Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.0% → 97.5%

Time: 9.4s

Alternatives: 14

Speedup: 5.3×

• 61.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot a}{b}, \frac{a}{b}, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-3, a, 1\right)}{b}, \frac{4}{b}, 1\right)\right) \cdot {b}^{4} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-39)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (-
    (*
     (fma (/ (* 2.0 a) b) (/ a b) (fma (/ (fma -3.0 a 1.0) b) (/ 4.0 b) 1.0))
     (pow b 4.0))
    1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-39) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else {
		tmp = (fma(((2.0 * a) / b), (a / b), fma((fma(-3.0, a, 1.0) / b), (4.0 / b), 1.0)) * pow(b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-39)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(2.0 * a) / b), Float64(a / b), fma(Float64(fma(-3.0, a, 1.0) / b), Float64(4.0 / b), 1.0)) * (b ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-39], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * a), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + N[(N[(N[(-3.0 * a + 1.0), $MachinePrecision] / b), $MachinePrecision] * N[(4.0 / b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999929e-40

   1. Initial program 80.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 97.8

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

   5. Applied rewrites 97.8%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 99.9%

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

   if 9.99999999999999929e-40 < (*.f64 b b)

   1. Initial program 55.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.8%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 94.2

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

   5. Applied rewrites 94.2%

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

   7. Applied rewrites 94.2%

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

Alternative 3: 93.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{4}{a} + 4}{a} - -1, {a}^{4}, -1\right)\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(4 + a\right) \cdot a\right) \cdot a, a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.5)
   (fma (- (/ (+ (/ 4.0 a) 4.0) a) -1.0) (pow a 4.0) -1.0)
   (if (<= a 1.15e+66)
     (- (fma (* b b) 4.0 (pow b 4.0)) 1.0)
     (fma (* (* (+ 4.0 a) a) a) a -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.5) {
		tmp = fma(((((4.0 / a) + 4.0) / a) - -1.0), pow(a, 4.0), -1.0);
	} else if (a <= 1.15e+66) {
		tmp = fma((b * b), 4.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma((((4.0 + a) * a) * a), a, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.5)
		tmp = fma(Float64(Float64(Float64(Float64(4.0 / a) + 4.0) / a) - -1.0), (a ^ 4.0), -1.0);
	elseif (a <= 1.15e+66)
		tmp = Float64(fma(Float64(b * b), 4.0, (b ^ 4.0)) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(4.0 + a) * a) * a), a, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.5], N[(N[(N[(N[(N[(4.0 / a), $MachinePrecision] + 4.0), $MachinePrecision] / a), $MachinePrecision] - -1.0), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 1.15e+66], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.5

   1. Initial program 31.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 91.1

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

   5. Applied rewrites 91.1%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 92.9%

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

   if -0.5 < a < 1.15e66

   1. Initial program 96.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 1.15e66 < a

   1. Initial program 53.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 98.1%

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

   12. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(\left({a}^{2} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)\right) \cdot a, a, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

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

Alternative 4: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(4 + a\right) \cdot a\right) \cdot a, a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.5)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 1.15e+66)
     (- (fma (* b b) 4.0 (pow b 4.0)) 1.0)
     (fma (* (* (+ 4.0 a) a) a) a -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.5) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 1.15e+66) {
		tmp = fma((b * b), 4.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma((((4.0 + a) * a) * a), a, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.5)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 1.15e+66)
		tmp = Float64(fma(Float64(b * b), 4.0, (b ^ 4.0)) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(4.0 + a) * a) * a), a, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.5], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 1.15e+66], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.5

   1. Initial program 31.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 91.1

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

   5. Applied rewrites 91.1%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 92.8%

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

   if -0.5 < a < 1.15e66

   1. Initial program 96.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 1.15e66 < a

   1. Initial program 53.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 98.1%

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

   12. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(\left({a}^{2} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)\right) \cdot a, a, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

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

Alternative 5: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+66}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(4, b, {b}^{3}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(4 + a\right) \cdot a\right) \cdot a, a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.5)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 1.15e+66)
     (- (* b (fma 4.0 b (pow b 3.0))) 1.0)
     (fma (* (* (+ 4.0 a) a) a) a -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.5) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 1.15e+66) {
		tmp = (b * fma(4.0, b, pow(b, 3.0))) - 1.0;
	} else {
		tmp = fma((((4.0 + a) * a) * a), a, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.5)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 1.15e+66)
		tmp = Float64(Float64(b * fma(4.0, b, (b ^ 3.0))) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(4.0 + a) * a) * a), a, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.5], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 1.15e+66], N[(N[(b * N[(4.0 * b + N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.5

   1. Initial program 31.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 91.1

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

   5. Applied rewrites 91.1%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 92.8%

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

   if -0.5 < a < 1.15e66

   1. Initial program 96.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 97.3%

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

   9. Applied rewrites 97.3%

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

   if 1.15e66 < a

   1. Initial program 53.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 98.1%

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

   12. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(\left({a}^{2} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)\right) \cdot a, a, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

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

Alternative 6: 93.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4100000000000 \lor \neg \left(a \leq 1.15 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(4 + a\right) \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4100000000000.0) (not (<= a 1.15e+66)))
   (fma (* (* (+ 4.0 a) a) a) a -1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -4100000000000.0) || !(a <= 1.15e+66)) {
		tmp = fma((((4.0 + a) * a) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -4100000000000.0) || !(a <= 1.15e+66))
		tmp = fma(Float64(Float64(Float64(4.0 + a) * a) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -4100000000000.0], N[Not[LessEqual[a, 1.15e+66]], $MachinePrecision]], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1e12 or 1.15e66 < a

   1. Initial program 37.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 95.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 95.4%

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

   12. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(\left({a}^{2} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)\right) \cdot a, a, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 95.4%

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

   if -4.1e12 < a < 1.15e66

   1. Initial program 96.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 54.3

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

   5. Applied rewrites 54.3%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 95.9

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

   11. Applied rewrites 95.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4100000000000 \lor \neg \left(a \leq 1.15 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(4 + a\right) \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(4 + a\right) \cdot a\right) \cdot a, a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.5)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 1.15e+66)
     (fma (* (fma b b 4.0) b) b -1.0)
     (fma (* (* (+ 4.0 a) a) a) a -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.5) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 1.15e+66) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma((((4.0 + a) * a) * a), a, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.5)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 1.15e+66)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(Float64(Float64(4.0 + a) * a) * a), a, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.5], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 1.15e+66], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.5

   1. Initial program 31.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 91.1

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

   5. Applied rewrites 91.1%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 92.8%

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

   if -0.5 < a < 1.15e66

   1. Initial program 96.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 55.1

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

   5. Applied rewrites 55.1%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 56.2%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 97.3

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

   11. Applied rewrites 97.3%

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

   if 1.15e66 < a

   1. Initial program 53.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 98.1%

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

   12. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(\left({a}^{2} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)\right) \cdot a, a, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

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

Alternative 8: 93.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7500000000000 \lor \neg \left(a \leq 1.15 \cdot 10^{+66}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -7500000000000.0) (not (<= a 1.15e+66)))
   (- (* (* a a) (* a a)) 1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -7500000000000.0) || !(a <= 1.15e+66)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -7500000000000.0) || !(a <= 1.15e+66))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -7500000000000.0], N[Not[LessEqual[a, 1.15e+66]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5e12 or 1.15e66 < a

   1. Initial program 37.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 95.4%

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

   if -7.5e12 < a < 1.15e66

   1. Initial program 96.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 54.3

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

   5. Applied rewrites 54.3%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 95.9

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

   11. Applied rewrites 95.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7500000000000 \lor \neg \left(a \leq 1.15 \cdot 10^{+66}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4100000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(4 + a\right) \cdot a, a \cdot a, -1\right)\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4100000000000.0)
   (fma (* (+ 4.0 a) a) (* a a) -1.0)
   (if (<= a 1.15e+66)
     (fma (* (fma b b 4.0) b) b -1.0)
     (- (* (* a a) (* a a)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4100000000000.0) {
		tmp = fma(((4.0 + a) * a), (a * a), -1.0);
	} else if (a <= 1.15e+66) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = ((a * a) * (a * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4100000000000.0)
		tmp = fma(Float64(Float64(4.0 + a) * a), Float64(a * a), -1.0);
	elseif (a <= 1.15e+66)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4100000000000.0], N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 1.15e+66], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.1e12

   1. Initial program 28.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.8

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

   5. Applied rewrites 93.8%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 93.8%

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

   if -4.1e12 < a < 1.15e66

   1. Initial program 96.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 54.3

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

   5. Applied rewrites 54.3%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 95.9

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

   11. Applied rewrites 95.9%

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

   if 1.15e66 < a

   1. Initial program 53.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.1%

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

Alternative 10: 86.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+132)
   (fma 4.0 (* a a) -1.0)
   (if (<= a 3.5e+102)
     (fma (* (fma b b 4.0) b) b -1.0)
     (fma (fma 4.0 a 4.0) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+132) {
		tmp = fma(4.0, (a * a), -1.0);
	} else if (a <= 3.5e+102) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma(fma(4.0, a, 4.0), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+132)
		tmp = fma(4.0, Float64(a * a), -1.0);
	elseif (a <= 3.5e+102)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(fma(4.0, a, 4.0), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+132], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 3.5e+102], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(4.0 * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.4000000000000001e132

   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. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   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 94.5%

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

   if -2.4000000000000001e132 < a < 3.50000000000000011e102

   1. Initial program 91.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 61.2

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

   5. Applied rewrites 61.2%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 84.3

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

   11. Applied rewrites 84.3%

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

   if 3.50000000000000011e102 < a

   1. Initial program 51.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

Alternative 11: 71.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.35e+106)
   (fma 4.0 (* a a) -1.0)
   (if (<= a 4.5e+96)
     (fma (* b b) 4.0 -1.0)
     (fma (fma 4.0 a 4.0) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.35e+106) {
		tmp = fma(4.0, (a * a), -1.0);
	} else if (a <= 4.5e+96) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = fma(fma(4.0, a, 4.0), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.35e+106)
		tmp = fma(4.0, Float64(a * a), -1.0);
	elseif (a <= 4.5e+96)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = fma(fma(4.0, a, 4.0), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.35e+106], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 4.5e+96], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(4.0 * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.35000000000000012e106

   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. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   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 84.8%

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

   if -2.35000000000000012e106 < a < 4.49999999999999957e96

   1. Initial program 95.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 85.7

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

   8. Applied rewrites 85.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 65.0%

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

   if 4.49999999999999957e96 < a

   1. Initial program 53.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 91.2%

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

Alternative 12: 69.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+303) (fma 4.0 (* a a) -1.0) (fma (* b b) 4.0 -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+303) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = fma((b * b), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+303)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = fma(Float64(b * b), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+303], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e303

   1. Initial program 73.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 81.8

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

   5. Applied rewrites 81.8%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 83.2%

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

   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 61.7%

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

   if 4e303 < (*.f64 b b)

   1. Initial program 49.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 46.5

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 98.4%

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

Alternative 13: 50.4% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.2%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
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 74.2

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

5. Applied rewrites 74.2%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 69.6

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

8. Applied rewrites 69.6%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 49.9%

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

Alternative 14: 24.2% accurate, 160.0× 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 68.2%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
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 74.2

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

5. Applied rewrites 74.2%

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

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

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt1-in N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. distribute-lft-in N/A

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

   10. associate-+r+ N/A

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

   11. unpow2 N/A

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

   12. distribute-rgt-in N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

8. Applied rewrites 75.3%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 27.3%

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

Reproduce

herbie shell --seed 2024319 
(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))