Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.8% → 98.2%

Time: 9.4s

Alternatives: 8

Speedup: 1.2×

• 61.9% 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: 73.8% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. fma-define 3.0%

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

      4. +-commutative 3.0%

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

      5. associate-*l* 3.0%

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

      6. cancel-sign-sub-inv 3.0%

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

      7. metadata-eval 3.0%

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

      8. fma-define 3.0%

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

      9. metadata-eval 3.0%

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

   3. Simplified 3.0%

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

   5. Taylor expanded in b around 0 33.9%

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

   6. Taylor expanded in a around 0 93.0%

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

      1. fma-neg 93.0%

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

      2. metadata-eval 93.0%

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

   8. Simplified 93.0%

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

   9. Taylor expanded in a around inf 93.0%

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

       1. associate-*r/ 93.0%

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

       2. metadata-eval 93.0%

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

   11. Simplified 93.0%

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

   12. Taylor expanded in a around 0 93.0%

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

4. Final simplification 98.1%

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

Alternative 2: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;{a}^{2} \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.45e+15)
   (+ (* (pow a 2.0) (+ 4.0 (* a (+ a 4.0)))) -1.0)
   (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.45e+15) {
		tmp = (pow(a, 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = pow(b, 4.0) + -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 (b <= 1.45d+15) then
        tmp = ((a ** 2.0d0) * (4.0d0 + (a * (a + 4.0d0)))) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.45e+15) {
		tmp = (Math.pow(a, 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.45e+15:
		tmp = (math.pow(a, 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.45e+15)
		tmp = Float64(Float64((a ^ 2.0) * Float64(4.0 + Float64(a * Float64(a + 4.0)))) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.45e+15)
		tmp = ((a ^ 2.0) * (4.0 + (a * (a + 4.0)))) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.45e+15], N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45e15

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. +-commutative 76.9%

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

      3. fma-define 77.4%

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

      4. +-commutative 77.4%

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

      5. associate-*l* 77.4%

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

      6. cancel-sign-sub-inv 77.4%

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

      7. metadata-eval 77.4%

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

      8. fma-define 77.4%

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

      9. metadata-eval 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in b around 0 62.1%

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

   6. Taylor expanded in a around 0 77.8%

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

   if 1.45e15 < b

   1. Initial program 64.9%

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

      1. sub-neg 64.9%

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

      2. +-commutative 64.9%

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

      3. fma-define 66.6%

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

      4. +-commutative 66.6%

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

      5. associate-*l* 66.6%

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

      6. cancel-sign-sub-inv 66.6%

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

      7. metadata-eval 66.6%

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

      8. fma-define 66.6%

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

      9. metadata-eval 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in b around inf 93.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;{a}^{2} \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.2e+15)
   (+ -1.0 (* (pow a 4.0) (+ 1.0 (/ 4.0 a))))
   (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.2e+15) {
		tmp = -1.0 + (pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else {
		tmp = pow(b, 4.0) + -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 (b <= 1.2d+15) then
        tmp = (-1.0d0) + ((a ** 4.0d0) * (1.0d0 + (4.0d0 / a)))
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.2e+15) {
		tmp = -1.0 + (Math.pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.2e+15:
		tmp = -1.0 + (math.pow(a, 4.0) * (1.0 + (4.0 / a)))
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.2e+15)
		tmp = Float64(-1.0 + Float64((a ^ 4.0) * Float64(1.0 + Float64(4.0 / a))));
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.2e+15)
		tmp = -1.0 + ((a ^ 4.0) * (1.0 + (4.0 / a)));
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.2e+15], N[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2e15

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. +-commutative 76.9%

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

      3. fma-define 77.4%

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

      4. +-commutative 77.4%

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

      5. associate-*l* 77.4%

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

      6. cancel-sign-sub-inv 77.4%

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

      7. metadata-eval 77.4%

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

      8. fma-define 77.4%

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

      9. metadata-eval 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in a around inf 76.1%

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

      1. associate-*r/ 76.1%

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

      2. metadata-eval 76.1%

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

   7. Simplified 76.1%

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

   if 1.2e15 < b

   1. Initial program 64.9%

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

      1. sub-neg 64.9%

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

      2. +-commutative 64.9%

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

      3. fma-define 66.6%

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

      4. +-commutative 66.6%

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

      5. associate-*l* 66.6%

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

      6. cancel-sign-sub-inv 66.6%

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

      7. metadata-eval 66.6%

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

      8. fma-define 66.6%

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

      9. metadata-eval 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in b around inf 93.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+14}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.2e+14) (+ (* (pow a 3.0) (+ a 4.0)) -1.0) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.2e+14) {
		tmp = (pow(a, 3.0) * (a + 4.0)) + -1.0;
	} else {
		tmp = pow(b, 4.0) + -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 (b <= 6.2d+14) then
        tmp = ((a ** 3.0d0) * (a + 4.0d0)) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.2e+14) {
		tmp = (Math.pow(a, 3.0) * (a + 4.0)) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.2e+14:
		tmp = (math.pow(a, 3.0) * (a + 4.0)) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.2e+14)
		tmp = Float64(Float64((a ^ 3.0) * Float64(a + 4.0)) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.2e+14)
		tmp = ((a ^ 3.0) * (a + 4.0)) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.2e+14], N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.2e14

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. +-commutative 76.9%

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

      3. fma-define 77.4%

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

      4. +-commutative 77.4%

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

      5. associate-*l* 77.4%

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

      6. cancel-sign-sub-inv 77.4%

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

      7. metadata-eval 77.4%

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

      8. fma-define 77.4%

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

      9. metadata-eval 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in a around inf 76.1%

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

      1. associate-*r/ 76.1%

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

      2. metadata-eval 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in a around 0 76.1%

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

   if 6.2e14 < b

   1. Initial program 64.9%

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

      1. sub-neg 64.9%

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

      2. +-commutative 64.9%

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

      3. fma-define 66.6%

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

      4. +-commutative 66.6%

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

      5. associate-*l* 66.6%

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

      6. cancel-sign-sub-inv 66.6%

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

      7. metadata-eval 66.6%

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

      8. fma-define 66.6%

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

      9. metadata-eval 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in b around inf 93.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+14}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.45 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.45) (not (<= a 0.41))) (pow a 4.0) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.45) || !(a <= 0.41)) {
		tmp = pow(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 <= (-2.45d0)) .or. (.not. (a <= 0.41d0))) then
        tmp = 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 <= -2.45) || !(a <= 0.41)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.45) or not (a <= 0.41):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.45) || !(a <= 0.41))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.45) || ~((a <= 0.41)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.45], N[Not[LessEqual[a, 0.41]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -2.4500000000000002 or 0.409999999999999976 < a

   1. Initial program 48.3%

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

      1. sub-neg 48.3%

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

      2. +-commutative 48.3%

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

      3. fma-define 49.9%

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

      4. +-commutative 49.9%

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

      5. associate-*l* 49.9%

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

      6. cancel-sign-sub-inv 49.9%

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

      7. metadata-eval 49.9%

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

      8. fma-define 49.9%

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

      9. metadata-eval 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in b around 0 57.7%

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

   6. Taylor expanded in a around 0 88.0%

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

      1. fma-neg 88.0%

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

      2. metadata-eval 88.0%

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

   8. Simplified 88.0%

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

   9. Taylor expanded in a around inf 85.9%

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

   if -2.4500000000000002 < a < 0.409999999999999976

   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. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. fma-define 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-*l* 99.9%

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

      6. cancel-sign-sub-inv 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 47.2%

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

   6. Taylor expanded in a around 0 45.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.45 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.6e+14) (+ -1.0 (pow a 4.0)) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.6e+14) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0) + -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 (b <= 4.6d+14) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.6e+14) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.6e+14:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.6e+14)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.6e+14)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.6e+14], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.6e14

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. +-commutative 76.9%

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

      3. fma-define 77.4%

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

      4. +-commutative 77.4%

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

      5. associate-*l* 77.4%

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

      6. cancel-sign-sub-inv 77.4%

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

      7. metadata-eval 77.4%

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

      8. fma-define 77.4%

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

      9. metadata-eval 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in a around inf 75.3%

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

   if 4.6e14 < b

   1. Initial program 64.9%

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

      1. sub-neg 64.9%

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

      2. +-commutative 64.9%

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

      3. fma-define 66.6%

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

      4. +-commutative 66.6%

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

      5. associate-*l* 66.6%

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

      6. cancel-sign-sub-inv 66.6%

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

      7. metadata-eval 66.6%

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

      8. fma-define 66.6%

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

      9. metadata-eval 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in b around inf 93.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 1.3× 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 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. sub-neg 74.1%

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

   2. +-commutative 74.1%

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

   3. fma-define 74.9%

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

   4. +-commutative 74.9%

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

   5. associate-*l* 74.9%

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

   6. cancel-sign-sub-inv 74.9%

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

   7. metadata-eval 74.9%

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

   8. fma-define 74.9%

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

   9. metadata-eval 74.9%

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

3. Simplified 74.9%

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

5. Taylor expanded in a around inf 65.7%

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

6. Final simplification 65.7%

   \[\leadsto -1 + {a}^{4} \]
7. Add Preprocessing

Alternative 8: 24.4% 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. sub-neg 74.1%

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

   2. +-commutative 74.1%

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

   3. fma-define 74.9%

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

   4. +-commutative 74.9%

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

   5. associate-*l* 74.9%

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

   6. cancel-sign-sub-inv 74.9%

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

   7. metadata-eval 74.9%

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

   8. fma-define 74.9%

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

   9. metadata-eval 74.9%

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

3. Simplified 74.9%

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

5. Taylor expanded in b around 0 52.4%

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

6. Taylor expanded in a around 0 23.1%

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

7. Final simplification 23.1%

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

Reproduce

herbie shell --seed 2024096 
(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))