Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.3% → 100.0%

Time: 7.3s

Alternatives: 10

Speedup: 1.2×

• 61.5% 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: 73.3% accurate, 1.0× speedup

Math

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

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: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+67}:\\ \;\;\;\;-1 + t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))))
   (if (<= t_0 5e+67) (+ -1.0 t_0) (+ (pow (hypot a b) 4.0) -1.0))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_0 <= 5e+67) {
		tmp = -1.0 + t_0;
	} else {
		tmp = pow(hypot(a, b), 4.0) + -1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_0 <= 5e+67) {
		tmp = -1.0 + t_0;
	} else {
		tmp = Math.pow(Math.hypot(a, b), 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))
	tmp = 0
	if t_0 <= 5e+67:
		tmp = -1.0 + t_0
	else:
		tmp = math.pow(math.hypot(a, b), 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= 5e+67)
		tmp = Float64(-1.0 + t_0);
	else
		tmp = Float64((hypot(a, b) ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	tmp = 0.0;
	if (t_0 <= 5e+67)
		tmp = -1.0 + t_0;
	else
		tmp = (hypot(a, b) ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+67], N[(-1.0 + t$95$0), $MachinePrecision], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 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 3 a))))) < 4.99999999999999976e67

   1. Initial program 99.9%

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

   if 4.99999999999999976e67 < (+.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 63.7%

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

      1. associate--l+ 63.7%

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

   3. Simplified 70.1%

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

      1. add-cube-cbrt 70.1%

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

      2. pow3 70.1%

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

      3. fma-udef 63.8%

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

      4. associate-*l* 63.8%

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

      5. fma-def 66.1%

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

   5. Applied egg-rr 66.1%

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

   6. Taylor expanded in a around 0 73.4%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\color{blue}{\left({\left(12 \cdot {b}^{2} - 1\right)}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\color{blue}{\left(\sqrt[3]{12 \cdot {b}^{2} - 1}\right)}}^{3} \]

      2. *-commutative 100.0%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\color{blue}{{b}^{2} \cdot 12} - 1}\right)}^{3} \]

      3. unpow2 100.0%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\color{blue}{\left(b \cdot b\right)} \cdot 12 - 1}\right)}^{3} \]

      4. associate-*l* 100.0%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\color{blue}{b \cdot \left(b \cdot 12\right)} - 1}\right)}^{3} \]

      5. fma-neg 100.0%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(b, b \cdot 12, -1\right)}}\right)}^{3} \]

      6. metadata-eval 100.0%

         \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\mathsf{fma}\left(b, b \cdot 12, \color{blue}{-1}\right)}\right)}^{3} \]

   8. Simplified 100.0%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b \cdot 12, -1\right)}\right)}}^{3} \]

   9. Taylor expanded in b around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 5 \cdot 10^{+67}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\mathsf{fma}\left(b, b \cdot 12, -1\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+ (pow (hypot a b) 4.0) (pow (cbrt (fma b (* b 12.0) -1.0)) 3.0)))

C

double code(double a, double b) {
	return pow(hypot(a, b), 4.0) + pow(cbrt(fma(b, (b * 12.0), -1.0)), 3.0);
}

Julia

function code(a, b)
	return Float64((hypot(a, b) ^ 4.0) + (cbrt(fma(b, Float64(b * 12.0), -1.0)) ^ 3.0))
end

Wolfram

code[a_, b_] := N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[Power[N[Power[N[(b * N[(b * 12.0), $MachinePrecision] + -1.0), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

   1. associate--l+ 74.9%

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

3. Simplified 79.3%

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

   1. add-cube-cbrt 79.3%

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

   2. pow3 79.3%

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

   3. fma-udef 75.0%

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

   4. associate-*l* 75.0%

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

   5. fma-def 76.5%

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

5. Applied egg-rr 76.5%

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

6. Taylor expanded in a around 0 52.7%

   \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\color{blue}{\left({\left(12 \cdot {b}^{2} - 1\right)}^{0.3333333333333333}\right)}}^{3} \]
7. Step-by-step derivation

   1. unpow1/3 99.2%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\color{blue}{\left(\sqrt[3]{12 \cdot {b}^{2} - 1}\right)}}^{3} \]

   2. *-commutative 99.2%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\color{blue}{{b}^{2} \cdot 12} - 1}\right)}^{3} \]

   3. unpow2 99.2%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\color{blue}{\left(b \cdot b\right)} \cdot 12 - 1}\right)}^{3} \]

   4. associate-*l* 99.2%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\color{blue}{b \cdot \left(b \cdot 12\right)} - 1}\right)}^{3} \]

   5. fma-neg 99.2%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(b, b \cdot 12, -1\right)}}\right)}^{3} \]

   6. metadata-eval 99.2%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\mathsf{fma}\left(b, b \cdot 12, \color{blue}{-1}\right)}\right)}^{3} \]

8. Simplified 99.2%

   \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b \cdot 12, -1\right)}\right)}}^{3} \]

9. Final simplification 99.2%

   \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + {\left(\sqrt[3]{\mathsf{fma}\left(b, b \cdot 12, -1\right)}\right)}^{3} \]

Alternative 3: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 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 = pow(a, 4.0);
	}
	return tmp;
}

Java

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 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 = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(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 = a ^ 4.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(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[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 3 a))))) < +inf.0

   1. Initial program 99.9%

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

   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%

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

   3. Simplified 17.2%

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

      1. add-cube-cbrt 17.2%

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

      2. pow3 17.2%

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

      3. fma-udef 0.0%

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

      4. associate-*l* 0.0%

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

      5. fma-def 6.3%

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

   5. Applied egg-rr 6.3%

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

   6. Taylor expanded in a around inf 91.1%

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

4. Final simplification 97.7%

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

Alternative 4: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+44}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 38000:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e+44)
   (pow a 4.0)
   (if (<= a 38000.0)
     (+ -1.0 (+ (pow b 4.0) (* b (* b 12.0))))
     (+ -1.0 (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.3e+44) {
		tmp = pow(a, 4.0);
	} else if (a <= 38000.0) {
		tmp = -1.0 + (pow(b, 4.0) + (b * (b * 12.0)));
	} else {
		tmp = -1.0 + pow(a, 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 (a <= (-3.3d+44)) then
        tmp = a ** 4.0d0
    else if (a <= 38000.0d0) then
        tmp = (-1.0d0) + ((b ** 4.0d0) + (b * (b * 12.0d0)))
    else
        tmp = (-1.0d0) + (a ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.3e+44) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 38000.0) {
		tmp = -1.0 + (Math.pow(b, 4.0) + (b * (b * 12.0)));
	} else {
		tmp = -1.0 + Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.3e+44:
		tmp = math.pow(a, 4.0)
	elif a <= 38000.0:
		tmp = -1.0 + (math.pow(b, 4.0) + (b * (b * 12.0)))
	else:
		tmp = -1.0 + math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.3e+44)
		tmp = a ^ 4.0;
	elseif (a <= 38000.0)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(b * Float64(b * 12.0))));
	else
		tmp = Float64(-1.0 + (a ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.3e+44)
		tmp = a ^ 4.0;
	elseif (a <= 38000.0)
		tmp = -1.0 + ((b ^ 4.0) + (b * (b * 12.0)));
	else
		tmp = -1.0 + (a ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.3e+44], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 38000.0], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.30000000000000013e44

   1. Initial program 51.9%

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

      1. associate--l+ 51.9%

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

   3. Simplified 73.1%

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

      1. add-cube-cbrt 73.1%

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

      2. pow3 73.1%

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

      3. fma-udef 51.9%

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

      4. associate-*l* 51.9%

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

      5. fma-def 59.6%

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

   5. Applied egg-rr 59.6%

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

   6. Taylor expanded in a around inf 94.6%

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

   if -3.30000000000000013e44 < a < 38000

   1. Initial program 97.1%

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

      1. sub-neg 97.1%

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

      2. fma-def 97.1%

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

      3. fma-def 97.1%

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

      4. +-commutative 97.1%

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

      5. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in a around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. +-commutative 83.9%

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

      3. associate-+l+ 83.9%

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

      4. +-commutative 83.9%

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

      5. associate-*r* 83.9%

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

      6. distribute-rgt-out 96.1%

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

      7. metadata-eval 96.1%

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

      8. distribute-lft-in 96.1%

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

      9. unpow2 96.1%

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

      10. distribute-rgt-in 96.1%

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

      11. metadata-eval 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in a around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. unpow2 98.9%

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

      3. associate-*l* 98.9%

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

   9. Simplified 98.9%

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

   if 38000 < a

   1. Initial program 45.1%

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

      1. sub-neg 45.1%

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

      2. fma-def 45.1%

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

      3. fma-def 49.7%

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

      4. +-commutative 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in a around inf 86.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+44}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 38000:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 5: 81.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-19}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+149}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-19)
   (+ -1.0 (* 4.0 (* a a)))
   (if (<= (* b b) 5e+149) (pow a 4.0) (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-19) {
		tmp = -1.0 + (4.0 * (a * a));
	} else if ((b * b) <= 5e+149) {
		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 * b) <= 1d-19) then
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    else if ((b * b) <= 5d+149) 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 * b) <= 1e-19) {
		tmp = -1.0 + (4.0 * (a * a));
	} else if ((b * b) <= 5e+149) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e-19:
		tmp = -1.0 + (4.0 * (a * a))
	elif (b * b) <= 5e+149:
		tmp = math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-19)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * a)));
	elseif (Float64(b * b) <= 5e+149)
		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 * b) <= 1e-19)
		tmp = -1.0 + (4.0 * (a * a));
	elseif ((b * b) <= 5e+149)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-19], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 5e+149], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 9.9999999999999998e-20

   1. Initial program 88.7%

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

      1. sub-neg 88.7%

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

      2. fma-def 88.7%

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

      3. fma-def 88.7%

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

      4. +-commutative 88.7%

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

      5. metadata-eval 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in b around 0 88.8%

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

      1. unpow2 88.8%

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

   6. Simplified 88.8%

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

   7. Taylor expanded in a around 0 77.2%

      \[\leadsto \color{blue}{4 \cdot {a}^{2}} + -1 \]
   8. Step-by-step derivation

      1. unpow2 77.2%

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

   9. Simplified 77.2%

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

   if 9.9999999999999998e-20 < (*.f64 b b) < 4.9999999999999999e149

   1. Initial program 63.7%

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

      1. associate--l+ 63.7%

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

   3. Simplified 69.2%

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

      1. add-cube-cbrt 69.2%

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

      2. pow3 69.2%

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

      3. fma-udef 64.1%

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

      4. associate-*l* 64.1%

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

      5. fma-def 69.2%

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

   5. Applied egg-rr 69.2%

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

   6. Taylor expanded in a around inf 60.3%

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

   if 4.9999999999999999e149 < (*.f64 b b)

   1. Initial program 63.4%

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

      1. associate--l+ 63.4%

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

   3. Simplified 72.3%

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

      1. add-cube-cbrt 72.3%

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

      2. pow3 72.3%

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

      3. fma-udef 63.4%

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

      4. associate-*l* 63.4%

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

      5. fma-def 65.3%

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

   5. Applied egg-rr 65.3%

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

   6. Taylor expanded in b around inf 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-19}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+149}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6: 91.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+149}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+149) (+ -1.0 (pow a 4.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+149) {
		tmp = -1.0 + 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 * b) <= 5d+149) then
        tmp = (-1.0d0) + (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 * b) <= 5e+149) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+149:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+149)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+149)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+149], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999999e149

   1. Initial program 82.4%

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

      1. sub-neg 82.4%

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

      2. fma-def 82.4%

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

      3. fma-def 82.4%

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

      4. +-commutative 82.4%

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

      5. metadata-eval 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in a around inf 88.9%

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

   if 4.9999999999999999e149 < (*.f64 b b)

   1. Initial program 63.4%

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

      1. associate--l+ 63.4%

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

   3. Simplified 72.3%

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

      1. add-cube-cbrt 72.3%

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

      2. pow3 72.3%

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

      3. fma-udef 63.4%

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

      4. associate-*l* 63.4%

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

      5. fma-def 65.3%

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

   5. Applied egg-rr 65.3%

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

   6. Taylor expanded in b around inf 100.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+149}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 7: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+44}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 35000:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e+44)
   (pow a 4.0)
   (if (<= a 35000.0) (+ -1.0 (* b (* b 12.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.3e+44) {
		tmp = pow(a, 4.0);
	} else if (a <= 35000.0) {
		tmp = -1.0 + (b * (b * 12.0));
	} else {
		tmp = pow(a, 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 (a <= (-3.3d+44)) then
        tmp = a ** 4.0d0
    else if (a <= 35000.0d0) then
        tmp = (-1.0d0) + (b * (b * 12.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.3e+44) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 35000.0) {
		tmp = -1.0 + (b * (b * 12.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.3e+44:
		tmp = math.pow(a, 4.0)
	elif a <= 35000.0:
		tmp = -1.0 + (b * (b * 12.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.3e+44)
		tmp = a ^ 4.0;
	elseif (a <= 35000.0)
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.3e+44)
		tmp = a ^ 4.0;
	elseif (a <= 35000.0)
		tmp = -1.0 + (b * (b * 12.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.3e+44], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 35000.0], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.30000000000000013e44 or 35000 < a

   1. Initial program 48.1%

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

      1. associate--l+ 48.1%

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

   3. Simplified 57.7%

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

      1. add-cube-cbrt 57.7%

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

      2. pow3 57.7%

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

      3. fma-udef 48.3%

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

      4. associate-*l* 48.3%

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

      5. fma-def 51.7%

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

   5. Applied egg-rr 51.7%

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

   6. Taylor expanded in a around inf 89.9%

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

   if -3.30000000000000013e44 < a < 35000

   1. Initial program 97.1%

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

      1. sub-neg 97.1%

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

      2. fma-def 97.1%

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

      3. fma-def 97.1%

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

      4. +-commutative 97.1%

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

      5. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in a around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. +-commutative 83.9%

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

      3. associate-+l+ 83.9%

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

      4. +-commutative 83.9%

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

      5. associate-*r* 83.9%

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

      6. distribute-rgt-out 96.1%

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

      7. metadata-eval 96.1%

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

      8. distribute-lft-in 96.1%

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

      9. unpow2 96.1%

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

      10. distribute-rgt-in 96.1%

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

      11. metadata-eval 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in b around 0 72.9%

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

      1. +-commutative 72.9%

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

      2. fma-def 72.9%

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

      3. *-commutative 72.9%

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

      4. unpow2 72.9%

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

      5. associate-*l* 72.9%

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

   9. Simplified 72.9%

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

   10. Taylor expanded in a around 0 75.9%

       \[\leadsto \color{blue}{12 \cdot {b}^{2}} + -1 \]
   11. Step-by-step derivation

       1. unpow2 75.9%

          \[\leadsto 12 \cdot \color{blue}{\left(b \cdot b\right)} + -1 \]

       2. *-commutative 75.9%

          \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot 12} + -1 \]

       3. associate-*l* 75.9%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot 12\right)} + -1 \]

   12. Simplified 75.9%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot 12\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+44}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 35000:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 8: 69.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+296}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+296) (+ -1.0 (* 4.0 (* a a))) (+ -1.0 (* b (* b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+296) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	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) <= 1d+296) then
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    else
        tmp = (-1.0d0) + (b * (b * 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+296) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+296:
		tmp = -1.0 + (4.0 * (a * a))
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+296)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+296)
		tmp = -1.0 + (4.0 * (a * a));
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+296], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999981e295

   1. Initial program 78.4%

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

      1. sub-neg 78.4%

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

      2. fma-def 78.4%

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

      3. fma-def 78.9%

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

      4. +-commutative 78.9%

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

      5. metadata-eval 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in b around 0 66.6%

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

      1. unpow2 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in a around 0 61.0%

      \[\leadsto \color{blue}{4 \cdot {a}^{2}} + -1 \]
   8. Step-by-step derivation

      1. unpow2 61.0%

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

   9. Simplified 61.0%

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

   if 9.99999999999999981e295 < (*.f64 b b)

   1. Initial program 64.6%

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

      1. sub-neg 64.6%

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

      2. fma-def 64.6%

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

      3. fma-def 67.7%

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

      4. +-commutative 67.7%

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

      5. metadata-eval 67.7%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in a around 0 49.2%

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

      1. +-commutative 49.2%

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

      2. +-commutative 49.2%

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

      3. associate-+l+ 49.2%

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

      4. +-commutative 49.2%

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

      5. associate-*r* 49.2%

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

      6. distribute-rgt-out 75.4%

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

      7. metadata-eval 75.4%

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

      8. distribute-lft-in 75.4%

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

      9. unpow2 75.4%

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

      10. distribute-rgt-in 75.4%

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

      11. metadata-eval 75.4%

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

   6. Simplified 75.4%

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

   7. Taylor expanded in b around 0 74.0%

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

      1. +-commutative 74.0%

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

      2. fma-def 74.0%

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

      3. *-commutative 74.0%

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

      4. unpow2 74.0%

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

      5. associate-*l* 74.0%

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

   9. Simplified 74.0%

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

   10. Taylor expanded in a around 0 97.4%

       \[\leadsto \color{blue}{12 \cdot {b}^{2}} + -1 \]
   11. Step-by-step derivation

       1. unpow2 97.4%

          \[\leadsto 12 \cdot \color{blue}{\left(b \cdot b\right)} + -1 \]

       2. *-commutative 97.4%

          \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot 12} + -1 \]

       3. associate-*l* 97.4%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot 12\right)} + -1 \]

   12. Simplified 97.4%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot 12\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+296}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 9: 51.5% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0 + (4.0 * (a * a))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. sub-neg 74.9%

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

   2. fma-def 74.9%

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

   3. fma-def 76.1%

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

   4. +-commutative 76.1%

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

   5. metadata-eval 76.1%

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

3. Simplified 76.1%

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

4. Taylor expanded in b around 0 55.1%

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

   1. unpow2 55.1%

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

6. Simplified 55.1%

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

7. Taylor expanded in a around 0 50.0%

   \[\leadsto \color{blue}{4 \cdot {a}^{2}} + -1 \]
8. Step-by-step derivation

   1. unpow2 50.0%

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

9. Simplified 50.0%

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

10. Final simplification 50.0%

    \[\leadsto -1 + 4 \cdot \left(a \cdot a\right) \]

Alternative 10: 24.6% 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 74.9%

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

   1. sub-neg 74.9%

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

   2. fma-def 74.9%

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

   3. fma-def 76.1%

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

   4. +-commutative 76.1%

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

   5. metadata-eval 76.1%

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

3. Simplified 76.1%

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

4. Taylor expanded in a around 0 56.4%

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

   1. +-commutative 56.4%

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

   2. +-commutative 56.4%

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

   3. associate-+l+ 56.4%

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

   4. +-commutative 56.4%

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

   5. associate-*r* 56.4%

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

   6. distribute-rgt-out 63.1%

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

   7. metadata-eval 63.1%

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

   8. distribute-lft-in 63.1%

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

   9. unpow2 63.1%

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

   10. distribute-rgt-in 63.1%

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

   11. metadata-eval 63.1%

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

6. Simplified 63.1%

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

7. Taylor expanded in b around 0 27.1%

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

8. Final simplification 27.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023258 
(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))