Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.2% → 99.8%

Time: 10.2s

Alternatives: 6

Speedup: 1.1×

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

Specification

Math

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

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: 74.2% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + {\left(\sqrt[3]{{b}^{2}}\right)}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.02e+74)
   (+ (pow a 4.0) -1.0)
   (+
    -1.0
    (+
     (pow (+ (* a a) (* b b)) 2.0)
     (* 4.0 (+ (* (* a a) (+ a 1.0)) (pow (cbrt (pow b 2.0)) 3.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.02e+74) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + pow(cbrt(pow(b, 2.0)), 3.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.02e+74) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + Math.pow(Math.cbrt(Math.pow(b, 2.0)), 3.0))));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.02e+74)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(-1.0 + Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + (cbrt((b ^ 2.0)) ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.02e+74], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + 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[Power[N[Power[N[Power[b, 2.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.02000000000000005e74

   1. Initial program 6.1%

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

      1. sub-neg 6.1%

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

      2. +-commutative 6.1%

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

      3. fma-define 16.3%

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

      4. +-commutative 16.3%

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

      5. associate-*l* 16.3%

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

      6. cancel-sign-sub-inv 16.3%

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

      7. metadata-eval 16.3%

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

      8. fma-define 16.3%

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

      9. metadata-eval 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in a around inf 100.0%

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

   if -1.02000000000000005e74 < a

   1. Initial program 85.9%

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

      1. add-cube-cbrt 85.9%

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

      2. pow3 85.9%

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

      3. pow2 85.9%

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

      4. cancel-sign-sub-inv 85.9%

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

      5. metadata-eval 85.9%

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

      6. +-commutative 85.9%

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

      7. *-commutative 85.9%

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

      8. fma-undefine 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in a around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + {\left(\sqrt[3]{{b}^{2}}\right)}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.02e+74)
   (+ (pow a 4.0) -1.0)
   (+
    -1.0
    (+
     (pow (+ (* a a) (* b b)) 2.0)
     (* 4.0 (+ (* b b) (* (* a a) (+ a 1.0))))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.02e+74) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + (pow(((a * a) + (b * b)), 2.0) + (4.0 * ((b * b) + ((a * a) * (a + 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 <= (-1.02d+74)) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (-1.0d0) + ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * ((b * b) + ((a * a) * (a + 1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.02e+74) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((b * b) + ((a * a) * (a + 1.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.02e+74:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = -1.0 + (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((b * b) + ((a * a) * (a + 1.0)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.02e+74)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(-1.0 + Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(b * b) + Float64(Float64(a * a) * Float64(a + 1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.02e+74)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = -1.0 + ((((a * a) + (b * b)) ^ 2.0) + (4.0 * ((b * b) + ((a * a) * (a + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.02e+74], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.02000000000000005e74

   1. Initial program 6.1%

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

      1. sub-neg 6.1%

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

      2. +-commutative 6.1%

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

      3. fma-define 16.3%

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

      4. +-commutative 16.3%

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

      5. associate-*l* 16.3%

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

      6. cancel-sign-sub-inv 16.3%

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

      7. metadata-eval 16.3%

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

      8. fma-define 16.3%

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

      9. metadata-eval 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in a around inf 100.0%

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

   if -1.02000000000000005e74 < a

   1. Initial program 85.9%

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

      1. add-cube-cbrt 85.9%

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

      2. pow3 85.9%

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

      3. pow2 85.9%

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

      4. cancel-sign-sub-inv 85.9%

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

      5. metadata-eval 85.9%

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

      6. +-commutative 85.9%

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

      7. *-commutative 85.9%

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

      8. fma-undefine 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in a around 0 99.9%

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

      1. pow2 99.9%

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

      2. rem-cube-cbrt 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7000000 \lor \neg \left(b \leq 3 \cdot 10^{+39}\right) \land b \leq 3.2 \cdot 10^{+69}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b 7000000.0) (and (not (<= b 3e+39)) (<= b 3.2e+69)))
   (+ (pow a 4.0) -1.0)
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= 7000000.0) || (!(b <= 3e+39) && (b <= 3.2e+69))) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= 7000000.0d0) .or. (.not. (b <= 3d+39)) .and. (b <= 3.2d+69)) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= 7000000.0) || (!(b <= 3e+39) && (b <= 3.2e+69))) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= 7000000.0) or (not (b <= 3e+39) and (b <= 3.2e+69)):
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= 7000000.0) || (!(b <= 3e+39) && (b <= 3.2e+69)))
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= 7000000.0) || (~((b <= 3e+39)) && (b <= 3.2e+69)))
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, 7000000.0], And[N[Not[LessEqual[b, 3e+39]], $MachinePrecision], LessEqual[b, 3.2e+69]]], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7e6 or 3e39 < b < 3.19999999999999985e69

   1. Initial program 75.0%

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

      1. sub-neg 75.0%

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

      2. +-commutative 75.0%

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

      3. fma-define 76.5%

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

      4. +-commutative 76.5%

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

      5. associate-*l* 76.5%

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

      6. cancel-sign-sub-inv 76.5%

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

      7. metadata-eval 76.5%

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

      8. fma-define 76.5%

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

      9. metadata-eval 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in a around inf 79.1%

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

   if 7e6 < b < 3e39 or 3.19999999999999985e69 < b

   1. Initial program 55.8%

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

      1. sub-neg 55.8%

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

      2. +-commutative 55.8%

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

      3. fma-define 59.2%

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

      4. +-commutative 59.2%

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

      5. associate-*l* 59.2%

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

      6. cancel-sign-sub-inv 59.2%

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

      7. metadata-eval 59.2%

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

      8. fma-define 59.2%

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

      9. metadata-eval 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in b around inf 100.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7000000 \lor \neg \left(b \leq 3 \cdot 10^{+39}\right) \land b \leq 3.2 \cdot 10^{+69}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10500000:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+40} \lor \neg \left(b \leq 9 \cdot 10^{+69}\right):\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 10500000.0)
   (+ -1.0 (* (pow a 3.0) (+ a 4.0)))
   (if (or (<= b 3.3e+40) (not (<= b 9e+69)))
     (+ -1.0 (pow b 4.0))
     (+ (pow a 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 10500000.0) {
		tmp = -1.0 + (pow(a, 3.0) * (a + 4.0));
	} else if ((b <= 3.3e+40) || !(b <= 9e+69)) {
		tmp = -1.0 + pow(b, 4.0);
	} else {
		tmp = pow(a, 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 <= 10500000.0d0) then
        tmp = (-1.0d0) + ((a ** 3.0d0) * (a + 4.0d0))
    else if ((b <= 3.3d+40) .or. (.not. (b <= 9d+69))) then
        tmp = (-1.0d0) + (b ** 4.0d0)
    else
        tmp = (a ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 10500000.0) {
		tmp = -1.0 + (Math.pow(a, 3.0) * (a + 4.0));
	} else if ((b <= 3.3e+40) || !(b <= 9e+69)) {
		tmp = -1.0 + Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 10500000.0:
		tmp = -1.0 + (math.pow(a, 3.0) * (a + 4.0))
	elif (b <= 3.3e+40) or not (b <= 9e+69):
		tmp = -1.0 + math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 10500000.0)
		tmp = Float64(-1.0 + Float64((a ^ 3.0) * Float64(a + 4.0)));
	elseif ((b <= 3.3e+40) || !(b <= 9e+69))
		tmp = Float64(-1.0 + (b ^ 4.0));
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 10500000.0)
		tmp = -1.0 + ((a ^ 3.0) * (a + 4.0));
	elseif ((b <= 3.3e+40) || ~((b <= 9e+69)))
		tmp = -1.0 + (b ^ 4.0);
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 10500000.0], N[(-1.0 + N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 3.3e+40], N[Not[LessEqual[b, 9e+69]], $MachinePrecision]], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.05e7

   1. Initial program 75.2%

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

      1. sub-neg 75.2%

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

      2. +-commutative 75.2%

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

      3. fma-define 76.7%

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

      4. +-commutative 76.7%

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

      5. associate-*l* 76.7%

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

      6. cancel-sign-sub-inv 76.7%

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

      7. metadata-eval 76.7%

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

      8. fma-define 76.7%

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

      9. metadata-eval 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in a around inf 79.4%

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

      1. associate-*r/ 79.4%

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

      2. metadata-eval 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in a around 0 79.3%

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

   if 1.05e7 < b < 3.2999999999999998e40 or 8.9999999999999999e69 < b

   1. Initial program 55.8%

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

      1. sub-neg 55.8%

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

      2. +-commutative 55.8%

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

      3. fma-define 59.2%

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

      4. +-commutative 59.2%

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

      5. associate-*l* 59.2%

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

      6. cancel-sign-sub-inv 59.2%

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

      7. metadata-eval 59.2%

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

      8. fma-define 59.2%

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

      9. metadata-eval 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 3.2999999999999998e40 < b < 8.9999999999999999e69

   1. Initial program 66.1%

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

      1. sub-neg 66.1%

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

      2. +-commutative 66.1%

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

      3. fma-define 66.1%

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

      4. +-commutative 66.1%

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

      5. associate-*l* 66.1%

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

      6. cancel-sign-sub-inv 66.1%

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

      7. metadata-eval 66.1%

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

      8. fma-define 66.1%

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

      9. metadata-eval 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in a around inf 66.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10500000:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+40} \lor \neg \left(b \leq 9 \cdot 10^{+69}\right):\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 33000000:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+40} \lor \neg \left(b \leq 8 \cdot 10^{+69}\right):\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 33000000.0)
   (+ -1.0 (* (pow a 4.0) (+ 1.0 (/ 4.0 a))))
   (if (or (<= b 2.8e+40) (not (<= b 8e+69)))
     (+ -1.0 (pow b 4.0))
     (+ (pow a 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 33000000.0) {
		tmp = -1.0 + (pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else if ((b <= 2.8e+40) || !(b <= 8e+69)) {
		tmp = -1.0 + pow(b, 4.0);
	} else {
		tmp = pow(a, 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 <= 33000000.0d0) then
        tmp = (-1.0d0) + ((a ** 4.0d0) * (1.0d0 + (4.0d0 / a)))
    else if ((b <= 2.8d+40) .or. (.not. (b <= 8d+69))) then
        tmp = (-1.0d0) + (b ** 4.0d0)
    else
        tmp = (a ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 33000000.0) {
		tmp = -1.0 + (Math.pow(a, 4.0) * (1.0 + (4.0 / a)));
	} else if ((b <= 2.8e+40) || !(b <= 8e+69)) {
		tmp = -1.0 + Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 33000000.0:
		tmp = -1.0 + (math.pow(a, 4.0) * (1.0 + (4.0 / a)))
	elif (b <= 2.8e+40) or not (b <= 8e+69):
		tmp = -1.0 + math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 33000000.0)
		tmp = Float64(-1.0 + Float64((a ^ 4.0) * Float64(1.0 + Float64(4.0 / a))));
	elseif ((b <= 2.8e+40) || !(b <= 8e+69))
		tmp = Float64(-1.0 + (b ^ 4.0));
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 33000000.0)
		tmp = -1.0 + ((a ^ 4.0) * (1.0 + (4.0 / a)));
	elseif ((b <= 2.8e+40) || ~((b <= 8e+69)))
		tmp = -1.0 + (b ^ 4.0);
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 33000000.0], N[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 2.8e+40], N[Not[LessEqual[b, 8e+69]], $MachinePrecision]], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.3e7

   1. Initial program 75.2%

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

      1. sub-neg 75.2%

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

      2. +-commutative 75.2%

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

      3. fma-define 76.7%

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

      4. +-commutative 76.7%

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

      5. associate-*l* 76.7%

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

      6. cancel-sign-sub-inv 76.7%

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

      7. metadata-eval 76.7%

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

      8. fma-define 76.7%

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

      9. metadata-eval 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in a around inf 79.4%

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

      1. associate-*r/ 79.4%

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

      2. metadata-eval 79.4%

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

   7. Simplified 79.4%

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

   if 3.3e7 < b < 2.8000000000000001e40 or 8.0000000000000006e69 < b

   1. Initial program 55.8%

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

      1. sub-neg 55.8%

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

      2. +-commutative 55.8%

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

      3. fma-define 59.2%

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

      4. +-commutative 59.2%

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

      5. associate-*l* 59.2%

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

      6. cancel-sign-sub-inv 59.2%

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

      7. metadata-eval 59.2%

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

      8. fma-define 59.2%

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

      9. metadata-eval 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 2.8000000000000001e40 < b < 8.0000000000000006e69

   1. Initial program 66.1%

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

      1. sub-neg 66.1%

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

      2. +-commutative 66.1%

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

      3. fma-define 66.1%

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

      4. +-commutative 66.1%

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

      5. associate-*l* 66.1%

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

      6. cancel-sign-sub-inv 66.1%

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

      7. metadata-eval 66.1%

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

      8. fma-define 66.1%

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

      9. metadata-eval 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in a around inf 66.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 33000000:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+40} \lor \neg \left(b \leq 8 \cdot 10^{+69}\right):\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. sub-neg 70.6%

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

   2. +-commutative 70.6%

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

   3. fma-define 72.6%

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

   4. +-commutative 72.6%

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

   5. associate-*l* 72.6%

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

   6. cancel-sign-sub-inv 72.6%

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

   7. metadata-eval 72.6%

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

   8. fma-define 72.6%

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

   9. metadata-eval 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in a around inf 69.7%

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

6. Final simplification 69.7%

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

Reproduce

herbie shell --seed 2024080 
(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))