Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 98.2%

Time: 6.9s

Alternatives: 11

Speedup: 8.6×

• 61.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

   3. Simplified 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 3.0%

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

   4. Taylor expanded in a around inf 95.7%

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

4. Final simplification 98.9%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 3.0%

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

   4. Taylor expanded in a around inf 95.7%

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

4. Final simplification 98.7%

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

Alternative 3: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (+ (* (* b b) 4.0) (pow b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (((b * b) * 4.0) + 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 * b) <= 0.005d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (((b * b) * 4.0d0) + (b ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (((b * b) * 4.0) + Math.pow(b, 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.005:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + (((b * b) * 4.0) + math.pow(b, 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(b * b) * 4.0) + (b ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.005)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (((b * b) * 4.0) + (b ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 80.3%

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

   3. Applied egg-rr 80.3%

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

   4. Taylor expanded in a around inf 98.3%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 68.3%

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

      1. associate--l+ 68.3%

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

      2. fma-def 68.3%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in a around 0 92.2%

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

      1. associate--l+ 92.2%

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

      2. fma-def 92.2%

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

      3. unpow2 92.2%

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

      4. sub-neg 92.2%

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

      5. metadata-eval 92.2%

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

   6. Simplified 92.2%

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

      1. fma-udef 92.2%

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

      2. associate-+r+ 92.2%

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

   8. Applied egg-rr 92.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right)\\ \end{array} \]

Alternative 4: 93.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005)
   (+ (pow a 4.0) (+ -1.0 (* (* a a) 4.0)))
   (+ -1.0 (+ (* (* b b) 4.0) (pow b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = pow(a, 4.0) + (-1.0 + ((a * a) * 4.0));
	} else {
		tmp = -1.0 + (((b * b) * 4.0) + 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 * b) <= 0.005d0) then
        tmp = (a ** 4.0d0) + ((-1.0d0) + ((a * a) * 4.0d0))
    else
        tmp = (-1.0d0) + (((b * b) * 4.0d0) + (b ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = Math.pow(a, 4.0) + (-1.0 + ((a * a) * 4.0));
	} else {
		tmp = -1.0 + (((b * b) * 4.0) + Math.pow(b, 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.005:
		tmp = math.pow(a, 4.0) + (-1.0 + ((a * a) * 4.0))
	else:
		tmp = -1.0 + (((b * b) * 4.0) + math.pow(b, 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a * a) * 4.0)));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(b * b) * 4.0) + (b ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.005)
		tmp = (a ^ 4.0) + (-1.0 + ((a * a) * 4.0));
	else
		tmp = -1.0 + (((b * b) * 4.0) + (b ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 80.3%

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

      1. associate--l+ 80.3%

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

      2. fma-def 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in b around 0 80.4%

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

      1. associate--l+ 80.4%

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

      2. associate-*r* 80.4%

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

      3. unpow2 80.4%

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

   6. Simplified 80.4%

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

   7. Taylor expanded in a around 0 98.4%

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

      1. unpow2 98.4%

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

   9. Simplified 98.4%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 68.3%

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

      1. associate--l+ 68.3%

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

      2. fma-def 68.3%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in a around 0 92.2%

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

      1. associate--l+ 92.2%

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

      2. fma-def 92.2%

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

      3. unpow2 92.2%

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

      4. sub-neg 92.2%

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

      5. metadata-eval 92.2%

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

   6. Simplified 92.2%

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

      1. fma-udef 92.2%

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

      2. associate-+r+ 92.2%

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

   8. Applied egg-rr 92.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right)\\ \end{array} \]

Alternative 5: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (* b (* b (fma b b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (b * (b * fma(b, b, 4.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 4.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 80.3%

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

   3. Applied egg-rr 80.3%

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

   4. Taylor expanded in a around inf 98.3%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 68.3%

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

      1. associate--l+ 68.3%

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

      2. fma-def 68.3%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in a around 0 92.2%

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

      1. associate--l+ 92.2%

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

      2. fma-def 92.2%

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

      3. unpow2 92.2%

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

      4. sub-neg 92.2%

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

      5. metadata-eval 92.2%

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

   6. Simplified 92.2%

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

      1. fma-udef 92.2%

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

      2. associate-+r+ 92.2%

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

   8. Applied egg-rr 92.2%

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

   9. Taylor expanded in b around 0 92.2%

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

       1. unpow2 92.2%

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

       2. metadata-eval 92.2%

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

       3. pow-sqr 92.1%

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

       4. unpow2 92.1%

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

       5. unpow2 92.1%

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

       6. distribute-rgt-in 92.1%

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

       7. fma-udef 92.1%

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

       8. associate-*l* 92.1%

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

   11. Simplified 92.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \]

Alternative 6: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (* (* b b) (+ (* b b) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.005d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.005:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.005)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 80.3%

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

   3. Applied egg-rr 80.3%

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

   4. Taylor expanded in a around inf 98.3%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 68.3%

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

      1. associate--l+ 68.3%

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

      2. fma-def 68.3%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in a around 0 92.2%

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

      1. associate--l+ 92.2%

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

      2. fma-def 92.2%

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

      3. unpow2 92.2%

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

      4. sub-neg 92.2%

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

      5. metadata-eval 92.2%

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

   6. Simplified 92.2%

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

      1. fma-udef 92.2%

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

      2. associate-+r+ 92.2%

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

   8. Applied egg-rr 92.2%

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

      1. sqr-pow 92.1%

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

      2. metadata-eval 92.1%

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

      3. pow2 92.1%

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

      4. metadata-eval 92.1%

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

      5. pow2 92.1%

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

      6. distribute-rgt-out 92.1%

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

   10. Applied egg-rr 92.1%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 7: 94.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+23}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+19}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+23)
   (pow a 4.0)
   (if (<= a 7.5e+19) (+ -1.0 (* (* b b) (+ (* b b) 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+23) {
		tmp = pow(a, 4.0);
	} else if (a <= 7.5e+19) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5d+23)) then
        tmp = a ** 4.0d0
    else if (a <= 7.5d+19) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+23) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 7.5e+19) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+23:
		tmp = math.pow(a, 4.0)
	elif a <= 7.5e+19:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+23)
		tmp = a ^ 4.0;
	elseif (a <= 7.5e+19)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+23)
		tmp = a ^ 4.0;
	elseif (a <= 7.5e+19)
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+23], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 7.5e+19], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.9999999999999999e23 or 7.5e19 < a

   1. Initial program 47.0%

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

      1. associate--l+ 47.0%

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

      2. fma-def 47.0%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in a around inf 92.4%

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

   if -4.9999999999999999e23 < a < 7.5e19

   1. Initial program 99.1%

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

      1. associate--l+ 99.1%

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

      2. fma-def 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 97.3%

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

      1. associate--l+ 97.3%

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

      2. fma-def 97.3%

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

      3. unpow2 97.3%

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

      4. sub-neg 97.3%

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

      5. metadata-eval 97.3%

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

   6. Simplified 97.3%

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

      1. fma-udef 97.3%

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

      2. associate-+r+ 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. sqr-pow 97.2%

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

      2. metadata-eval 97.2%

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

      3. pow2 97.2%

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

      4. metadata-eval 97.2%

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

      5. pow2 97.2%

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

      6. distribute-rgt-out 97.2%

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

   10. Applied egg-rr 97.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+23}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+19}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 8: 84.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.3 \cdot 10^{+153} \lor \neg \left(a \leq 6.8 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -6.3e+153) (not (<= a 6.8e+153)))
   (* (* a a) 4.0)
   (+ -1.0 (* (* b b) (+ (* b b) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -6.3e+153) || !(a <= 6.8e+153)) {
		tmp = (a * a) * 4.0;
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-6.3d+153)) .or. (.not. (a <= 6.8d+153))) then
        tmp = (a * a) * 4.0d0
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -6.3e+153) || !(a <= 6.8e+153)) {
		tmp = (a * a) * 4.0;
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -6.3e+153) or not (a <= 6.8e+153):
		tmp = (a * a) * 4.0
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -6.3e+153) || !(a <= 6.8e+153))
		tmp = Float64(Float64(a * a) * 4.0);
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -6.3e+153) || ~((a <= 6.8e+153)))
		tmp = (a * a) * 4.0;
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -6.3e+153], N[Not[LessEqual[a, 6.8e+153]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.3000000000000001e153 or 6.7999999999999995e153 < a

   1. Initial program 24.6%

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

      1. associate--l+ 24.6%

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

      2. fma-def 24.6%

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

   3. Simplified 24.6%

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

   4. Taylor expanded in b around 0 44.6%

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

      1. associate--l+ 44.6%

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

      2. associate-*r* 44.6%

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

      3. unpow2 44.6%

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

   6. Simplified 44.6%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in a around inf 100.0%

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

       1. +-commutative 100.0%

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

       2. metadata-eval 100.0%

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

       3. pow-sqr 100.0%

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

       4. unpow2 100.0%

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

       5. unpow2 100.0%

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

       6. unpow2 100.0%

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

       7. distribute-rgt-in 100.0%

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

       8. fma-udef 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in a around 0 100.0%

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

       1. unpow2 100.0%

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

   15. Simplified 100.0%

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

   if -6.3000000000000001e153 < a < 6.7999999999999995e153

   1. Initial program 90.9%

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

      1. associate--l+ 90.9%

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

      2. fma-def 90.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in a around 0 78.0%

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

      1. associate--l+ 78.0%

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

      2. fma-def 78.0%

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

      3. unpow2 78.0%

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

      4. sub-neg 78.0%

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

      5. metadata-eval 78.0%

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

   6. Simplified 78.0%

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

      1. fma-udef 78.0%

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

      2. associate-+r+ 78.0%

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

   8. Applied egg-rr 78.0%

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

      1. sqr-pow 77.9%

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

      2. metadata-eval 77.9%

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

      3. pow2 77.9%

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

      4. metadata-eval 77.9%

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

      5. pow2 77.9%

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

      6. distribute-rgt-out 77.9%

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

   10. Applied egg-rr 77.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.3 \cdot 10^{+153} \lor \neg \left(a \leq 6.8 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 9: 68.9% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+152} \lor \neg \left(a \leq 6.8 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.35e+152) (not (<= a 6.8e+153)))
   (* (* a a) 4.0)
   (+ -1.0 (* b (* b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.35e+152) || !(a <= 6.8e+153)) {
		tmp = (a * a) * 4.0;
	} else {
		tmp = -1.0 + (b * (b * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.35d+152)) .or. (.not. (a <= 6.8d+153))) then
        tmp = (a * a) * 4.0d0
    else
        tmp = (-1.0d0) + (b * (b * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.35e+152) || !(a <= 6.8e+153)) {
		tmp = (a * a) * 4.0;
	} else {
		tmp = -1.0 + (b * (b * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.35e+152) or not (a <= 6.8e+153):
		tmp = (a * a) * 4.0
	else:
		tmp = -1.0 + (b * (b * 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.35e+152) || !(a <= 6.8e+153))
		tmp = Float64(Float64(a * a) * 4.0);
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.35e+152) || ~((a <= 6.8e+153)))
		tmp = (a * a) * 4.0;
	else
		tmp = -1.0 + (b * (b * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.35e+152], N[Not[LessEqual[a, 6.8e+153]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35000000000000007e152 or 6.7999999999999995e153 < a

   1. Initial program 23.9%

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

      1. associate--l+ 23.9%

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

      2. fma-def 23.9%

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

   3. Simplified 25.4%

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

   4. Taylor expanded in b around 0 43.3%

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

      1. associate--l+ 43.3%

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

      2. associate-*r* 43.3%

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

      3. unpow2 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in a around inf 100.0%

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

       1. +-commutative 100.0%

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

       2. metadata-eval 100.0%

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

       3. pow-sqr 100.0%

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

       4. unpow2 100.0%

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

       5. unpow2 100.0%

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

       6. unpow2 100.0%

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

       7. distribute-rgt-in 100.0%

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

       8. fma-udef 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in a around 0 97.4%

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

       1. unpow2 97.4%

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

   15. Simplified 97.4%

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

   if -1.35000000000000007e152 < a < 6.7999999999999995e153

   1. Initial program 91.9%

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

      1. associate--l+ 91.9%

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

      2. fma-def 91.9%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in a around 0 78.3%

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

      1. associate--l+ 78.3%

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

      2. fma-def 78.3%

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

      3. unpow2 78.3%

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

      4. sub-neg 78.3%

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

      5. metadata-eval 78.3%

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

   6. Simplified 78.3%

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

      1. fma-udef 78.3%

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

      2. associate-+r+ 78.3%

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

   8. Applied egg-rr 78.3%

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

   9. Taylor expanded in b around 0 78.3%

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

       1. unpow2 78.3%

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

       2. metadata-eval 78.3%

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

       3. pow-sqr 78.2%

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

       4. unpow2 78.2%

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

       5. unpow2 78.2%

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

       6. distribute-rgt-in 78.2%

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

       7. fma-udef 78.2%

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

       8. associate-*l* 78.2%

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

   11. Simplified 78.2%

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

   12. Taylor expanded in b around 0 52.7%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+152} \lor \neg \left(a \leq 6.8 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 10: 50.5% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00045 \lor \neg \left(a \leq 0.09\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.00045) (not (<= a 0.09))) (* (* a a) 4.0) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.00045) || !(a <= 0.09)) {
		tmp = (a * a) * 4.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-0.00045d0)) .or. (.not. (a <= 0.09d0))) then
        tmp = (a * a) * 4.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -0.00045) || !(a <= 0.09)) {
		tmp = (a * a) * 4.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -0.00045) or not (a <= 0.09):
		tmp = (a * a) * 4.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.00045) || !(a <= 0.09))
		tmp = Float64(Float64(a * a) * 4.0);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -0.00045) || ~((a <= 0.09)))
		tmp = (a * a) * 4.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.00045], N[Not[LessEqual[a, 0.09]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -4.4999999999999999e-4 or 0.089999999999999997 < a

   1. Initial program 50.2%

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

      1. associate--l+ 50.2%

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

      2. fma-def 50.2%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in b around 0 53.3%

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

      1. associate--l+ 53.3%

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

      2. associate-*r* 53.3%

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

      3. unpow2 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in a around 0 86.7%

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

      1. unpow2 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in a around inf 86.8%

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

       1. +-commutative 86.8%

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

       2. metadata-eval 86.8%

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

       3. pow-sqr 86.6%

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

       4. unpow2 86.6%

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

       5. unpow2 86.6%

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

       6. unpow2 86.6%

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

       7. distribute-rgt-in 86.6%

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

       8. fma-udef 86.6%

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

   12. Simplified 86.6%

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

   13. Taylor expanded in a around 0 52.2%

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

       1. unpow2 52.2%

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

   15. Simplified 52.2%

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

   if -4.4999999999999999e-4 < a < 0.089999999999999997

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 43.5%

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

      1. associate--l+ 43.5%

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

      2. associate-*r* 43.5%

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

      3. unpow2 43.5%

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

   6. Simplified 43.5%

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

   7. Taylor expanded in a around 0 43.0%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00045 \lor \neg \left(a \leq 0.09\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11: 24.5% accurate, 130.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.1%

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

   1. associate--l+ 74.1%

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

   2. fma-def 74.1%

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

3. Simplified 74.8%

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

4. Taylor expanded in b around 0 48.6%

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

   1. associate--l+ 48.6%

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

   2. associate-*r* 48.6%

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

   3. unpow2 48.6%

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

6. Simplified 48.6%

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

7. Taylor expanded in a around 0 21.0%

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

8. Final simplification 21.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023227 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))