

      6. associate-*l* 0.1%

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

      7. metadata-eval 0.1%

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

      8. distribute-rgt-neg-in 0.1%

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

      9. metadata-eval 0.1%

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

      10. swap-sqr 0.1%

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

      11. unpow2 0.1%

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

      12. distribute-lft-out 0.1%

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

      13. unpow2 0.1%

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

      14. swap-sqr 0.1%

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

      15. metadata-eval 0.1%

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

      16. associate-*r* 0.1%

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

      17. distribute-lft-neg-in 0.1%

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

      18. distribute-rgt1-in 0.1%

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

      19. +-commutative 0.1%

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

      20. sub-neg 0.1%

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

      21. *-commutative 0.1%

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

      22. *-commutative 0.1%

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

   9. Simplified 0.1%

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

       1. associate-*r* 0.1%

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

       2. flip-- 0.1%

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

       3. associate-*r/ 0.1%

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

       4. pow2 0.1%

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

       5. metadata-eval 0.1%

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

       6. pow-prod-down 0.1%

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

       7. pow-prod-up 0.1%

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

       8. metadata-eval 0.1%

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

       9. metadata-eval 0.1%

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

       10. metadata-eval 0.1%

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

       11. distribute-rgt1-in 0.1%

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

   11. Applied egg-rr 0.1%

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

       1. associate-/l* 0.1%

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

       2. *-commutative 0.1%

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

       3. associate-/l* 0.1%

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

       4. metadata-eval 0.1%

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

       5. pow-plus 0.1%

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

       6. unpow3 0.1%

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

       7. associate-*r* 0.1%

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

       8. difference-of-squares 0.1%

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

       9. distribute-rgt1-in 0.1%

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

       10. associate-/r* 0.1%

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

       11. *-inverses 0.1%

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

   13. Simplified 0.1%

       \[\leadsto \color{blue}{\frac{4}{\frac{\frac{1}{a - a \cdot a}}{a}}} + -1 \]
   14. Step-by-step derivation

       1. div-inv 0.1%

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

       2. associate-/l/ 0.1%

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

       3. remove-double-div 0.1%

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

       4. associate-*r* 0.1%

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

       5. *-un-lft-identity 0.1%

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

       6. distribute-rgt-out-- 0.1%

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

       7. associate-*r* 0.1%

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

       8. add-sqr-sqrt 0.1%

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

       9. sqrt-prod 0.1%

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

       10. sqr-neg 0.1%

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

       11. sqrt-unprod 0.0%

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

       12. add-sqr-sqrt 66.9%

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

       13. associate-*r* 66.9%

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

       14. *-commutative 66.9%

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

       15. add-sqr-sqrt 0.0%

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

       16. sqrt-unprod 0.1%

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

       17. sqr-neg 0.1%

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

       18. sqrt-prod 0.1%

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

       19. add-sqr-sqrt 0.1%

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

       20. sub-neg 0.1%

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

       21. add-sqr-sqrt 0.0%

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

       22. sqrt-unprod 66.9%

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

       23. sqr-neg 66.9%

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

   15. Applied egg-rr 66.9%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{elif}\;a \leq 2.4:\\ \;\;\;\;-1 + a \cdot \left(4 \cdot \frac{a \cdot a - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{a + a \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - \left(\left(a \cdot a\right) \cdot 4\right) \cdot \left(-1 - a\right)\\ \end{array} \]

Alternative 6: 55.3% accurate, 9.8× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.4) (- -1.0 (* a (* 4.0 (- (* a a) a)))) (+ -1.0 (* (* a a) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.4) {
		tmp = -1.0 - (a * (4.0 * ((a * a) - a)));
	} 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 <= 2.4d0) then
        tmp = (-1.0d0) - (a * (4.0d0 * ((a * a) - a)))
    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 <= 2.4) {
		tmp = -1.0 - (a * (4.0 * ((a * a) - a)));
	} else {
		tmp = -1.0 + ((a * a) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2.4:
		tmp = -1.0 - (a * (4.0 * ((a * a) - a)))
	else:
		tmp = -1.0 + ((a * a) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.4)
		tmp = Float64(-1.0 - Float64(a * Float64(4.0 * Float64(Float64(a * a) - a))));
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 2.4)
		tmp = -1.0 - (a * (4.0 * ((a * a) - a)));
	else
		tmp = -1.0 + ((a * a) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.4], N[(-1.0 - N[(a * N[(4.0 * N[(N[(a * a), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.39999999999999991

   1. Initial program 85.7%

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

      1. sub-neg 85.7%

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

      2. fma-def 85.7%

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

      3. fma-def 85.7%

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

      4. +-commutative 85.7%

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

      5. metadata-eval 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in b around 0 58.5%

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

      1. associate-*r* 58.5%

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

      2. unpow2 58.5%

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

   6. Simplified 58.5%

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

   7. Taylor expanded in a around 0 51.9%

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

      1. unpow2 51.9%

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

      2. *-commutative 51.9%

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

      3. associate-*r* 51.9%

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

      4. *-commutative 51.9%

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

      5. cube-mult 51.9%

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

      6. associate-*l* 51.9%

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

      7. metadata-eval 51.9%

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

      8. distribute-rgt-neg-in 51.9%

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

      9. metadata-eval 51.9%

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

      10. swap-sqr 51.9%

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

      11. unpow2 51.9%

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

      12. distribute-lft-out 51.9%

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

      13. unpow2 51.9%

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

      14. swap-sqr 51.9%

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

      15. metadata-eval 51.9%

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

      16. associate-*r* 51.9%

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

      17. distribute-lft-neg-in 51.9%

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

      18. distribute-rgt1-in 51.9%

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

      19. +-commutative 51.9%

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

      20. sub-neg 51.9%

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

      21. *-commutative 51.9%

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

      22. *-commutative 51.9%

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

   9. Simplified 51.9%

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

   if 2.39999999999999991 < a

   1. Initial program 35.5%

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

      1. sub-neg 35.5%

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

      2. fma-def 35.5%

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

      3. fma-def 42.3%

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

      4. +-commutative 42.3%

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

      5. metadata-eval 42.3%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in b around 0 26.6%

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

      1. associate-*r* 26.6%

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

      2. unpow2 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in a around 0 51.0%

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

      1. unpow2 51.0%

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

   9. Simplified 51.0%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.4:\\ \;\;\;\;-1 - a \cdot \left(4 \cdot \left(a \cdot a - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \end{array} \]

Alternative 7: 59.4% accurate, 9.8× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.4)
   (- -1.0 (* a (* 4.0 (- (* a a) a))))
   (- -1.0 (* (* (* a a) 4.0) (- -1.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.4) {
		tmp = -1.0 - (a * (4.0 * ((a * a) - a)));
	} else {
		tmp = -1.0 - (((a * a) * 4.0) * (-1.0 - a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 2.4d0) then
        tmp = (-1.0d0) - (a * (4.0d0 * ((a * a) - a)))
    else
        tmp = (-1.0d0) - (((a * a) * 4.0d0) * ((-1.0d0) - a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 2.4) {
		tmp = -1.0 - (a * (4.0 * ((a * a) - a)));
	} else {
		tmp = -1.0 - (((a * a) * 4.0) * (-1.0 - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2.4:
		tmp = -1.0 - (a * (4.0 * ((a * a) - a)))
	else:
		tmp = -1.0 - (((a * a) * 4.0) * (-1.0 - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.4)
		tmp = Float64(-1.0 - Float64(a * Float64(4.0 * Float64(Float64(a * a) - a))));
	else
		tmp = Float64(-1.0 - Float64(Float64(Float64(a * a) * 4.0) * Float64(-1.0 - a)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[a, 2.4], N[(-1.0 - N[(a * N[(4.0 * N[(N[(a * a), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] * N[(-1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.39999999999999991

   1. Initial program 85.7%

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

      1. sub-neg 85.7%

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

      2. fma-def 85.7%

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

      3. fma-def 85.7%

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

      4. +-commutative 85.7%

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

      5. metadata-eval 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in b around 0 58.5%

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

      1. associate-*r* 58.5%

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

      2. unpow2 58.5%

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

   6. Simplified 58.5%

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

   7. Taylor expanded in a around 0 51.9%

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

      1. unpow2 51.9%

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

      2. *-commutative 51.9%

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

      3. associate-*r* 51.9%

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

      4. *-commutative 51.9%

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

      5. cube-mult 51.9%

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

      6. associate-*l* 51.9%

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

      7. metadata-eval 51.9%

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

      8. distribute-rgt-neg-in 51.9%

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

      9. metadata-eval 51.9%

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

      10. swap-sqr 51.9%

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

      11. unpow2 51.9%

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

      12. distribute-lft-out 51.9%

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

      13. unpow2 51.9%

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

      14. swap-sqr 51.9%

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

      15. metadata-eval 51.9%

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

      16. associate-*r* 51.9%

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

      17. distribute-lft-neg-in 51.9%

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

      18. distribute-rgt1-in 51.9%

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

      19. +-commutative 51.9%

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

      20. sub-neg 51.9%

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

      21. *-commutative 51.9%

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

      22. *-commutative 51.9%

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

   9. Simplified 51.9%

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

   if 2.39999999999999991 < a

   1. Initial program 35.5%

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

      1. sub-neg 35.5%

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

      2. fma-def 35.5%

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

      3. fma-def 42.3%

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

      4. +-commutative 42.3%

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

      5. metadata-eval 42.3%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in b around 0 26.6%

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

      1. associate-*r* 26.6%

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

      2. unpow2 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in a around 0 0.1%

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

      1. unpow2 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-*r* 0.1%

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

      4. *-commutative 0.1%

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

      5. cube-mult 0.1%

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

      6. associate-*l* 0.1%

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

      7. metadata-eval 0.1%

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

      8. distribute-rgt-neg-in 0.1%

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

      9. metadata-eval 0.1%

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

      10. swap-sqr 0.1%

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

      11. unpow2 0.1%

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

      12. distribute-lft-out 0.1%

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

      13. unpow2 0.1%

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

      14. swap-sqr 0.1%

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

      15. metadata-eval 0.1%

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

      16. associate-*r* 0.1%

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

      17. distribute-lft-neg-in 0.1%

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

      18. distribute-rgt1-in 0.1%

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

      19. +-commutative 0.1%

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

      20. sub-neg 0.1%

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

      21. *-commutative 0.1%

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

      22. *-commutative 0.1%

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

   9. Simplified 0.1%

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

       1. associate-*r* 0.1%

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

       2. flip-- 0.1%

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

       3. associate-*r/ 0.1%

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

       4. pow2 0.1%

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

       5. metadata-eval 0.1%

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

       6. pow-prod-down 0.1%

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

       7. pow-prod-up 0.1%

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

       8. metadata-eval 0.1%

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

       9. metadata-eval 0.1%

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

       10. metadata-eval 0.1%

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

       11. distribute-rgt1-in 0.1%

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

   11. Applied egg-rr 0.1%

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

       1. associate-/l* 0.1%

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

       2. *-commutative 0.1%

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

       3. associate-/l* 0.1%

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

       4. metadata-eval 0.1%

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

       5. pow-plus 0.1%

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

       6. unpow3 0.1%

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

       7. associate-*r* 0.1%

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

       8. difference-of-squares 0.1%

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

       9. distribute-rgt1-in 0.1%

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

       10. associate-/r* 0.1%

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

       11. *-inverses 0.1%

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

   13. Simplified 0.1%

       \[\leadsto \color{blue}{\frac{4}{\frac{\frac{1}{a - a \cdot a}}{a}}} + -1 \]
   14. Step-by-step derivation

       1. div-inv 0.1%

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

       2. associate-/l/ 0.1%

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

       3. remove-double-div 0.1%

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

       4. associate-*r* 0.1%

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

       5. *-un-lft-identity 0.1%

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

       6. distribute-rgt-out-- 0.1%

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

       7. associate-*r* 0.1%

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

       8. add-sqr-sqrt 0.1%

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

       9. sqrt-prod 0.1%

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

       10. sqr-neg 0.1%

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

       11. sqrt-unprod 0.0%

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

       12. add-sqr-sqrt 66.9%

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

       13. associate-*r* 66.9%

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

       14. *-commutative 66.9%

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

       15. add-sqr-sqrt 0.0%

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

       16. sqrt-unprod 0.1%

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

       17. sqr-neg 0.1%

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

       18. sqrt-prod 0.1%

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

       19. add-sqr-sqrt 0.1%

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

       20. sub-neg 0.1%

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

       21. add-sqr-sqrt 0.0%

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

       22. sqrt-unprod 66.9%

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

       23. sqr-neg 66.9%

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

   15. Applied egg-rr 66.9%

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

4. Final simplification 56.1%

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

Alternative 8: 55.2% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = -1.0 + (a * (a * (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 <= (-1.0d0)) then
        tmp = (-1.0d0) + (a * (a * (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 <= -1.0) {
		tmp = -1.0 + (a * (a * (a * -4.0)));
	} else {
		tmp = -1.0 + ((a * a) * 4.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. fma-def 55.0%

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

      3. fma-def 55.0%

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

      4. +-commutative 55.0%

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

      5. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in b around 0 88.8%

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

      1. associate-*r* 88.8%

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

      2. unpow2 88.8%

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

   6. Simplified 88.8%

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

   7. Taylor expanded in a around 0 68.0%

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

      1. unpow2 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-*r* 68.0%

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

      4. *-commutative 68.0%

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

      5. cube-mult 68.0%

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

      6. associate-*l* 68.0%

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

      7. metadata-eval 68.0%

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

      8. distribute-rgt-neg-in 68.0%

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

      9. metadata-eval 68.0%

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

      10. swap-sqr 68.0%

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

      11. unpow2 68.0%

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

      12. distribute-lft-out 68.0%

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

      13. unpow2 68.0%

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

      14. swap-sqr 68.0%

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

      15. metadata-eval 68.0%

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

      16. associate-*r* 68.0%

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

      17. distribute-lft-neg-in 68.0%

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

      18. distribute-rgt1-in 68.0%

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

      19. +-commutative 68.0%

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

      20. sub-neg 68.0%

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

      21. *-commutative 68.0%

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

      22. *-commutative 68.0%

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

   9. Simplified 68.0%

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

   10. Taylor expanded in a around inf 68.0%

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

       1. unpow2 68.0%

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

       2. mul-1-neg 68.0%

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

       3. distribute-rgt-neg-out 68.0%

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

   12. Simplified 68.0%

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

   13. Taylor expanded in a around 0 68.0%

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

       1. unpow2 68.0%

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

       2. *-commutative 68.0%

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

       3. associate-*l* 68.0%

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

   15. Simplified 68.0%

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

   if -1 < a

   1. Initial program 76.2%

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

      1. sub-neg 76.2%

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

      2. fma-def 76.2%

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

      3. fma-def 78.7%

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

      4. +-commutative 78.7%

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

      5. metadata-eval 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in b around 0 37.8%

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

      1. associate-*r* 37.8%

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

      2. unpow2 37.8%

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

   6. Simplified 37.8%

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

   7. Taylor expanded in a around 0 46.7%

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

      1. unpow2 46.7%

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

   9. Simplified 46.7%

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

4. Final simplification 51.6%

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

Alternative 9: 51.3% accurate, 18.3× speedup

Math

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

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(Float64(a * 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[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.4%

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

   1. sub-neg 71.4%

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

   2. fma-def 71.4%

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

   3. fma-def 73.3%

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

   4. +-commutative 73.3%

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

   5. metadata-eval 73.3%

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

3. Simplified 73.3%

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

4. Taylor expanded in b around 0 49.4%

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

   1. associate-*r* 49.4%

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

   2. unpow2 49.4%

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

6. Simplified 49.4%

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

7. Taylor expanded in a around 0 47.4%

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

   1. unpow2 47.4%

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

9. Simplified 47.4%

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

10. Final simplification 47.4%

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

Reproduce

herbie shell --seed 2023196 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))