Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.5% → 98.2%

Time: 8.4s

Alternatives: 7

Speedup: 1.2×

• 60.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(-12, a \cdot {b}^{2}, \mathsf{fma}\left(4, \mathsf{fma}\left(b, b, {a}^{2}\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1.05e+139)
   (+
    (pow (hypot a b) 4.0)
    (fma -12.0 (* a (pow b 2.0)) (fma 4.0 (fma b b (pow a 2.0)) -1.0)))
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1.05e+139) {
		tmp = pow(hypot(a, b), 4.0) + fma(-12.0, (a * pow(b, 2.0)), fma(4.0, fma(b, b, pow(a, 2.0)), -1.0));
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1.05e+139)
		tmp = Float64((hypot(a, b) ^ 4.0) + fma(-12.0, Float64(a * (b ^ 2.0)), fma(4.0, fma(b, b, (a ^ 2.0)), -1.0)));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1.05e+139], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(-12.0 * N[(a * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(b * b + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.0499999999999999e139

   1. Initial program 77.1%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in a around 0 97.4%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \color{blue}{\left(\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {a}^{2} + 4 \cdot {b}^{2}\right)\right) - 1\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 97.4%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(\left(4 \cdot {a}^{2} + 4 \cdot {b}^{2}\right) - 1\right)\right)} \]

      2. fma-def 97.4%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \color{blue}{\mathsf{fma}\left(-12, a \cdot {b}^{2}, \left(4 \cdot {a}^{2} + 4 \cdot {b}^{2}\right) - 1\right)} \]

      3. *-commutative 97.4%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(-12, \color{blue}{{b}^{2} \cdot a}, \left(4 \cdot {a}^{2} + 4 \cdot {b}^{2}\right) - 1\right) \]

      4. distribute-lft-out 97.4%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(-12, {b}^{2} \cdot a, \color{blue}{4 \cdot \left({a}^{2} + {b}^{2}\right)} - 1\right) \]

      5. fma-neg 97.4%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(-12, {b}^{2} \cdot a, \color{blue}{\mathsf{fma}\left(4, {a}^{2} + {b}^{2}, -1\right)}\right) \]

      6. +-commutative 97.4%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(-12, {b}^{2} \cdot a, \mathsf{fma}\left(4, \color{blue}{{b}^{2} + {a}^{2}}, -1\right)\right) \]

      7. unpow2 97.4%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(-12, {b}^{2} \cdot a, \mathsf{fma}\left(4, \color{blue}{b \cdot b} + {a}^{2}, -1\right)\right) \]

      8. fma-def 97.4%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(-12, {b}^{2} \cdot a, \mathsf{fma}\left(4, \color{blue}{\mathsf{fma}\left(b, b, {a}^{2}\right)}, -1\right)\right) \]

      9. metadata-eval 97.4%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(-12, {b}^{2} \cdot a, \mathsf{fma}\left(4, \mathsf{fma}\left(b, b, {a}^{2}\right), \color{blue}{-1}\right)\right) \]

   6. Simplified 97.4%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \color{blue}{\mathsf{fma}\left(-12, {b}^{2} \cdot a, \mathsf{fma}\left(4, \mathsf{fma}\left(b, b, {a}^{2}\right), -1\right)\right)} \]

   if 1.0499999999999999e139 < (*.f64 b b)

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in b around inf 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(-12, a \cdot {b}^{2}, \mathsf{fma}\left(4, \mathsf{fma}\left(b, b, {a}^{2}\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \left(4 \cdot {a}^{2} + \left(4 \cdot \left({b}^{2} \cdot \left(1 + a \cdot -3\right)\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1.05e+139)
   (+
    (pow (hypot a b) 4.0)
    (+
     (* 4.0 (pow a 2.0))
     (+ (* 4.0 (* (pow b 2.0) (+ 1.0 (* a -3.0)))) -1.0)))
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1.05e+139) {
		tmp = pow(hypot(a, b), 4.0) + ((4.0 * pow(a, 2.0)) + ((4.0 * (pow(b, 2.0) * (1.0 + (a * -3.0)))) + -1.0));
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1.05e+139) {
		tmp = Math.pow(Math.hypot(a, b), 4.0) + ((4.0 * Math.pow(a, 2.0)) + ((4.0 * (Math.pow(b, 2.0) * (1.0 + (a * -3.0)))) + -1.0));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1.05e+139:
		tmp = math.pow(math.hypot(a, b), 4.0) + ((4.0 * math.pow(a, 2.0)) + ((4.0 * (math.pow(b, 2.0) * (1.0 + (a * -3.0)))) + -1.0))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1.05e+139)
		tmp = Float64((hypot(a, b) ^ 4.0) + Float64(Float64(4.0 * (a ^ 2.0)) + Float64(Float64(4.0 * Float64((b ^ 2.0) * Float64(1.0 + Float64(a * -3.0)))) + -1.0)));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1.05e+139)
		tmp = (hypot(a, b) ^ 4.0) + ((4.0 * (a ^ 2.0)) + ((4.0 * ((b ^ 2.0) * (1.0 + (a * -3.0)))) + -1.0));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1.05e+139], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(4.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(N[Power[b, 2.0], $MachinePrecision] * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.0499999999999999e139

   1. Initial program 77.1%

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

   3. Simplified 77.2%

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

      1. fma-udef 77.2%

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

      2. fma-udef 77.2%

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

      3. *-commutative 77.2%

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

      4. +-commutative 77.2%

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

      5. metadata-eval 77.2%

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

      6. cancel-sign-sub-inv 77.2%

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

      7. associate-*l* 77.2%

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

      8. fma-udef 77.2%

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

      9. *-un-lft-identity 77.2%

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

      10. cube-mult 77.2%

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

      11. distribute-rgt-in 77.2%

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

      12. +-commutative 77.2%

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

      13. metadata-eval 77.2%

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

      14. fma-neg 77.2%

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

      15. distribute-rgt-in 77.2%

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

      16. associate--l+ 77.3%

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

   5. Applied egg-rr 77.3%

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

   6. Taylor expanded in a around 0 97.4%

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

   7. Taylor expanded in b around 0 97.4%

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

   if 1.0499999999999999e139 < (*.f64 b b)

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in b around inf 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \left(4 \cdot {a}^{2} + \left(4 \cdot \left({b}^{2} \cdot \left(1 + a \cdot -3\right)\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.5× speedup

Math

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

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))))

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);
	}
	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);
	}
	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)
	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 = a ^ 4.0;
	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;
	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[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in a around inf 93.9%

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

4. Final simplification 98.0%

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

Alternative 4: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1150000000 \lor \neg \left(a \leq 1.16 \cdot 10^{+21}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {b}^{2} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1150000000.0) (not (<= a 1.16e+21)))
   (pow a 4.0)
   (+ (* 4.0 (pow b 2.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1150000000.0) || !(a <= 1.16e+21)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (4.0 * pow(b, 2.0)) + -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 ((a <= (-1150000000.0d0)) .or. (.not. (a <= 1.16d+21))) then
        tmp = a ** 4.0d0
    else
        tmp = (4.0d0 * (b ** 2.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1150000000.0) || !(a <= 1.16e+21)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (4.0 * Math.pow(b, 2.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1150000000.0) or not (a <= 1.16e+21):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (4.0 * math.pow(b, 2.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1150000000.0) || !(a <= 1.16e+21))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(4.0 * (b ^ 2.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1150000000.0) || ~((a <= 1.16e+21)))
		tmp = a ^ 4.0;
	else
		tmp = (4.0 * (b ^ 2.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1150000000.0], N[Not[LessEqual[a, 1.16e+21]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(4.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15e9 or 1.16e21 < a

   1. Initial program 43.1%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in a around inf 93.4%

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

   if -1.15e9 < a < 1.16e21

   1. Initial program 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in a around 0 97.9%

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

   5. Taylor expanded in b around 0 72.5%

      \[\leadsto \color{blue}{4 \cdot {b}^{2}} - 1 \]
   6. Step-by-step derivation

      1. *-commutative 72.5%

         \[\leadsto \color{blue}{{b}^{2} \cdot 4} - 1 \]

   7. Simplified 72.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1150000000 \lor \neg \left(a \leq 1.16 \cdot 10^{+21}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {b}^{2} + -1\\ \end{array} \]

Alternative 5: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.5) (not (<= a 0.41))) (pow a 4.0) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.5) || !(a <= 0.41)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-2.5d0)) .or. (.not. (a <= 0.41d0))) then
        tmp = a ** 4.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.5) || !(a <= 0.41)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.5) or not (a <= 0.41):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.5) || !(a <= 0.41))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.5) || ~((a <= 0.41)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.5], N[Not[LessEqual[a, 0.41]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -2.5 or 0.409999999999999976 < a

   1. Initial program 43.0%

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

   3. Simplified 43.1%

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

   4. Taylor expanded in a around inf 90.3%

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

   if -2.5 < a < 0.409999999999999976

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 98.6%

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

   5. Taylor expanded in b around 0 45.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 56.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1620000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1620000000.0) (pow a 4.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1620000000.0) {
		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 <= 1620000000.0d0) 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 <= 1620000000.0) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1620000000.0:
		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 <= 1620000000.0)
		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 <= 1620000000.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1620000000.0], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.62e9

   1. Initial program 72.0%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in a around inf 53.5%

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

   if 1.62e9 < b

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in b around inf 95.7%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1620000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 7: 25.4% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 69.4%

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

3. Simplified 69.5%

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

4. Taylor expanded in a around 0 67.4%

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

5. Taylor expanded in b around 0 21.3%

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

6. Final simplification 21.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023299 
(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))