Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.3% → 97.1%

Time: 7.5s

Alternatives: 6

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(1 + a \cdot -3\right)\right) + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2000000.0)
   (+ (pow a 4.0) -1.0)
   (+
    (* (* b b) (+ (* 2.0 (pow a 2.0)) (* 4.0 (+ 1.0 (* a -3.0)))))
    (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2000000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = ((b * b) * ((2.0 * pow(a, 2.0)) + (4.0 * (1.0 + (a * -3.0))))) + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2000000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = ((b * b) * ((2.0d0 * (a ** 2.0d0)) + (4.0d0 * (1.0d0 + (a * (-3.0d0)))))) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2000000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = ((b * b) * ((2.0 * Math.pow(a, 2.0)) + (4.0 * (1.0 + (a * -3.0))))) + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2000000.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = ((b * b) * ((2.0 * math.pow(a, 2.0)) + (4.0 * (1.0 + (a * -3.0))))) + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2000000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(2.0 * (a ^ 2.0)) + Float64(4.0 * Float64(1.0 + Float64(a * -3.0))))) + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2000000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = ((b * b) * ((2.0 * (a ^ 2.0)) + (4.0 * (1.0 + (a * -3.0))))) + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2000000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e6

   1. Initial program 81.6%

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

      1. associate--l+ 81.6%

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

      2. fma-def 81.6%

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

      3. distribute-rgt-in 81.6%

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

      4. sqr-neg 81.6%

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

      5. distribute-rgt-in 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in b around 0 81.4%

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

      1. add-cbrt-cube 70.9%

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

      2. pow3 70.9%

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

      3. fma-def 70.9%

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

      4. distribute-rgt-in 70.9%

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

      5. *-un-lft-identity 70.9%

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

      6. unpow2 70.9%

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

      7. unpow2 70.9%

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

      8. cube-mult 70.9%

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

      9. fma-udef 70.9%

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in a around inf 96.7%

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

   if 2e6 < (*.f64 b b)

   1. Initial program 66.0%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in b around inf 98.5%

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

      1. unpow2 98.5%

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

   6. Applied egg-rr 98.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(1 + a \cdot -3\right)\right) + {b}^{4}\\ \end{array} \]

Alternative 2: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 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 ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

   3. Simplified 10.4%

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

   4. Taylor expanded in b around 0 23.1%

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

      1. add-cbrt-cube 32.9%

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

      2. pow3 32.9%

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

      3. fma-def 32.9%

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

      4. distribute-rgt-in 32.9%

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

      5. *-un-lft-identity 32.9%

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

      6. unpow2 32.9%

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

      7. unpow2 32.9%

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

      8. cube-mult 32.9%

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

      9. fma-udef 32.9%

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

   6. Applied egg-rr 32.9%

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

   7. Taylor expanded in a around inf 87.3%

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

4. Final simplification 96.5%

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

Alternative 3: 82.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 10000.0) (+ (pow a 4.0) -1.0) (+ (pow b 4.0) (* (* b b) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 10000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0) + ((b * b) * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 10000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + ((b * b) * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 10000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + ((b * b) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 10000.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + ((b * b) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 10000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 10000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + ((b * b) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 10000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1e4

   1. Initial program 76.0%

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

      1. associate--l+ 76.0%

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

      2. fma-def 76.0%

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

      3. distribute-rgt-in 76.0%

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

      4. sqr-neg 76.0%

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

      5. distribute-rgt-in 76.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in b around 0 58.7%

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

      1. add-cbrt-cube 55.3%

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

      2. pow3 55.3%

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

      3. fma-def 55.3%

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

      4. distribute-rgt-in 55.3%

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

      5. *-un-lft-identity 55.3%

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

      6. unpow2 55.3%

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

      7. unpow2 55.3%

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

      8. cube-mult 55.3%

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

      9. fma-udef 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in a around inf 73.4%

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

   if 1e4 < b

   1. Initial program 66.5%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. unpow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 87.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot 4\\ \end{array} \]

Alternative 4: 82.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4700:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4700.0) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4700.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = 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 <= 4700.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4700.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4700.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4700.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4700.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4700.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4700

   1. Initial program 76.0%

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

      1. associate--l+ 76.0%

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

      2. fma-def 76.0%

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

      3. distribute-rgt-in 76.0%

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

      4. sqr-neg 76.0%

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

      5. distribute-rgt-in 76.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in b around 0 58.7%

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

      1. add-cbrt-cube 55.3%

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

      2. pow3 55.3%

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

      3. fma-def 55.3%

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

      4. distribute-rgt-in 55.3%

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

      5. *-un-lft-identity 55.3%

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

      6. unpow2 55.3%

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

      7. unpow2 55.3%

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

      8. cube-mult 55.3%

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

      9. fma-udef 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in a around inf 73.4%

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

   if 4700 < b

   1. Initial program 66.5%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in b around inf 86.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4700:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 58.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 19000:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 19000.0) (pow a 4.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 19000.0) {
		tmp = pow(a, 4.0);
	} else {
		tmp = 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 <= 19000.0d0) then
        tmp = a ** 4.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 19000.0) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 19000.0:
		tmp = math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 19000.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 19000.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 19000.0], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 19000

   1. Initial program 76.0%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in a around inf 43.7%

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

   if 19000 < b

   1. Initial program 66.5%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in b around inf 86.9%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 19000:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6: 46.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (pow a 4.0))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b)
	return a ^ 4.0
end

MATLAB

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

Wolfram

code[a_, b_] := N[Power[a, 4.0], $MachinePrecision]

Derivation

1. Initial program 73.7%

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

3. Simplified 73.8%

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

4. Taylor expanded in a around inf 42.7%

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

5. Final simplification 42.7%

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

Reproduce

herbie shell --seed 2023321 
(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))