Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.3% → 98.3%

Time: 6.6s

Alternatives: 7

Speedup: 1.2×

• 61.7% 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: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = (math.pow(a, 3.0) * (a + 4.0)) + -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(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(Float64((a ^ 3.0) * Float64(a + 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = ((a ^ 3.0) * (a + 4.0)) + -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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

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

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in a around inf 23.3%

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

      1. expm1-log1p-u 23.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(4 \cdot {a}^{3} + {a}^{4}\right)\right)} + -1 \]

      2. expm1-udef 23.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(4 \cdot {a}^{3} + {a}^{4}\right)} - 1\right)} + -1 \]

      3. fma-def 23.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(4, {a}^{3}, {a}^{4}\right)}\right)} - 1\right) + -1 \]

   7. Applied egg-rr 23.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(4, {a}^{3}, {a}^{4}\right)\right)} - 1\right)} + -1 \]
   8. Step-by-step derivation

      1. expm1-def 23.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, {a}^{3}, {a}^{4}\right)\right)\right)} + -1 \]

      2. expm1-log1p 23.3%

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

      3. fma-udef 23.3%

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

      4. metadata-eval 23.3%

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

      5. pow-sqr 23.3%

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

      6. unpow2 23.3%

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

      7. associate-*l* 23.3%

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

      8. unpow2 23.3%

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

      9. cube-mult 23.3%

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

      10. distribute-rgt-out 95.3%

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

   9. Simplified 95.3%

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

4. Final simplification 98.8%

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

Alternative 2: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 46000000:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 46000000.0)
   (+ (* (pow a 3.0) (+ a 4.0)) -1.0)
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 46000000.0) {
		tmp = (pow(a, 3.0) * (a + 4.0)) + -1.0;
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 46000000.0d0) then
        tmp = ((a ** 3.0d0) * (a + 4.0d0)) + (-1.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 46000000.0) {
		tmp = (Math.pow(a, 3.0) * (a + 4.0)) + -1.0;
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 46000000.0:
		tmp = (math.pow(a, 3.0) * (a + 4.0)) + -1.0
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 46000000.0)
		tmp = Float64(Float64((a ^ 3.0) * Float64(a + 4.0)) + -1.0);
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 46000000.0)
		tmp = ((a ^ 3.0) * (a + 4.0)) + -1.0;
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 46000000.0], N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.6e7

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in a around inf 60.6%

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

      1. expm1-log1p-u 59.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(4 \cdot {a}^{3} + {a}^{4}\right)\right)} + -1 \]

      2. expm1-udef 59.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(4 \cdot {a}^{3} + {a}^{4}\right)} - 1\right)} + -1 \]

      3. fma-def 59.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(4, {a}^{3}, {a}^{4}\right)}\right)} - 1\right) + -1 \]

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(4, {a}^{3}, {a}^{4}\right)\right)} - 1\right)} + -1 \]
   8. Step-by-step derivation

      1. expm1-def 59.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, {a}^{3}, {a}^{4}\right)\right)\right)} + -1 \]

      2. expm1-log1p 60.6%

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

      3. fma-udef 60.6%

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

      4. metadata-eval 60.6%

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

      5. pow-sqr 60.5%

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

      6. unpow2 60.5%

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

      7. associate-*l* 60.6%

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

      8. unpow2 60.6%

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

      9. cube-mult 60.6%

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

      10. distribute-rgt-out 78.6%

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

   9. Simplified 78.6%

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

   if 4.6e7 < b

   1. Initial program 80.3%

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

      1. sub-neg 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in b around inf 93.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 46000000:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 11000000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 11000000.0) (+ -1.0 (pow a 4.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 11000000.0) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 11000000.0d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 11000000.0) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 11000000.0:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 11000000.0)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 11000000.0)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 11000000.0], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1e7

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in a around inf 78.2%

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

   if 1.1e7 < b

   1. Initial program 80.3%

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

      1. associate--l+ 80.3%

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

      2. +-commutative 80.3%

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

      3. +-commutative 80.3%

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

      4. sub-neg 80.3%

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

      5. associate-+l+ 80.3%

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

      6. +-commutative 80.3%

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

      7. fma-def 80.3%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in a around 0 93.3%

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

      1. associate--l+ 93.3%

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

      2. unpow2 93.3%

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

      3. associate-*r* 93.3%

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

      4. fma-def 93.3%

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

      5. sub-neg 93.3%

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

      6. metadata-eval 93.3%

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

   7. Applied egg-rr 93.3%

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

   8. Taylor expanded in b around inf 93.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11000000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15000000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 15000000.0) (+ -1.0 (pow a 4.0)) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 15000000.0) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 15000000.0d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 15000000.0) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 15000000.0:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 15000000.0)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 15000000.0)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 15000000.0], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e7

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in a around inf 78.2%

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

   if 1.5e7 < b

   1. Initial program 80.3%

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

      1. sub-neg 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in b around inf 93.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15000000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.29:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(-1 + b \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.29) (* (+ 1.0 (* b 2.0)) (+ -1.0 (* b 2.0))) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.29) {
		tmp = (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.29d0) then
        tmp = (1.0d0 + (b * 2.0d0)) * ((-1.0d0) + (b * 2.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.29) {
		tmp = (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.29:
		tmp = (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.29)
		tmp = Float64(Float64(1.0 + Float64(b * 2.0)) * Float64(-1.0 + Float64(b * 2.0)));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.29)
		tmp = (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.29], N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.28999999999999998

   1. Initial program 77.6%

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

      1. associate--l+ 77.5%

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

      2. +-commutative 77.5%

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

      3. +-commutative 77.5%

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

      4. sub-neg 77.5%

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

      5. associate-+l+ 77.5%

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

      6. +-commutative 77.5%

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

      7. fma-def 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in a around 0 68.6%

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

   6. Taylor expanded in b around 0 55.4%

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

      1. add-sqr-sqrt 55.4%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2}} \cdot \sqrt{4 \cdot {b}^{2}}} - 1 \]

      2. difference-of-sqr-1 55.4%

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

      3. *-commutative 55.4%

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

      4. sqrt-prod 55.4%

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

      5. unpow2 55.4%

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

      6. sqrt-prod 21.2%

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

      7. add-sqr-sqrt 36.4%

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

      8. metadata-eval 36.4%

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

      9. *-commutative 36.4%

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

      10. sqrt-prod 36.4%

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

      11. unpow2 36.4%

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

      12. sqrt-prod 21.2%

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

      13. add-sqr-sqrt 55.4%

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

      14. metadata-eval 55.4%

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

   8. Applied egg-rr 55.4%

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

   if 0.28999999999999998 < b

   1. Initial program 77.9%

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

      1. associate--l+ 77.9%

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

      2. +-commutative 77.9%

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

      3. +-commutative 77.9%

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

      4. sub-neg 77.9%

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

      5. associate-+l+ 77.9%

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

      6. +-commutative 77.9%

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

      7. fma-def 77.9%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in a around 0 88.8%

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

      1. associate--l+ 88.8%

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

      2. unpow2 88.8%

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

      3. associate-*r* 88.8%

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

      4. fma-def 88.8%

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

      5. sub-neg 88.8%

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

      6. metadata-eval 88.8%

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

   7. Applied egg-rr 88.8%

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

   8. Taylor expanded in b around inf 88.7%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.29:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(-1 + b \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
}

Python

def code(a, b):
	return (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. associate--l+ 77.6%

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

   2. +-commutative 77.6%

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

   3. +-commutative 77.6%

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

   4. sub-neg 77.6%

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

   5. associate-+l+ 77.6%

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

   6. +-commutative 77.6%

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

   7. fma-def 77.6%

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

3. Simplified 78.4%

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

5. Taylor expanded in a around 0 73.3%

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

6. Taylor expanded in b around 0 55.1%

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

   1. add-sqr-sqrt 55.1%

      \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2}} \cdot \sqrt{4 \cdot {b}^{2}}} - 1 \]

   2. difference-of-sqr-1 55.1%

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

   3. *-commutative 55.1%

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

   4. sqrt-prod 55.1%

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

   5. unpow2 55.1%

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

   6. sqrt-prod 28.8%

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

   7. add-sqr-sqrt 40.5%

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

   8. metadata-eval 40.5%

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

   9. *-commutative 40.5%

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

   10. sqrt-prod 40.5%

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

   11. unpow2 40.5%

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

   12. sqrt-prod 28.8%

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

   13. add-sqr-sqrt 55.1%

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

   14. metadata-eval 55.1%

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

8. Applied egg-rr 55.1%

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

9. Final simplification 55.1%

   \[\leadsto \left(1 + b \cdot 2\right) \cdot \left(-1 + b \cdot 2\right) \]
10. Add Preprocessing

Alternative 7: 24.7% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 77.6%

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

   1. associate--l+ 77.6%

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

   2. +-commutative 77.6%

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

   3. +-commutative 77.6%

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

   4. sub-neg 77.6%

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

   5. associate-+l+ 77.6%

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

   6. +-commutative 77.6%

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

   7. fma-def 77.6%

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

3. Simplified 78.4%

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

5. Taylor expanded in a around 0 73.3%

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

6. Taylor expanded in b around 0 28.4%

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

7. Final simplification 28.4%

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

Reproduce

herbie shell --seed 2024040 
(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))