Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.1% → 99.9%

Time: 7.6s

Alternatives: 10

Speedup: 5.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(N[(3.0 + a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 + a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 - 1.0), $MachinePrecision], N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 - N[(N[(4.0 - N[(N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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 3 binary64) a))))) < +inf.0

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 3 binary64) a)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {a}^{4} \cdot \left(1 - \frac{4 - \frac{\mathsf{fma}\left(b \cdot b, 2, 4\right)}{a}}{a}\right) - 1\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (-
          (* (pow a 4.0) (- 1.0 (/ (- 4.0 (/ (fma (* b b) 2.0 4.0) a)) a)))
          1.0)))
   (if (<= a -7.2e-6)
     t_0
     (if (<= a 1.15e-40) (- (fma (* b b) 12.0 (pow b 4.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (pow(a, 4.0) * (1.0 - ((4.0 - (fma((b * b), 2.0, 4.0) / a)) / a))) - 1.0;
	double tmp;
	if (a <= -7.2e-6) {
		tmp = t_0;
	} else if (a <= 1.15e-40) {
		tmp = fma((b * b), 12.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64((a ^ 4.0) * Float64(1.0 - Float64(Float64(4.0 - Float64(fma(Float64(b * b), 2.0, 4.0) / a)) / a))) - 1.0)
	tmp = 0.0
	if (a <= -7.2e-6)
		tmp = t_0;
	elseif (a <= 1.15e-40)
		tmp = Float64(fma(Float64(b * b), 12.0, (b ^ 4.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 - N[(N[(4.0 - N[(N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -7.2e-6], t$95$0, If[LessEqual[a, 1.15e-40], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.19999999999999967e-6 or 1.15e-40 < a

   1. Initial program 48.8%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.0%

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

   if -7.19999999999999967e-6 < a < 1.15e-40

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;{a}^{4} \cdot \left(1 - \frac{4 - \frac{\mathsf{fma}\left(b \cdot b, 2, 4\right)}{a}}{a}\right) - 1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} \cdot \left(1 - \frac{4 - \frac{\mathsf{fma}\left(b \cdot b, 2, 4\right)}{a}}{a}\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+45) (fma (* (fma (- a 4.0) a 4.0) a) a -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+45) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+45)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+45], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999999e45

   1. Initial program 84.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 97.6%

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

   10. Applied rewrites 97.6%

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

   if 1.9999999999999999e45 < (*.f64 b b)

   1. Initial program 58.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. distribute-lft-in N/A

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

   8. Applied rewrites 42.7%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 96.7

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

   11. Applied rewrites 96.7%

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

Alternative 4: 94.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+45)
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+45) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+45)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+45], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999999e45

   1. Initial program 84.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 97.6%

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

   10. Applied rewrites 97.6%

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

   if 1.9999999999999999e45 < (*.f64 b b)

   1. Initial program 58.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 96.7

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

   8. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -4.35 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -4.35e+36)
     t_0
     (if (<= a 2.4e+61) (fma (* (fma b b 12.0) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -4.35e+36) {
		tmp = t_0;
	} else if (a <= 2.4e+61) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -4.35e+36)
		tmp = t_0;
	elseif (a <= 2.4e+61)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.35e+36], t$95$0, If[LessEqual[a, 2.4e+61], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.35000000000000003e36 or 2.3999999999999999e61 < a

   1. Initial program 38.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 42.5

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

   5. Applied rewrites 42.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. distribute-lft-in N/A

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in a around inf

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

       1. lower-pow.f64 98.4

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

   11. Applied rewrites 98.4%

       \[\leadsto \color{blue}{{a}^{4}} \]
   12. Step-by-step derivation

   13. Applied rewrites 98.3%

       \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   if -4.35000000000000003e36 < a < 2.3999999999999999e61

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 95.2

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

   8. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+45)
   (- (* (* a a) (* a a)) 1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+45) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+45)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+45], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999999e45

   1. Initial program 84.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 94.9%

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

   if 1.9999999999999999e45 < (*.f64 b b)

   1. Initial program 58.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 96.7

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

   8. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -2.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(12 \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -2.55) t_0 (if (<= a 1.8e+61) (fma (* 12.0 b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -2.55) {
		tmp = t_0;
	} else if (a <= 1.8e+61) {
		tmp = fma((12.0 * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -2.55)
		tmp = t_0;
	elseif (a <= 1.8e+61)
		tmp = fma(Float64(12.0 * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.55], t$95$0, If[LessEqual[a, 1.8e+61], N[(N[(12.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5499999999999998 or 1.80000000000000005e61 < a

   1. Initial program 40.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. distribute-lft-in N/A

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in a around inf

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

       1. lower-pow.f64 93.7

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

   11. Applied rewrites 93.7%

       \[\leadsto \color{blue}{{a}^{4}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.6%

       \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   if -2.5499999999999998 < a < 1.80000000000000005e61

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 97.8

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

   8. Applied rewrites 97.8%

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

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 75.8%

       \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(12 \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e+153) (fma 4.0 (* a a) -1.0) (fma (* 12.0 b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+153) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = fma((12.0 * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e+153)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = fma(Float64(12.0 * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e+153], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(12.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.50000000000000009e153

   1. Initial program 78.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 75.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 60.0%

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

   if 1.50000000000000009e153 < b

   1. Initial program 37.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 50.8% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 72.6

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

5. Applied rewrites 72.6%

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

6. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

8. Applied rewrites 72.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 56.0%

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

Alternative 10: 25.2% accurate, 155.0× 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 72.5%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 72.6

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

5. Applied rewrites 72.6%

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

6. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-rgt-in N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 72.6

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

8. Applied rewrites 72.6%

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

9. Taylor expanded in b around 0

   \[\leadsto -1 \]
10. Step-by-step derivation

11. Applied rewrites 28.1%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024290 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))