Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.5% → 98.1%

Time: 8.6s

Alternatives: 7

Speedup: 1.2×

• 60.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(2 - a\right)\\ \mathbf{if}\;a \leq 5.4 \cdot 10^{+50}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + 4 \cdot \left({a}^{2} - {a}^{3}\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{{t_0}^{4}}, {\left(\sqrt[3]{t_0}\right)}^{2}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (- 2.0 a))))
   (if (<= a 5.4e+50)
     (+
      (+ (pow (fma a a (* b b)) 2.0) (* 4.0 (- (pow a 2.0) (pow a 3.0))))
      -1.0)
     (fma (cbrt (pow t_0 4.0)) (pow (cbrt t_0) 2.0) -1.0))))

C

double code(double a, double b) {
	double t_0 = a * (2.0 - a);
	double tmp;
	if (a <= 5.4e+50) {
		tmp = (pow(fma(a, a, (b * b)), 2.0) + (4.0 * (pow(a, 2.0) - pow(a, 3.0)))) + -1.0;
	} else {
		tmp = fma(cbrt(pow(t_0, 4.0)), pow(cbrt(t_0), 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(2.0 - a))
	tmp = 0.0
	if (a <= 5.4e+50)
		tmp = Float64(Float64((fma(a, a, Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64((a ^ 2.0) - (a ^ 3.0)))) + -1.0);
	else
		tmp = fma(cbrt((t_0 ^ 4.0)), (cbrt(t_0) ^ 2.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(2.0 - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5.4e+50], N[(N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[N[Power[t$95$0, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.4e50

   1. Initial program 86.6%

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

      1. sub-neg 86.6%

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

      2. sqr-neg 86.6%

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

      3. +-commutative 86.6%

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

      4. sqr-neg 86.6%

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

      5. +-commutative 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in a around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

   6. Simplified 99.1%

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

   if 5.4e50 < a

   1. Initial program 18.3%

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

      1. associate--l+ 18.3%

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

      2. sqr-pow 18.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in b around 0 12.2%

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

      1. add-sqr-sqrt 12.2%

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

      2. pow2 12.2%

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

      3. +-commutative 12.2%

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

      4. metadata-eval 12.2%

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

      5. pow-prod-up 12.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. hypot-def 68.3%

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

      8. sqrt-prod 68.3%

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

      9. metadata-eval 68.3%

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

      10. sqrt-prod 68.3%

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

      11. unpow2 68.3%

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

      12. sqrt-prod 68.3%

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

      13. add-sqr-sqrt 68.3%

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

   6. Applied egg-rr 68.3%

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

   7. Taylor expanded in a around 0 93.8%

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

      1. +-commutative 93.8%

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

      2. *-commutative 93.8%

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

      3. mul-1-neg 93.8%

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

      4. unsub-neg 93.8%

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

   9. Simplified 93.8%

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

       1. add-cube-cbrt 93.8%

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

       2. fma-neg 93.8%

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

       3. metadata-eval 93.8%

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

       4. cbrt-unprod 100.0%

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

       5. pow-prod-up 100.0%

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

       6. metadata-eval 100.0%

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

       7. unpow2 100.0%

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

       8. cbrt-prod 100.0%

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

       9. pow2 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(a \cdot 2 - {a}^{2}\right)}^{4}}, {\left(\sqrt[3]{a \cdot 2 - {a}^{2}}\right)}^{2}, -1\right)} \]
   12. Step-by-step derivation

       1. unpow2 100.0%

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

       2. distribute-lft-out-- 100.0%

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

       3. unpow2 100.0%

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

       4. distribute-lft-out-- 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{+50}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + 4 \cdot \left({a}^{2} - {a}^{3}\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{{\left(a \cdot \left(2 - a\right)\right)}^{4}}, {\left(\sqrt[3]{a \cdot \left(2 - a\right)}\right)}^{2}, -1\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + 4 \cdot \left({a}^{2} - {a}^{3}\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2e+76)
   (+ (+ (pow (fma a a (* b b)) 2.0) (* 4.0 (- (pow a 2.0) (pow a 3.0)))) -1.0)
   (+ -1.0 (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2e+76) {
		tmp = (pow(fma(a, a, (b * b)), 2.0) + (4.0 * (pow(a, 2.0) - pow(a, 3.0)))) + -1.0;
	} else {
		tmp = -1.0 + pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2e+76)
		tmp = Float64(Float64((fma(a, a, Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64((a ^ 2.0) - (a ^ 3.0)))) + -1.0);
	else
		tmp = Float64(-1.0 + (a ^ 4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2e+76], N[(N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.0000000000000001e76

   1. Initial program 86.8%

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

      1. sub-neg 86.8%

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

      2. sqr-neg 86.8%

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

      3. +-commutative 86.8%

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

      4. sqr-neg 86.8%

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

      5. +-commutative 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in a around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

   6. Simplified 99.1%

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

   if 2.0000000000000001e76 < a

   1. Initial program 12.5%

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

      1. sub-neg 12.5%

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

      2. sqr-neg 12.5%

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

      3. +-commutative 12.5%

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

      4. sqr-neg 12.5%

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

      5. +-commutative 12.5%

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

   3. Simplified 17.9%

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

   4. Taylor expanded in a around inf 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + 4 \cdot \left({a}^{2} - {a}^{3}\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 3: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	t_0 = math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = -1.0 + t_0
	else:
		tmp = -1.0 + math.pow((a * (2.0 - a)), 2.0)
	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(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(-1.0 + t_0);
	else
		tmp = Float64(-1.0 + (Float64(a * Float64(2.0 - a)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((b * b) + (a * a)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = -1.0 + t_0;
	else
		tmp = -1.0 + ((a * (2.0 - a)) ^ 2.0);
	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[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(-1.0 + t$95$0), $MachinePrecision], N[(-1.0 + N[Power[N[(a * N[(2.0 - a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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 3 a)))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. sqr-pow 0.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in b around 0 27.1%

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

      1. add-sqr-sqrt 27.1%

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

      2. pow2 27.1%

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

      3. +-commutative 27.1%

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

      4. metadata-eval 27.1%

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

      5. pow-prod-up 27.1%

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

      6. add-sqr-sqrt 27.1%

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

      7. hypot-def 81.8%

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

      8. sqrt-prod 81.8%

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

      9. metadata-eval 81.8%

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

      10. sqrt-prod 81.8%

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

      11. unpow2 81.8%

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

      12. sqrt-prod 72.0%

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

      13. add-sqr-sqrt 81.8%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in a around 0 92.5%

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

      1. +-commutative 92.5%

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

      2. *-commutative 92.5%

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

      3. mul-1-neg 92.5%

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

      4. unsub-neg 92.5%

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

   9. Simplified 92.5%

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

   10. Taylor expanded in a around 0 92.5%

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

       1. neg-mul-1 92.5%

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

       2. *-commutative 92.5%

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

       3. +-commutative 92.5%

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

       4. sub-neg 92.5%

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

       5. unpow2 92.5%

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

       6. distribute-lft-out-- 92.5%

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

   12. Simplified 92.5%

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

4. Final simplification 97.7%

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

Alternative 4: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2350000000:\\ \;\;\;\;-1 + {\left(a \cdot \left(2 - a\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2350000000.0)
   (+ -1.0 (pow (* a (- 2.0 a)) 2.0))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2350000000.0) {
		tmp = -1.0 + pow((a * (2.0 - a)), 2.0);
	} 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 <= 2350000000.0d0) then
        tmp = (-1.0d0) + ((a * (2.0d0 - a)) ** 2.0d0)
    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 <= 2350000000.0) {
		tmp = -1.0 + Math.pow((a * (2.0 - a)), 2.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2350000000.0:
		tmp = -1.0 + math.pow((a * (2.0 - a)), 2.0)
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2350000000.0)
		tmp = Float64(-1.0 + (Float64(a * Float64(2.0 - a)) ^ 2.0));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2350000000.0)
		tmp = -1.0 + ((a * (2.0 - a)) ^ 2.0);
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2350000000.0], N[(-1.0 + N[Power[N[(a * N[(2.0 - a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.35e9

   1. Initial program 73.2%

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

      1. associate--l+ 73.2%

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

      2. sqr-pow 73.2%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in b around 0 61.5%

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

      1. add-sqr-sqrt 61.5%

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

      2. pow2 61.5%

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

      3. +-commutative 61.5%

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

      4. metadata-eval 61.5%

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

      5. pow-prod-up 61.5%

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

      6. add-sqr-sqrt 54.8%

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

      7. hypot-def 71.7%

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

      8. sqrt-prod 71.7%

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

      9. metadata-eval 71.7%

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

      10. sqrt-prod 71.7%

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

      11. unpow2 71.7%

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

      12. sqrt-prod 45.6%

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

      13. add-sqr-sqrt 71.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in a around 0 82.0%

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

      1. +-commutative 82.0%

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

      2. *-commutative 82.0%

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

      3. mul-1-neg 82.0%

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

      4. unsub-neg 82.0%

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

   9. Simplified 82.0%

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

   10. Taylor expanded in a around 0 82.0%

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

       1. neg-mul-1 82.0%

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

       2. *-commutative 82.0%

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

       3. +-commutative 82.0%

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

       4. sub-neg 82.0%

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

       5. unpow2 82.0%

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

       6. distribute-lft-out-- 82.0%

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

   12. Simplified 82.0%

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

   if 2.35e9 < b

   1. Initial program 62.1%

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

      1. sub-neg 62.1%

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

      2. sqr-neg 62.1%

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

      3. +-commutative 62.1%

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

      4. sqr-neg 62.1%

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

      5. +-commutative 62.1%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in b around inf 93.7%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2350000000:\\ \;\;\;\;-1 + {\left(a \cdot \left(2 - a\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 5: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6000000000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6000000000.0) (+ -1.0 (pow a 4.0)) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6000000000.0) {
		tmp = -1.0 + pow(a, 4.0);
	} 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 <= 6000000000.0d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    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 <= 6000000000.0) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6000000000.0:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6000000000.0)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6000000000.0)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6000000000.0], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6e9

   1. Initial program 73.2%

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

      1. sub-neg 73.2%

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

      2. sqr-neg 73.2%

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

      3. +-commutative 73.2%

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

      4. sqr-neg 73.2%

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

      5. +-commutative 73.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in a around inf 79.7%

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

   if 6e9 < b

   1. Initial program 62.1%

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

      1. sub-neg 62.1%

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

      2. sqr-neg 62.1%

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

      3. +-commutative 62.1%

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

      4. sqr-neg 62.1%

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

      5. +-commutative 62.1%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in b around inf 93.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6000000000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 6: 69.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. sub-neg 70.6%

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

   2. sqr-neg 70.6%

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

   3. +-commutative 70.6%

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

   4. sqr-neg 70.6%

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

   5. +-commutative 70.6%

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

3. Simplified 71.7%

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

4. Taylor expanded in a around inf 70.0%

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

5. Final simplification 70.0%

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

Alternative 7: 25.2% accurate, 128.0× 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 70.6%

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

   1. sub-neg 70.6%

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

   2. sqr-neg 70.6%

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

   3. +-commutative 70.6%

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

   4. sqr-neg 70.6%

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

   5. +-commutative 70.6%

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

3. Simplified 71.7%

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

4. Taylor expanded in a around inf 70.0%

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

5. Taylor expanded in a around 0 23.1%

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

6. Final simplification 23.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023314 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))