Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 6.1s

Alternatives: 14

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. unpow2 99.9%

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

   3. unpow1 99.9%

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

   4. sqr-pow 99.9%

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

   5. associate-*r* 99.9%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005)
   (+ -1.0 (pow a 4.0))
   (+ (pow b 4.0) (* (* b b) (+ 4.0 (* (* a a) 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.005d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (b ** 4.0d0) + ((b * b) * (4.0d0 + ((a * a) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.005:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64((b ^ 4.0) + Float64(Float64(b * b) * Float64(4.0 + Float64(Float64(a * a) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.005)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = (b ^ 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 96.6%

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

      1. *-commutative 96.6%

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

      2. unpow2 96.6%

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

      3. unpow2 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 4: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (+ (pow b 4.0) (* b (* b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (pow(b, 4.0) + (b * (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 * b) <= 0.005d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b ** 4.0d0) + (b * (b * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (Math.pow(b, 4.0) + (b * (b * 4.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.005:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + (math.pow(b, 4.0) + (b * (b * 4.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(b * Float64(b * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.005)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b ^ 4.0) + (b * (b * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 51.2%

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

      1. fma-udef 51.2%

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

      2. fma-udef 51.2%

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

      3. associate-+r+ 51.2%

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

      4. fma-def 51.2%

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

      5. associate-*l* 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. fma-udef 51.2%

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

      2. hypot-udef 51.9%

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

      3. +-commutative 51.9%

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

      4. hypot-def 51.2%

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

   8. Applied egg-rr 51.2%

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

   9. Taylor expanded in b around inf 92.2%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 4\right)\right)\\ \end{array} \]

Alternative 5: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (* b (* b (fma b b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (b * (b * fma(b, b, 4.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 4.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 51.2%

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

      1. fma-udef 51.2%

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

      2. fma-udef 51.2%

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

      3. associate-+r+ 51.2%

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

      4. fma-def 51.2%

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

      5. associate-*l* 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. fma-udef 51.2%

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

      2. hypot-udef 51.9%

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

      3. +-commutative 51.9%

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

      4. hypot-def 51.2%

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

   8. Applied egg-rr 51.2%

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

   9. Taylor expanded in a around 0 92.2%

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

       1. metadata-eval 92.2%

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

       2. pow-plus 92.1%

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

       3. unpow2 92.1%

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

       4. *-commutative 92.1%

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

       5. associate-*r* 92.1%

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

       6. *-commutative 92.1%

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

       7. distribute-rgt-in 92.1%

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

       8. cube-mult 92.1%

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

       9. distribute-lft-out 92.1%

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

       10. fma-udef 92.1%

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

   11. Simplified 92.1%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \]

Alternative 6: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005) (+ -1.0 (pow a 4.0)) (+ (* 4.0 (* b b)) (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = (4.0 * (b * b)) + 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 * b) <= 0.005d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (4.0d0 * (b * b)) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = (4.0 * (b * b)) + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.005:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = (4.0 * (b * b)) + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(Float64(4.0 * Float64(b * b)) + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.005)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = (4.0 * (b * b)) + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 96.6%

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

      1. *-commutative 96.6%

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

      2. unpow2 96.6%

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

      3. unpow2 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in a around 0 90.9%

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

      1. unpow2 90.9%

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

   9. Simplified 90.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + {b}^{4}\\ \end{array} \]

Alternative 7: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005) (+ -1.0 (pow a 4.0)) (* b (* b (fma b b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = b * (b * fma(b, b, 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(b * Float64(b * fma(b, b, 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 96.6%

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

      1. *-commutative 96.6%

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

      2. unpow2 96.6%

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

      3. unpow2 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in a around 0 90.9%

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

      1. unpow2 90.9%

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

   9. Simplified 90.9%

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

   10. Taylor expanded in b around 0 90.9%

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

       1. metadata-eval 90.9%

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

       2. pow-sqr 90.8%

          \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} + 4 \cdot {b}^{2} \]

       3. unpow2 90.8%

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

       4. unpow2 90.8%

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

       5. unpow2 90.8%

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

       6. distribute-rgt-in 90.8%

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

       7. fma-udef 90.8%

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

       8. associate-*l* 90.8%

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

   12. Simplified 90.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \]

Alternative 8: 63.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{-156}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-108}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{elif}\;a \leq 8:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 7.8e-156)
   (+ -1.0 (* b (* b 4.0)))
   (if (<= a 8.5e-108)
     (* (* b b) (+ 4.0 (* b b)))
     (if (<= a 8.0) (+ -1.0 (* (* b b) (+ 4.0 (* a b)))) (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 7.8e-156) {
		tmp = -1.0 + (b * (b * 4.0));
	} else if (a <= 8.5e-108) {
		tmp = (b * b) * (4.0 + (b * b));
	} else if (a <= 8.0) {
		tmp = -1.0 + ((b * b) * (4.0 + (a * b)));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 7.8d-156) then
        tmp = (-1.0d0) + (b * (b * 4.0d0))
    else if (a <= 8.5d-108) then
        tmp = (b * b) * (4.0d0 + (b * b))
    else if (a <= 8.0d0) then
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + (a * b)))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 7.8e-156) {
		tmp = -1.0 + (b * (b * 4.0));
	} else if (a <= 8.5e-108) {
		tmp = (b * b) * (4.0 + (b * b));
	} else if (a <= 8.0) {
		tmp = -1.0 + ((b * b) * (4.0 + (a * b)));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 7.8e-156:
		tmp = -1.0 + (b * (b * 4.0))
	elif a <= 8.5e-108:
		tmp = (b * b) * (4.0 + (b * b))
	elif a <= 8.0:
		tmp = -1.0 + ((b * b) * (4.0 + (a * b)))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 7.8e-156)
		tmp = Float64(-1.0 + Float64(b * Float64(b * 4.0)));
	elseif (a <= 8.5e-108)
		tmp = Float64(Float64(b * b) * Float64(4.0 + Float64(b * b)));
	elseif (a <= 8.0)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(a * b))));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 7.8e-156)
		tmp = -1.0 + (b * (b * 4.0));
	elseif (a <= 8.5e-108)
		tmp = (b * b) * (4.0 + (b * b));
	elseif (a <= 8.0)
		tmp = -1.0 + ((b * b) * (4.0 + (a * b)));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 7.8e-156], N[(-1.0 + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.5e-108], N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.0], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < 7.8000000000000002e-156

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 55.7%

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

      1. fma-udef 55.7%

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

      2. fma-udef 55.7%

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

      3. associate-+r+ 55.7%

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

      4. fma-def 55.7%

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

      5. associate-*l* 55.7%

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

   6. Applied egg-rr 55.7%

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

      1. fma-udef 55.7%

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

      2. hypot-udef 57.4%

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

      3. +-commutative 57.4%

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

      4. hypot-def 55.7%

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

   8. Applied egg-rr 55.7%

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

   9. Taylor expanded in b around 0 53.8%

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

       1. unpow2 53.8%

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

       2. *-commutative 53.8%

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

       3. associate-*r* 53.8%

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

   11. Simplified 53.8%

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

   if 7.8000000000000002e-156 < a < 8.49999999999999986e-108

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. unpow2 99.6%

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

      3. unpow1 99.6%

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

      4. sqr-pow 99.6%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

       1. +-commutative 100.0%

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

       2. sqr-pow 99.6%

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

       3. metadata-eval 99.6%

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

       4. pow2 99.6%

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

       5. metadata-eval 99.6%

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

       6. pow2 99.6%

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

       7. distribute-rgt-out 99.6%

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

   11. Applied egg-rr 99.6%

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

   if 8.49999999999999986e-108 < a < 8

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow2 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. associate-*l* 100.0%

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

      6. fma-def 100.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 79.6%

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

      1. fma-udef 79.6%

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

      2. fma-udef 79.6%

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

      3. associate-+r+ 79.6%

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

      4. fma-def 79.6%

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

      5. associate-*l* 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. fma-udef 79.6%

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

      2. hypot-udef 79.6%

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

      3. +-commutative 79.6%

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

      4. hypot-def 79.6%

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

   8. Applied egg-rr 79.6%

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

   9. Taylor expanded in b around 0 73.7%

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

       1. *-commutative 73.7%

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

       2. unpow3 73.7%

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

       3. associate-*l* 68.2%

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

       4. unpow2 68.2%

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

       5. *-commutative 68.2%

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

       6. distribute-lft-out 68.2%

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

   11. Simplified 68.2%

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

   if 8 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around inf 89.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{-156}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-108}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{elif}\;a \leq 8:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 9: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.005) (+ -1.0 (pow a 4.0)) (* (* b b) (+ 4.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.005d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (b * b) * (4.0d0 + (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.005) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.005:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = (b * b) * (4.0 + (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.005)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(Float64(b * b) * Float64(4.0 + Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.005)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = (b * b) * (4.0 + (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.005], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0050000000000000001

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   if 0.0050000000000000001 < (*.f64 b b)

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 96.6%

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

      1. *-commutative 96.6%

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

      2. unpow2 96.6%

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

      3. unpow2 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in a around 0 90.9%

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

      1. unpow2 90.9%

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

   9. Simplified 90.9%

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

       1. +-commutative 90.9%

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

       2. sqr-pow 90.8%

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

       3. metadata-eval 90.8%

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

       4. pow2 90.8%

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

       5. metadata-eval 90.8%

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

       6. pow2 90.8%

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

       7. distribute-rgt-out 90.8%

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

   11. Applied egg-rr 90.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.005:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 10: 55.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.22)
   (+ -1.0 (* (* b b) (+ 4.0 (* a b))))
   (* (* b b) (+ 4.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.22) {
		tmp = -1.0 + ((b * b) * (4.0 + (a * b)));
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.22d0) then
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + (a * b)))
    else
        tmp = (b * b) * (4.0d0 + (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.22) {
		tmp = -1.0 + ((b * b) * (4.0 + (a * b)));
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.22:
		tmp = -1.0 + ((b * b) * (4.0 + (a * b)))
	else:
		tmp = (b * b) * (4.0 + (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.22)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(a * b))));
	else
		tmp = Float64(Float64(b * b) * Float64(4.0 + Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.22)
		tmp = -1.0 + ((b * b) * (4.0 + (a * b)));
	else
		tmp = (b * b) * (4.0 + (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.22], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.21999999999999997

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 28.6%

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

      1. fma-udef 28.6%

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

      2. fma-udef 28.6%

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

      3. associate-+r+ 28.6%

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

      4. fma-def 28.6%

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

      5. associate-*l* 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. fma-udef 28.6%

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

      2. hypot-udef 30.6%

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

      3. +-commutative 30.6%

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

      4. hypot-def 28.6%

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

   8. Applied egg-rr 28.6%

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

   9. Taylor expanded in b around 0 42.0%

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

       1. *-commutative 42.0%

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

       2. unpow3 42.0%

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

       3. associate-*l* 40.6%

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

       4. unpow2 40.6%

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

       5. *-commutative 40.6%

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

       6. distribute-lft-out 40.6%

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

   11. Simplified 40.6%

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

   if 1.21999999999999997 < b

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. unpow2 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in a around 0 90.7%

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

      1. unpow2 90.7%

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

   9. Simplified 90.7%

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

       1. +-commutative 90.7%

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

       2. sqr-pow 90.6%

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

       3. metadata-eval 90.6%

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

       4. pow2 90.6%

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

       5. metadata-eval 90.6%

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

       6. pow2 90.6%

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

       7. distribute-rgt-out 90.6%

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

   11. Applied egg-rr 90.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 11: 59.1% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.057:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.057) (+ -1.0 (* b (* b 4.0))) (* (* b b) (+ 4.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.057) {
		tmp = -1.0 + (b * (b * 4.0));
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.057d0) then
        tmp = (-1.0d0) + (b * (b * 4.0d0))
    else
        tmp = (b * b) * (4.0d0 + (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.057) {
		tmp = -1.0 + (b * (b * 4.0));
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.057:
		tmp = -1.0 + (b * (b * 4.0))
	else:
		tmp = (b * b) * (4.0 + (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.057)
		tmp = Float64(-1.0 + Float64(b * Float64(b * 4.0)));
	else
		tmp = Float64(Float64(b * b) * Float64(4.0 + Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.057)
		tmp = -1.0 + (b * (b * 4.0));
	else
		tmp = (b * b) * (4.0 + (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.057], N[(-1.0 + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0570000000000000021

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 28.6%

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

      1. fma-udef 28.6%

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

      2. fma-udef 28.6%

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

      3. associate-+r+ 28.6%

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

      4. fma-def 28.6%

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

      5. associate-*l* 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. fma-udef 28.6%

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

      2. hypot-udef 30.6%

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

      3. +-commutative 30.6%

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

      4. hypot-def 28.6%

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

   8. Applied egg-rr 28.6%

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

   9. Taylor expanded in b around 0 44.1%

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

       1. unpow2 44.1%

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

       2. *-commutative 44.1%

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

       3. associate-*r* 44.1%

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

   11. Simplified 44.1%

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

   if 0.0570000000000000021 < b

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. unpow2 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in a around 0 90.7%

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

      1. unpow2 90.7%

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

   9. Simplified 90.7%

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

       1. +-commutative 90.7%

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

       2. sqr-pow 90.6%

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

       3. metadata-eval 90.6%

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

       4. pow2 90.6%

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

       5. metadata-eval 90.6%

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

       6. pow2 90.6%

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

       7. distribute-rgt-out 90.6%

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

   11. Applied egg-rr 90.6%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.057:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 12: 37.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.057:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.057) -1.0 (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.057) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.057d0) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.057) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.057:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.057)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.057)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.057], -1.0, N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0570000000000000021

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 28.6%

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

   5. Taylor expanded in b around 0 28.7%

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

   if 0.0570000000000000021 < b

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. unpow2 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in a around 0 90.7%

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

      1. unpow2 90.7%

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

   9. Simplified 90.7%

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

   10. Taylor expanded in b around 0 54.1%

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

       1. unpow2 54.1%

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

   12. Simplified 54.1%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.057:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 13: 50.5% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. unpow2 99.9%

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

   3. unpow1 99.9%

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

   4. sqr-pow 99.9%

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

   5. associate-*l* 99.9%

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

   6. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in a around 0 47.5%

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

   1. fma-udef 47.5%

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

   2. fma-udef 47.5%

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

   3. associate-+r+ 47.5%

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

   4. fma-def 47.5%

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

   5. associate-*l* 47.5%

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

6. Applied egg-rr 47.5%

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

   1. fma-udef 47.5%

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

   2. hypot-udef 49.3%

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

   3. +-commutative 49.3%

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

   4. hypot-def 47.5%

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

8. Applied egg-rr 47.5%

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

9. Taylor expanded in b around 0 46.8%

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

    1. unpow2 46.8%

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

    2. *-commutative 46.8%

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

    3. associate-*r* 46.8%

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

11. Simplified 46.8%

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

12. Final simplification 46.8%

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

Alternative 14: 24.6% accurate, 116.0× 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 99.9%

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

   1. associate--l+ 99.9%

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

   2. unpow2 99.9%

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

   3. unpow1 99.9%

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

   4. sqr-pow 99.9%

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

   5. associate-*l* 99.9%

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

   6. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in a around 0 47.5%

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

5. Taylor expanded in b around 0 21.2%

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

6. Final simplification 21.2%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023227 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))