Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.2% → 99.9%

Time: 9.2s

Alternatives: 8

Speedup: 1.2×

• 60.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% 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: 99.9% accurate, 0.3× speedup

Math

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

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)
   (+
    (+
     (* 4.0 (fma (* a a) (+ a 1.0) (* b (* b (+ 1.0 (* a -3.0))))))
     (pow (fma a a (* b b)) 2.0))
    -1.0)
   (+
    -1.0
    (* (pow a 4.0) (+ 1.0 (/ (+ 4.0 (/ (+ 4.0 (* (* b b) 2.0)) a)) a))))))

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 = ((4.0 * fma((a * a), (a + 1.0), (b * (b * (1.0 + (a * -3.0)))))) + pow(fma(a, a, (b * b)), 2.0)) + -1.0;
	} else {
		tmp = -1.0 + (pow(a, 4.0) * (1.0 + ((4.0 + ((4.0 + ((b * b) * 2.0)) / a)) / a)));
	}
	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(Float64(Float64(4.0 * fma(Float64(a * a), Float64(a + 1.0), Float64(b * Float64(b * Float64(1.0 + Float64(a * -3.0)))))) + (fma(a, a, Float64(b * b)) ^ 2.0)) + -1.0);
	else
		tmp = Float64(-1.0 + Float64((a ^ 4.0) * Float64(1.0 + Float64(Float64(4.0 + Float64(Float64(4.0 + Float64(Float64(b * b) * 2.0)) / a)) / a))));
	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[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision] + N[(b * N[(b * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(N[(4.0 + N[(N[(4.0 + N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) 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 99.9%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) 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. sub-neg 0.0%

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

   3. Simplified 10.1%

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

   5. Taylor expanded in a around -inf 100.0%

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

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4 + \frac{4 + \left(b \cdot b\right) \cdot 2}{a}}{a}\right)\\ \end{array} \end{array} \]

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)
     (+
      -1.0
      (* (pow a 4.0) (+ 1.0 (/ (+ 4.0 (/ (+ 4.0 (* (* b b) 2.0)) a)) a)))))))

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 = -1.0 + (pow(a, 4.0) * (1.0 + ((4.0 + ((4.0 + ((b * b) * 2.0)) / a)) / a)));
	}
	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 = -1.0 + (Math.pow(a, 4.0) * (1.0 + ((4.0 + ((4.0 + ((b * b) * 2.0)) / a)) / a)));
	}
	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 = -1.0 + (math.pow(a, 4.0) * (1.0 + ((4.0 + ((4.0 + ((b * b) * 2.0)) / a)) / a)))
	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(-1.0 + Float64((a ^ 4.0) * Float64(1.0 + Float64(Float64(4.0 + Float64(Float64(4.0 + Float64(Float64(b * b) * 2.0)) / a)) / a))));
	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 = -1.0 + ((a ^ 4.0) * (1.0 + ((4.0 + ((4.0 + ((b * b) * 2.0)) / a)) / a)));
	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[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(N[(4.0 + N[(N[(4.0 + N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) 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. Add Preprocessing

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) 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. sub-neg 0.0%

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

   3. Simplified 10.1%

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

   5. Taylor expanded in a around -inf 100.0%

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

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-5} \lor \neg \left(a \leq 6.5 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4 + \frac{4 + \left(b \cdot b\right) \cdot 2}{a}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -5.4e-5) (not (<= a 6.5e-7)))
   (+ -1.0 (* (pow a 4.0) (+ 1.0 (/ (+ 4.0 (/ (+ 4.0 (* (* b b) 2.0)) a)) a))))
   (+ -1.0 (+ (* (* b b) 4.0) (pow b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -5.4e-5) || !(a <= 6.5e-7)) {
		tmp = -1.0 + (pow(a, 4.0) * (1.0 + ((4.0 + ((4.0 + ((b * b) * 2.0)) / a)) / a)));
	} else {
		tmp = -1.0 + (((b * b) * 4.0) + pow(b, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-5.4d-5)) .or. (.not. (a <= 6.5d-7))) then
        tmp = (-1.0d0) + ((a ** 4.0d0) * (1.0d0 + ((4.0d0 + ((4.0d0 + ((b * b) * 2.0d0)) / a)) / a)))
    else
        tmp = (-1.0d0) + (((b * b) * 4.0d0) + (b ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -5.4e-5) || !(a <= 6.5e-7)) {
		tmp = -1.0 + (Math.pow(a, 4.0) * (1.0 + ((4.0 + ((4.0 + ((b * b) * 2.0)) / a)) / a)));
	} else {
		tmp = -1.0 + (((b * b) * 4.0) + Math.pow(b, 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -5.4e-5) or not (a <= 6.5e-7):
		tmp = -1.0 + (math.pow(a, 4.0) * (1.0 + ((4.0 + ((4.0 + ((b * b) * 2.0)) / a)) / a)))
	else:
		tmp = -1.0 + (((b * b) * 4.0) + math.pow(b, 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -5.4e-5) || !(a <= 6.5e-7))
		tmp = Float64(-1.0 + Float64((a ^ 4.0) * Float64(1.0 + Float64(Float64(4.0 + Float64(Float64(4.0 + Float64(Float64(b * b) * 2.0)) / a)) / a))));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(b * b) * 4.0) + (b ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -5.4e-5) || ~((a <= 6.5e-7)))
		tmp = -1.0 + ((a ^ 4.0) * (1.0 + ((4.0 + ((4.0 + ((b * b) * 2.0)) / a)) / a)));
	else
		tmp = -1.0 + (((b * b) * 4.0) + (b ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -5.4e-5], N[Not[LessEqual[a, 6.5e-7]], $MachinePrecision]], N[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(N[(4.0 + N[(N[(4.0 + N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.3999999999999998e-5 or 6.50000000000000024e-7 < a

   1. Initial program 45.1%

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

      1. sub-neg 45.1%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in a around -inf 98.2%

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

      1. unpow2 98.2%

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

   7. Applied egg-rr 98.2%

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

   if -5.3999999999999998e-5 < a < 6.50000000000000024e-7

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 100.0%

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

      1. unpow2 34.9%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-5} \lor \neg \left(a \leq 6.5 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + {a}^{4} \cdot \left(1 + \frac{4 + \frac{4 + \left(b \cdot b\right) \cdot 2}{a}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 32000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 32000.0)
   (+ -1.0 (* (* a a) (+ 4.0 (* a (+ a 4.0)))))
   (+ -1.0 (+ (* (* b b) 4.0) (pow b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 32000.0) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	} else {
		tmp = -1.0 + (((b * b) * 4.0) + pow(b, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 32000.0d0) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (a * (a + 4.0d0))))
    else
        tmp = (-1.0d0) + (((b * b) * 4.0d0) + (b ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 32000.0) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	} else {
		tmp = -1.0 + (((b * b) * 4.0) + Math.pow(b, 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 32000.0:
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))))
	else:
		tmp = -1.0 + (((b * b) * 4.0) + math.pow(b, 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 32000.0)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(b * b) * 4.0) + (b ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 32000.0)
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	else
		tmp = -1.0 + (((b * b) * 4.0) + (b ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 32000.0], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 32000

   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 77.5%

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

   5. Taylor expanded in a around -inf 69.2%

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

   6. Taylor expanded in b around 0 61.2%

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

      1. associate-*r/ 61.2%

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

      2. associate-*r/ 61.2%

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

      3. metadata-eval 61.2%

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

      4. distribute-lft-in 61.2%

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

      5. metadata-eval 61.2%

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

      6. associate-*r/ 61.2%

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

      7. metadata-eval 61.2%

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

   8. Simplified 61.2%

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

   9. Taylor expanded in a around 0 76.6%

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

       1. unpow2 76.6%

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

   11. Applied egg-rr 76.6%

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

   if 32000 < b

   1. Initial program 64.3%

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

      1. associate--l+ 64.3%

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

      2. +-commutative 64.3%

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

      3. +-commutative 64.3%

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

      4. sub-neg 64.3%

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

      5. associate-+l+ 64.3%

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

      6. +-commutative 64.3%

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

      7. fma-define 64.3%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in a around 0 92.3%

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

      1. unpow2 58.2%

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

   7. Applied egg-rr 92.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 32000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 210000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 210000.0)
   (+ -1.0 (* (* a a) (+ 4.0 (* a (+ a 4.0)))))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 210000.0) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 210000.0d0) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (a * (a + 4.0d0))))
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 210000.0) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 210000.0:
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 210000.0)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 210000.0)
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 210000.0], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1e5

   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 77.5%

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

   5. Taylor expanded in a around -inf 69.2%

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

   6. Taylor expanded in b around 0 61.2%

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

      1. associate-*r/ 61.2%

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

      2. associate-*r/ 61.2%

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

      3. metadata-eval 61.2%

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

      4. distribute-lft-in 61.2%

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

      5. metadata-eval 61.2%

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

      6. associate-*r/ 61.2%

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

      7. metadata-eval 61.2%

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

   8. Simplified 61.2%

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

   9. Taylor expanded in a around 0 76.6%

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

       1. unpow2 76.6%

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

   11. Applied egg-rr 76.6%

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

   if 2.1e5 < b

   1. Initial program 64.3%

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

      1. sub-neg 64.3%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in b around inf 92.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 210000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4)
   (+ -1.0 (* (* a a) 4.0))
   (+ -1.0 (* (* a a) (+ 4.0 (* a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + ((a * a) * (4.0 + (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 <= (-2.4d0)) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (a * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.4) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + ((a * a) * (4.0 + (a * 4.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.4:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = -1.0 + ((a * a) * (4.0 + (a * 4.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(a * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.4)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = -1.0 + ((a * a) * (4.0 + (a * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.39999999999999991

   1. Initial program 27.8%

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

      1. sub-neg 27.8%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in a around -inf 98.5%

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

   6. Taylor expanded in b around 0 90.6%

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

      1. associate-*r/ 90.6%

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

      2. associate-*r/ 90.6%

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

      3. metadata-eval 90.6%

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

      4. distribute-lft-in 90.6%

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

      5. metadata-eval 90.6%

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

      6. associate-*r/ 90.6%

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

      7. metadata-eval 90.6%

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

   8. Simplified 90.6%

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

   9. Taylor expanded in a around 0 52.7%

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

       1. unpow2 90.6%

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

   11. Applied egg-rr 52.7%

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

   if -2.39999999999999991 < a

   1. Initial program 87.0%

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

      1. sub-neg 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in a around -inf 55.9%

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

   6. Taylor expanded in b around 0 42.6%

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

      1. associate-*r/ 42.6%

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

      2. associate-*r/ 42.6%

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

      3. metadata-eval 42.6%

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

      4. distribute-lft-in 42.6%

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

      5. metadata-eval 42.6%

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

      6. associate-*r/ 42.6%

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

      7. metadata-eval 42.6%

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

   8. Simplified 42.6%

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

   9. Taylor expanded in a around 0 50.1%

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

       1. unpow2 57.1%

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

   11. Applied egg-rr 50.1%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. sub-neg 72.9%

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

3. Simplified 75.7%

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

5. Taylor expanded in a around -inf 66.1%

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

6. Taylor expanded in b around 0 54.0%

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

   1. associate-*r/ 54.0%

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

   2. associate-*r/ 54.0%

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

   3. metadata-eval 54.0%

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

   4. distribute-lft-in 54.0%

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

   5. metadata-eval 54.0%

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

   6. associate-*r/ 54.0%

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

   7. metadata-eval 54.0%

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

8. Simplified 54.0%

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

9. Taylor expanded in a around 0 65.1%

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

    1. unpow2 65.1%

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

11. Applied egg-rr 65.1%

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

12. Final simplification 65.1%

    \[\leadsto -1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) \]
13. Add Preprocessing

Alternative 8: 51.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. sub-neg 72.9%

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

3. Simplified 75.7%

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

5. Taylor expanded in a around -inf 66.1%

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

6. Taylor expanded in b around 0 54.0%

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

   1. associate-*r/ 54.0%

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

   2. associate-*r/ 54.0%

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

   3. metadata-eval 54.0%

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

   4. distribute-lft-in 54.0%

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

   5. metadata-eval 54.0%

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

   6. associate-*r/ 54.0%

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

   7. metadata-eval 54.0%

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

8. Simplified 54.0%

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

9. Taylor expanded in a around 0 46.5%

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

    1. unpow2 65.1%

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

11. Applied egg-rr 46.5%

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

12. Final simplification 46.5%

    \[\leadsto -1 + \left(a \cdot a\right) \cdot 4 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024092 
(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))