Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.0% → 98.3%

Time: 5.4s

Alternatives: 9

Speedup: 1.2×

• 61.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

   3. Simplified 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in a around inf 25.1%

      \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   5. Step-by-step derivation

      1. *-commutative 25.1%

         \[\leadsto \color{blue}{{a}^{3} \cdot 4} + {a}^{4} \]

      2. metadata-eval 25.1%

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

      3. pow-plus 25.1%

         \[\leadsto {a}^{3} \cdot 4 + \color{blue}{{a}^{3} \cdot a} \]

      4. distribute-lft-out 96.8%

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

   6. Simplified 96.8%

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

4. Final simplification 99.2%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = math.pow(a, 3.0) * (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(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64((a ^ 3.0) * Float64(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) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = (a ^ 3.0) * (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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in a around inf 25.1%

      \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   5. Step-by-step derivation

      1. *-commutative 25.1%

         \[\leadsto \color{blue}{{a}^{3} \cdot 4} + {a}^{4} \]

      2. metadata-eval 25.1%

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

      3. pow-plus 25.1%

         \[\leadsto {a}^{3} \cdot 4 + \color{blue}{{a}^{3} \cdot a} \]

      4. distribute-lft-out 96.8%

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

   6. Simplified 96.8%

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

4. Final simplification 99.1%

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

Alternative 3: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -26:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+20}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -26.0)
   (* (pow a 3.0) (+ a 4.0))
   (if (<= a 1.65e+20)
     (+ -1.0 (+ (pow b 4.0) (* (* b b) 4.0)))
     (+ (pow a 4.0) (+ -1.0 (* (+ a 1.0) (* (* a a) 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -26.0) {
		tmp = pow(a, 3.0) * (a + 4.0);
	} else if (a <= 1.65e+20) {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 4.0));
	} else {
		tmp = pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * 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 <= (-26.0d0)) then
        tmp = (a ** 3.0d0) * (a + 4.0d0)
    else if (a <= 1.65d+20) then
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * 4.0d0))
    else
        tmp = (a ** 4.0d0) + ((-1.0d0) + ((a + 1.0d0) * ((a * a) * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -26.0) {
		tmp = Math.pow(a, 3.0) * (a + 4.0);
	} else if (a <= 1.65e+20) {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * 4.0));
	} else {
		tmp = Math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -26.0:
		tmp = math.pow(a, 3.0) * (a + 4.0)
	elif a <= 1.65e+20:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * 4.0))
	else:
		tmp = math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -26.0)
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	elseif (a <= 1.65e+20)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0)));
	else
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a + 1.0) * Float64(Float64(a * a) * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -26.0)
		tmp = (a ^ 3.0) * (a + 4.0);
	elseif (a <= 1.65e+20)
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * 4.0));
	else
		tmp = (a ^ 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -26.0], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.65e+20], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -26

   1. Initial program 35.7%

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

      1. associate--l+ 35.7%

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

      2. fma-def 35.7%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in a around inf 26.4%

      \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   5. Step-by-step derivation

      1. *-commutative 26.4%

         \[\leadsto \color{blue}{{a}^{3} \cdot 4} + {a}^{4} \]

      2. metadata-eval 26.4%

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

      3. pow-plus 26.3%

         \[\leadsto {a}^{3} \cdot 4 + \color{blue}{{a}^{3} \cdot a} \]

      4. distribute-lft-out 90.5%

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

   6. Simplified 90.5%

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

   if -26 < a < 1.65e20

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. associate-+l+ 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-*r* 82.6%

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

      5. *-commutative 82.6%

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

      6. distribute-lft-out 97.8%

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

      7. unpow2 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in a around 0 98.6%

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

   if 1.65e20 < a

   1. Initial program 68.5%

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

      1. associate--l+ 68.5%

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

      2. fma-def 68.5%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in b around 0 94.4%

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

      1. associate--l+ 94.4%

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

      2. associate-*r* 94.4%

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

      3. unpow2 94.4%

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

   6. Simplified 94.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -26:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+20}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \end{array} \]

Alternative 4: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -26:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;a \leq 1.36 \cdot 10^{+20}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -26.0)
   (* (pow a 3.0) (+ a 4.0))
   (if (<= a 1.36e+20) (+ -1.0 (+ (pow b 4.0) (* (* b b) 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -26.0) {
		tmp = pow(a, 3.0) * (a + 4.0);
	} else if (a <= 1.36e+20) {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 4.0));
	} 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 <= (-26.0d0)) then
        tmp = (a ** 3.0d0) * (a + 4.0d0)
    else if (a <= 1.36d+20) then
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * 4.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -26.0) {
		tmp = Math.pow(a, 3.0) * (a + 4.0);
	} else if (a <= 1.36e+20) {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * 4.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -26.0:
		tmp = math.pow(a, 3.0) * (a + 4.0)
	elif a <= 1.36e+20:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * 4.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -26.0)
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	elseif (a <= 1.36e+20)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -26.0)
		tmp = (a ^ 3.0) * (a + 4.0);
	elseif (a <= 1.36e+20)
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * 4.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -26.0], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.36e+20], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -26

   1. Initial program 35.7%

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

      1. associate--l+ 35.7%

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

      2. fma-def 35.7%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in a around inf 26.4%

      \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   5. Step-by-step derivation

      1. *-commutative 26.4%

         \[\leadsto \color{blue}{{a}^{3} \cdot 4} + {a}^{4} \]

      2. metadata-eval 26.4%

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

      3. pow-plus 26.3%

         \[\leadsto {a}^{3} \cdot 4 + \color{blue}{{a}^{3} \cdot a} \]

      4. distribute-lft-out 90.5%

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

   6. Simplified 90.5%

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

   if -26 < a < 1.36e20

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. associate-+l+ 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-*r* 82.6%

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

      5. *-commutative 82.6%

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

      6. distribute-lft-out 97.8%

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

      7. unpow2 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in a around 0 98.6%

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

   if 1.36e20 < a

   1. Initial program 68.5%

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

      1. associate--l+ 68.5%

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

      2. fma-def 68.5%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in a around inf 94.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -26:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;a \leq 1.36 \cdot 10^{+20}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5: 93.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -26:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{+20}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -26.0)
   (* (pow a 3.0) (+ a 4.0))
   (if (<= a 4.9e+20) (+ -1.0 (* (* b b) (+ (* b b) 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -26.0) {
		tmp = pow(a, 3.0) * (a + 4.0);
	} else if (a <= 4.9e+20) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} 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 <= (-26.0d0)) then
        tmp = (a ** 3.0d0) * (a + 4.0d0)
    else if (a <= 4.9d+20) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -26.0) {
		tmp = Math.pow(a, 3.0) * (a + 4.0);
	} else if (a <= 4.9e+20) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -26.0:
		tmp = math.pow(a, 3.0) * (a + 4.0)
	elif a <= 4.9e+20:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -26.0)
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	elseif (a <= 4.9e+20)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -26.0)
		tmp = (a ^ 3.0) * (a + 4.0);
	elseif (a <= 4.9e+20)
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -26.0], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.9e+20], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -26

   1. Initial program 35.7%

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

      1. associate--l+ 35.7%

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

      2. fma-def 35.7%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in a around inf 26.4%

      \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   5. Step-by-step derivation

      1. *-commutative 26.4%

         \[\leadsto \color{blue}{{a}^{3} \cdot 4} + {a}^{4} \]

      2. metadata-eval 26.4%

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

      3. pow-plus 26.3%

         \[\leadsto {a}^{3} \cdot 4 + \color{blue}{{a}^{3} \cdot a} \]

      4. distribute-lft-out 90.5%

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

   6. Simplified 90.5%

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

   if -26 < a < 4.9e20

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. associate-+l+ 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-*r* 82.6%

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

      5. *-commutative 82.6%

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

      6. distribute-lft-out 97.8%

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

      7. unpow2 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in a around 0 98.6%

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

      1. sqr-pow 98.4%

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

      2. metadata-eval 98.4%

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

      3. pow2 98.4%

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

      4. metadata-eval 98.4%

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

      5. pow2 98.4%

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

      6. distribute-lft-out 98.4%

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

   9. Applied egg-rr 98.4%

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

   if 4.9e20 < a

   1. Initial program 68.5%

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

      1. associate--l+ 68.5%

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

      2. fma-def 68.5%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in a around inf 94.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -26:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{+20}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6: 93.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -26:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6.7 \cdot 10^{+19}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -26.0)
   (pow a 4.0)
   (if (<= a 6.7e+19) (+ -1.0 (* (* b b) (+ (* b b) 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -26.0) {
		tmp = pow(a, 4.0);
	} else if (a <= 6.7e+19) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} 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 <= (-26.0d0)) then
        tmp = a ** 4.0d0
    else if (a <= 6.7d+19) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -26.0) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 6.7e+19) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -26.0:
		tmp = math.pow(a, 4.0)
	elif a <= 6.7e+19:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -26.0)
		tmp = a ^ 4.0;
	elseif (a <= 6.7e+19)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -26.0)
		tmp = a ^ 4.0;
	elseif (a <= 6.7e+19)
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -26.0], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 6.7e+19], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -26 or 6.7e19 < a

   1. Initial program 49.9%

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

      1. associate--l+ 49.9%

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

      2. fma-def 49.9%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in a around inf 92.0%

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

   if -26 < a < 6.7e19

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. associate-+l+ 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-*r* 82.6%

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

      5. *-commutative 82.6%

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

      6. distribute-lft-out 97.8%

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

      7. unpow2 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in a around 0 98.6%

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

      1. sqr-pow 98.4%

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

      2. metadata-eval 98.4%

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

      3. pow2 98.4%

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

      4. metadata-eval 98.4%

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

      5. pow2 98.4%

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

      6. distribute-lft-out 98.4%

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

   9. Applied egg-rr 98.4%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -26:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6.7 \cdot 10^{+19}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7: 69.4% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. sub-neg 76.4%

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

3. Simplified 78.0%

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

4. Taylor expanded in a around 0 55.3%

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

   1. +-commutative 55.3%

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

   2. associate-+l+ 55.3%

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

   3. *-commutative 55.3%

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

   4. associate-*r* 55.3%

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

   5. *-commutative 55.3%

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

   6. distribute-lft-out 63.5%

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

   7. unpow2 63.5%

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

6. Simplified 63.5%

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

7. Taylor expanded in a around 0 68.6%

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

   1. sqr-pow 68.5%

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

   2. metadata-eval 68.5%

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

   3. pow2 68.5%

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

   4. metadata-eval 68.5%

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

   5. pow2 68.5%

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

   6. distribute-lft-out 68.5%

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

9. Applied egg-rr 68.5%

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

10. Final simplification 68.5%

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

Alternative 8: 50.8% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. sub-neg 76.4%

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

3. Simplified 78.0%

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

4. Taylor expanded in a around 0 55.3%

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

   1. +-commutative 55.3%

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

   2. associate-+l+ 55.3%

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

   3. *-commutative 55.3%

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

   4. associate-*r* 55.3%

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

   5. *-commutative 55.3%

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

   6. distribute-lft-out 63.5%

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

   7. unpow2 63.5%

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

6. Simplified 63.5%

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

7. Taylor expanded in a around 0 68.6%

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

   1. sqr-pow 68.5%

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

   2. metadata-eval 68.5%

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

   3. pow2 68.5%

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

   4. metadata-eval 68.5%

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

   5. pow2 68.5%

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

   6. distribute-lft-out 68.5%

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

9. Applied egg-rr 68.5%

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

10. Taylor expanded in b around 0 48.9%

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

    1. unpow2 48.9%

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

    2. *-commutative 48.9%

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

    3. associate-*r* 48.9%

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

    4. *-commutative 48.9%

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

12. Simplified 48.9%

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

13. Final simplification 48.9%

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

Alternative 9: 24.4% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 76.4%

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

   1. associate--l+ 76.4%

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

   2. fma-def 76.4%

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

3. Simplified 77.6%

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

4. Taylor expanded in b around 0 48.9%

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

   1. associate--l+ 48.9%

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

   2. associate-*r* 48.9%

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

   3. unpow2 48.9%

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

6. Simplified 48.9%

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

7. Taylor expanded in a around 0 22.4%

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

8. Final simplification 22.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))