Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.3% → 99.1%

Time: 18.6s

Alternatives: 15

Speedup: 7.5×

• 61.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot a + b \cdot b\\ \left(\left(\frac{b}{\frac{1}{b \cdot t\_0}} + a \cdot \left(a \cdot t\_0\right)\right) + 4 \cdot \left(b \cdot \left(b \cdot 3\right)\right)\right) - 1 \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* a a) (* b b))))
   (-
    (+ (+ (/ b (/ 1.0 (* b t_0))) (* a (* a t_0))) (* 4.0 (* b (* b 3.0))))
    1.0)))

C

double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return (((b / (1.0 / (b * t_0))) + (a * (a * t_0))) + (4.0 * (b * (b * 3.0)))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    t_0 = (a * a) + (b * b)
    code = (((b / (1.0d0 / (b * t_0))) + (a * (a * t_0))) + (4.0d0 * (b * (b * 3.0d0)))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return (((b / (1.0 / (b * t_0))) + (a * (a * t_0))) + (4.0 * (b * (b * 3.0)))) - 1.0;
}

Python

def code(a, b):
	t_0 = (a * a) + (b * b)
	return (((b / (1.0 / (b * t_0))) + (a * (a * t_0))) + (4.0 * (b * (b * 3.0)))) - 1.0

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b))
	return Float64(Float64(Float64(Float64(b / Float64(1.0 / Float64(b * t_0))) + Float64(a * Float64(a * t_0))) + Float64(4.0 * Float64(b * Float64(b * 3.0)))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	t_0 = (a * a) + (b * b);
	tmp = (((b / (1.0 / (b * t_0))) + (a * (a * t_0))) + (4.0 * (b * (b * 3.0)))) - 1.0;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(b / N[(1.0 / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(b * N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Initial program 72.9%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]

9. Taylor expanded in a around 0 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]
12. Add Preprocessing

Alternative 2: 97.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\left(4 + a \cdot \left(-4 + a\right)\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\left(\left(1 + \frac{1}{b \cdot b} \cdot \left(\left(a \cdot a\right) \cdot 2 + \left(12 + a \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-63)
   (- (* (+ 4.0 (* a (+ -4.0 a))) (* a a)) 1.0)
   (if (<= (* b b) 2e+133)
     (-
      (*
       (*
        (+ 1.0 (* (/ 1.0 (* b b)) (+ (* (* a a) 2.0) (+ 12.0 (* a 4.0)))))
        (* b (* b b)))
       b)
      1.0)
     (* (* b b) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-63) {
		tmp = ((4.0 + (a * (-4.0 + a))) * (a * a)) - 1.0;
	} else if ((b * b) <= 2e+133) {
		tmp = (((1.0 + ((1.0 / (b * b)) * (((a * a) * 2.0) + (12.0 + (a * 4.0))))) * (b * (b * b))) * b) - 1.0;
	} else {
		tmp = (b * b) * (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) <= 2d-63) then
        tmp = ((4.0d0 + (a * ((-4.0d0) + a))) * (a * a)) - 1.0d0
    else if ((b * b) <= 2d+133) then
        tmp = (((1.0d0 + ((1.0d0 / (b * b)) * (((a * a) * 2.0d0) + (12.0d0 + (a * 4.0d0))))) * (b * (b * b))) * b) - 1.0d0
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-63) {
		tmp = ((4.0 + (a * (-4.0 + a))) * (a * a)) - 1.0;
	} else if ((b * b) <= 2e+133) {
		tmp = (((1.0 + ((1.0 / (b * b)) * (((a * a) * 2.0) + (12.0 + (a * 4.0))))) * (b * (b * b))) * b) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-63:
		tmp = ((4.0 + (a * (-4.0 + a))) * (a * a)) - 1.0
	elif (b * b) <= 2e+133:
		tmp = (((1.0 + ((1.0 / (b * b)) * (((a * a) * 2.0) + (12.0 + (a * 4.0))))) * (b * (b * b))) * b) - 1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-63)
		tmp = Float64(Float64(Float64(4.0 + Float64(a * Float64(-4.0 + a))) * Float64(a * a)) - 1.0);
	elseif (Float64(b * b) <= 2e+133)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / Float64(b * b)) * Float64(Float64(Float64(a * a) * 2.0) + Float64(12.0 + Float64(a * 4.0))))) * Float64(b * Float64(b * b))) * b) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-63)
		tmp = ((4.0 + (a * (-4.0 + a))) * (a * a)) - 1.0;
	elseif ((b * b) <= 2e+133)
		tmp = (((1.0 + ((1.0 / (b * b)) * (((a * a) * 2.0) + (12.0 + (a * 4.0))))) * (b * (b * b))) * b) - 1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-63], N[(N[(N[(4.0 + N[(a * N[(-4.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 2e+133], N[(N[(N[(N[(1.0 + N[(N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] + N[(12.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 2.00000000000000013e-63

   1. Initial program 83.5%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if 2.00000000000000013e-63 < (*.f64 b b) < 2e133

   1. Initial program 68.0%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if 2e133 < (*.f64 b b)

   1. Initial program 59.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 69.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq -1.5 \cdot 10^{+50}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 5.7 \cdot 10^{-291}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-120}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.16 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.45:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot a\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) (* b b))))
   (if (<= a -1.5e+50)
     (* (* a a) (* a a))
     (if (<= a 5.7e-291)
       t_0
       (if (<= a 5.5e-120)
         -1.0
         (if (<= a 1.16e-35)
           t_0
           (if (<= a 2.45) -1.0 (* (* a (* a a)) a))))))))

C

double code(double a, double b) {
	double t_0 = (b * b) * (b * b);
	double tmp;
	if (a <= -1.5e+50) {
		tmp = (a * a) * (a * a);
	} else if (a <= 5.7e-291) {
		tmp = t_0;
	} else if (a <= 5.5e-120) {
		tmp = -1.0;
	} else if (a <= 1.16e-35) {
		tmp = t_0;
	} else if (a <= 2.45) {
		tmp = -1.0;
	} else {
		tmp = (a * (a * a)) * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) * (b * b)
    if (a <= (-1.5d+50)) then
        tmp = (a * a) * (a * a)
    else if (a <= 5.7d-291) then
        tmp = t_0
    else if (a <= 5.5d-120) then
        tmp = -1.0d0
    else if (a <= 1.16d-35) then
        tmp = t_0
    else if (a <= 2.45d0) then
        tmp = -1.0d0
    else
        tmp = (a * (a * a)) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (b * b) * (b * b);
	double tmp;
	if (a <= -1.5e+50) {
		tmp = (a * a) * (a * a);
	} else if (a <= 5.7e-291) {
		tmp = t_0;
	} else if (a <= 5.5e-120) {
		tmp = -1.0;
	} else if (a <= 1.16e-35) {
		tmp = t_0;
	} else if (a <= 2.45) {
		tmp = -1.0;
	} else {
		tmp = (a * (a * a)) * a;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b * b) * (b * b)
	tmp = 0
	if a <= -1.5e+50:
		tmp = (a * a) * (a * a)
	elif a <= 5.7e-291:
		tmp = t_0
	elif a <= 5.5e-120:
		tmp = -1.0
	elif a <= 1.16e-35:
		tmp = t_0
	elif a <= 2.45:
		tmp = -1.0
	else:
		tmp = (a * (a * a)) * a
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * Float64(b * b))
	tmp = 0.0
	if (a <= -1.5e+50)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 5.7e-291)
		tmp = t_0;
	elseif (a <= 5.5e-120)
		tmp = -1.0;
	elseif (a <= 1.16e-35)
		tmp = t_0;
	elseif (a <= 2.45)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * Float64(a * a)) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b * b) * (b * b);
	tmp = 0.0;
	if (a <= -1.5e+50)
		tmp = (a * a) * (a * a);
	elseif (a <= 5.7e-291)
		tmp = t_0;
	elseif (a <= 5.5e-120)
		tmp = -1.0;
	elseif (a <= 1.16e-35)
		tmp = t_0;
	elseif (a <= 2.45)
		tmp = -1.0;
	else
		tmp = (a * (a * a)) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.5e+50], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.7e-291], t$95$0, If[LessEqual[a, 5.5e-120], -1.0, If[LessEqual[a, 1.16e-35], t$95$0, If[LessEqual[a, 2.45], -1.0, N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.4999999999999999e50

   1. Initial program 63.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if -1.4999999999999999e50 < a < 5.70000000000000034e-291 or 5.5000000000000001e-120 < a < 1.16000000000000005e-35

   1. Initial program 96.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 5.70000000000000034e-291 < a < 5.5000000000000001e-120 or 1.16000000000000005e-35 < a < 2.4500000000000002

   1. Initial program 99.8%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 2.4500000000000002 < a

   1. Initial program 31.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 38.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;b \leq 1.85 \cdot 10^{-120}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 10^{+136}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* -4.0 (* a (* a a)))))
   (if (<= b 1.85e-120)
     -1.0
     (if (<= b 3e-99)
       t_0
       (if (<= b 5.5e-6) -1.0 (if (<= b 1e+136) t_0 (* b (* b 12.0))))))))

C

double code(double a, double b) {
	double t_0 = -4.0 * (a * (a * a));
	double tmp;
	if (b <= 1.85e-120) {
		tmp = -1.0;
	} else if (b <= 3e-99) {
		tmp = t_0;
	} else if (b <= 5.5e-6) {
		tmp = -1.0;
	} else if (b <= 1e+136) {
		tmp = t_0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-4.0d0) * (a * (a * a))
    if (b <= 1.85d-120) then
        tmp = -1.0d0
    else if (b <= 3d-99) then
        tmp = t_0
    else if (b <= 5.5d-6) then
        tmp = -1.0d0
    else if (b <= 1d+136) then
        tmp = t_0
    else
        tmp = b * (b * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = -4.0 * (a * (a * a));
	double tmp;
	if (b <= 1.85e-120) {
		tmp = -1.0;
	} else if (b <= 3e-99) {
		tmp = t_0;
	} else if (b <= 5.5e-6) {
		tmp = -1.0;
	} else if (b <= 1e+136) {
		tmp = t_0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = -4.0 * (a * (a * a))
	tmp = 0
	if b <= 1.85e-120:
		tmp = -1.0
	elif b <= 3e-99:
		tmp = t_0
	elif b <= 5.5e-6:
		tmp = -1.0
	elif b <= 1e+136:
		tmp = t_0
	else:
		tmp = b * (b * 12.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(-4.0 * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (b <= 1.85e-120)
		tmp = -1.0;
	elseif (b <= 3e-99)
		tmp = t_0;
	elseif (b <= 5.5e-6)
		tmp = -1.0;
	elseif (b <= 1e+136)
		tmp = t_0;
	else
		tmp = Float64(b * Float64(b * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = -4.0 * (a * (a * a));
	tmp = 0.0;
	if (b <= 1.85e-120)
		tmp = -1.0;
	elseif (b <= 3e-99)
		tmp = t_0;
	elseif (b <= 5.5e-6)
		tmp = -1.0;
	elseif (b <= 1e+136)
		tmp = t_0;
	else
		tmp = b * (b * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(-4.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.85e-120], -1.0, If[LessEqual[b, 3e-99], t$95$0, If[LessEqual[b, 5.5e-6], -1.0, If[LessEqual[b, 1e+136], t$95$0, N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.85e-120 or 3.00000000000000006e-99 < b < 5.4999999999999999e-6

   1. Initial program 76.4%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 1.85e-120 < b < 3.00000000000000006e-99 or 5.4999999999999999e-6 < b < 1.00000000000000006e136

   1. Initial program 66.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if 1.00000000000000006e136 < b

   1. Initial program 57.6%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   12. Taylor expanded in b around inf 0

       \[\leadsto expr\]

   14. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+55}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 90000000:\\ \;\;\;\;b \cdot \left(b \cdot \left(12 + b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.8e+55)
   (* (* a a) (* a a))
   (if (<= a 90000000.0)
     (- (* b (* b (+ 12.0 (* b b)))) 1.0)
     (* a (* (* a a) (+ a -4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+55) {
		tmp = (a * a) * (a * a);
	} else if (a <= 90000000.0) {
		tmp = (b * (b * (12.0 + (b * b)))) - 1.0;
	} else {
		tmp = a * ((a * a) * (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+55)) then
        tmp = (a * a) * (a * a)
    else if (a <= 90000000.0d0) then
        tmp = (b * (b * (12.0d0 + (b * b)))) - 1.0d0
    else
        tmp = a * ((a * a) * (a + (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+55) {
		tmp = (a * a) * (a * a);
	} else if (a <= 90000000.0) {
		tmp = (b * (b * (12.0 + (b * b)))) - 1.0;
	} else {
		tmp = a * ((a * a) * (a + -4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7.8e+55:
		tmp = (a * a) * (a * a)
	elif a <= 90000000.0:
		tmp = (b * (b * (12.0 + (b * b)))) - 1.0
	else:
		tmp = a * ((a * a) * (a + -4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.8e+55)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 90000000.0)
		tmp = Float64(Float64(b * Float64(b * Float64(12.0 + Float64(b * b)))) - 1.0);
	else
		tmp = Float64(a * Float64(Float64(a * a) * Float64(a + -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.8e+55)
		tmp = (a * a) * (a * a);
	elseif (a <= 90000000.0)
		tmp = (b * (b * (12.0 + (b * b)))) - 1.0;
	else
		tmp = a * ((a * a) * (a + -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.8e+55], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 90000000.0], N[(N[(b * N[(b * N[(12.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.80000000000000054e55

   1. Initial program 63.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if -7.80000000000000054e55 < a < 9e7

   1. Initial program 97.7%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 9e7 < a

   1. Initial program 28.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-6}:\\ \;\;\;\;\left(4 + a \cdot \left(-4 + a\right)\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(12 + b \cdot b\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-6)
   (- (* (+ 4.0 (* a (+ -4.0 a))) (* a a)) 1.0)
   (- (* b (* b (+ 12.0 (* b b)))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-6) {
		tmp = ((4.0 + (a * (-4.0 + a))) * (a * a)) - 1.0;
	} else {
		tmp = (b * (b * (12.0 + (b * b)))) - 1.0;
	}
	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) <= 1d-6) then
        tmp = ((4.0d0 + (a * ((-4.0d0) + a))) * (a * a)) - 1.0d0
    else
        tmp = (b * (b * (12.0d0 + (b * b)))) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-6) {
		tmp = ((4.0 + (a * (-4.0 + a))) * (a * a)) - 1.0;
	} else {
		tmp = (b * (b * (12.0 + (b * b)))) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e-6:
		tmp = ((4.0 + (a * (-4.0 + a))) * (a * a)) - 1.0
	else:
		tmp = (b * (b * (12.0 + (b * b)))) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-6)
		tmp = Float64(Float64(Float64(4.0 + Float64(a * Float64(-4.0 + a))) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(b * Float64(b * Float64(12.0 + Float64(b * b)))) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e-6)
		tmp = ((4.0 + (a * (-4.0 + a))) * (a * a)) - 1.0;
	else
		tmp = (b * (b * (12.0 + (b * b)))) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-6], N[(N[(N[(4.0 + N[(a * N[(-4.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * N[(b * N[(12.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999955e-7

   1. Initial program 83.0%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if 9.99999999999999955e-7 < (*.f64 b b)

   1. Initial program 61.5%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{+53}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 160000000:\\ \;\;\;\;\left(b \cdot \left(b \cdot b\right)\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.6e+53)
   (* (* a a) (* a a))
   (if (<= a 160000000.0)
     (- (* (* b (* b b)) b) 1.0)
     (* a (* (* a a) (+ a -4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.6e+53) {
		tmp = (a * a) * (a * a);
	} else if (a <= 160000000.0) {
		tmp = ((b * (b * b)) * b) - 1.0;
	} else {
		tmp = a * ((a * a) * (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 <= (-1.6d+53)) then
        tmp = (a * a) * (a * a)
    else if (a <= 160000000.0d0) then
        tmp = ((b * (b * b)) * b) - 1.0d0
    else
        tmp = a * ((a * a) * (a + (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.6e+53) {
		tmp = (a * a) * (a * a);
	} else if (a <= 160000000.0) {
		tmp = ((b * (b * b)) * b) - 1.0;
	} else {
		tmp = a * ((a * a) * (a + -4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.6e+53:
		tmp = (a * a) * (a * a)
	elif a <= 160000000.0:
		tmp = ((b * (b * b)) * b) - 1.0
	else:
		tmp = a * ((a * a) * (a + -4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.6e+53)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 160000000.0)
		tmp = Float64(Float64(Float64(b * Float64(b * b)) * b) - 1.0);
	else
		tmp = Float64(a * Float64(Float64(a * a) * Float64(a + -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.6e+53)
		tmp = (a * a) * (a * a);
	elseif (a <= 160000000.0)
		tmp = ((b * (b * b)) * b) - 1.0;
	else
		tmp = a * ((a * a) * (a + -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.6e+53], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 160000000.0], N[(N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.6e53

   1. Initial program 63.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if -1.6e53 < a < 1.6e8

   1. Initial program 97.7%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if 1.6e8 < a

   1. Initial program 28.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+50}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 6000000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+50)
   (* (* a a) (* a a))
   (if (<= a 6000000.0)
     (- (* (* b b) (* b b)) 1.0)
     (* a (* (* a a) (+ a -4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+50) {
		tmp = (a * a) * (a * a);
	} else if (a <= 6000000.0) {
		tmp = ((b * b) * (b * b)) - 1.0;
	} else {
		tmp = a * ((a * a) * (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 <= (-1.5d+50)) then
        tmp = (a * a) * (a * a)
    else if (a <= 6000000.0d0) then
        tmp = ((b * b) * (b * b)) - 1.0d0
    else
        tmp = a * ((a * a) * (a + (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+50) {
		tmp = (a * a) * (a * a);
	} else if (a <= 6000000.0) {
		tmp = ((b * b) * (b * b)) - 1.0;
	} else {
		tmp = a * ((a * a) * (a + -4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.5e+50:
		tmp = (a * a) * (a * a)
	elif a <= 6000000.0:
		tmp = ((b * b) * (b * b)) - 1.0
	else:
		tmp = a * ((a * a) * (a + -4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+50)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 6000000.0)
		tmp = Float64(Float64(Float64(b * b) * Float64(b * b)) - 1.0);
	else
		tmp = Float64(a * Float64(Float64(a * a) * Float64(a + -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.5e+50)
		tmp = (a * a) * (a * a);
	elseif (a <= 6000000.0)
		tmp = ((b * b) * (b * b)) - 1.0;
	else
		tmp = a * ((a * a) * (a + -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+50], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6000000.0], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.4999999999999999e50

   1. Initial program 63.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if -1.4999999999999999e50 < a < 6e6

   1. Initial program 97.7%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 6e6 < a

   1. Initial program 28.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 82.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7600:\\ \;\;\;\;\left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 510000000:\\ \;\;\;\;12 \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7600.0)
   (* (+ a -4.0) (* a (* a a)))
   (if (<= a 510000000.0)
     (- (* 12.0 (* b b)) 1.0)
     (* a (* (* a a) (+ a -4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7600.0) {
		tmp = (a + -4.0) * (a * (a * a));
	} else if (a <= 510000000.0) {
		tmp = (12.0 * (b * b)) - 1.0;
	} else {
		tmp = a * ((a * a) * (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 <= (-7600.0d0)) then
        tmp = (a + (-4.0d0)) * (a * (a * a))
    else if (a <= 510000000.0d0) then
        tmp = (12.0d0 * (b * b)) - 1.0d0
    else
        tmp = a * ((a * a) * (a + (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7600.0) {
		tmp = (a + -4.0) * (a * (a * a));
	} else if (a <= 510000000.0) {
		tmp = (12.0 * (b * b)) - 1.0;
	} else {
		tmp = a * ((a * a) * (a + -4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7600.0:
		tmp = (a + -4.0) * (a * (a * a))
	elif a <= 510000000.0:
		tmp = (12.0 * (b * b)) - 1.0
	else:
		tmp = a * ((a * a) * (a + -4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7600.0)
		tmp = Float64(Float64(a + -4.0) * Float64(a * Float64(a * a)));
	elseif (a <= 510000000.0)
		tmp = Float64(Float64(12.0 * Float64(b * b)) - 1.0);
	else
		tmp = Float64(a * Float64(Float64(a * a) * Float64(a + -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7600.0)
		tmp = (a + -4.0) * (a * (a * a));
	elseif (a <= 510000000.0)
		tmp = (12.0 * (b * b)) - 1.0;
	else
		tmp = a * ((a * a) * (a + -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7600.0], N[(N[(a + -4.0), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 510000000.0], N[(N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7600

   1. Initial program 63.4%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]

   if -7600 < a < 5.1e8

   1. Initial program 99.1%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 5.1e8 < a

   1. Initial program 28.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 82.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(\left(a \cdot a\right) \cdot \left(a + -4\right)\right)\\ \mathbf{if}\;a \leq -1600:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4200000:\\ \;\;\;\;12 \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* (* a a) (+ a -4.0)))))
   (if (<= a -1600.0) t_0 (if (<= a 4200000.0) (- (* 12.0 (* b b)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * ((a * a) * (a + -4.0));
	double tmp;
	if (a <= -1600.0) {
		tmp = t_0;
	} else if (a <= 4200000.0) {
		tmp = (12.0 * (b * b)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * ((a * a) * (a + (-4.0d0)))
    if (a <= (-1600.0d0)) then
        tmp = t_0
    else if (a <= 4200000.0d0) then
        tmp = (12.0d0 * (b * b)) - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * ((a * a) * (a + -4.0));
	double tmp;
	if (a <= -1600.0) {
		tmp = t_0;
	} else if (a <= 4200000.0) {
		tmp = (12.0 * (b * b)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * ((a * a) * (a + -4.0))
	tmp = 0
	if a <= -1600.0:
		tmp = t_0
	elif a <= 4200000.0:
		tmp = (12.0 * (b * b)) - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(Float64(a * a) * Float64(a + -4.0)))
	tmp = 0.0
	if (a <= -1600.0)
		tmp = t_0;
	elseif (a <= 4200000.0)
		tmp = Float64(Float64(12.0 * Float64(b * b)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * ((a * a) * (a + -4.0));
	tmp = 0.0;
	if (a <= -1600.0)
		tmp = t_0;
	elseif (a <= 4200000.0)
		tmp = (12.0 * (b * b)) - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(N[(a * a), $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1600.0], t$95$0, If[LessEqual[a, 4200000.0], N[(N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1600 or 4.2e6 < a

   1. Initial program 43.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   8. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if -1600 < a < 4.2e6

   1. Initial program 99.1%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 82.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1250000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+21}:\\ \;\;\;\;12 \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot a\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1250000.0)
   (* (* a a) (* a a))
   (if (<= a 3.6e+21) (- (* 12.0 (* b b)) 1.0) (* (* a (* a a)) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1250000.0) {
		tmp = (a * a) * (a * a);
	} else if (a <= 3.6e+21) {
		tmp = (12.0 * (b * b)) - 1.0;
	} else {
		tmp = (a * (a * a)) * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1250000.0d0)) then
        tmp = (a * a) * (a * a)
    else if (a <= 3.6d+21) then
        tmp = (12.0d0 * (b * b)) - 1.0d0
    else
        tmp = (a * (a * a)) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1250000.0) {
		tmp = (a * a) * (a * a);
	} else if (a <= 3.6e+21) {
		tmp = (12.0 * (b * b)) - 1.0;
	} else {
		tmp = (a * (a * a)) * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1250000.0:
		tmp = (a * a) * (a * a)
	elif a <= 3.6e+21:
		tmp = (12.0 * (b * b)) - 1.0
	else:
		tmp = (a * (a * a)) * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1250000.0)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 3.6e+21)
		tmp = Float64(Float64(12.0 * Float64(b * b)) - 1.0);
	else
		tmp = Float64(Float64(a * Float64(a * a)) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1250000.0)
		tmp = (a * a) * (a * a);
	elseif (a <= 3.6e+21)
		tmp = (12.0 * (b * b)) - 1.0;
	else
		tmp = (a * (a * a)) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1250000.0], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.6e+21], N[(N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.25e6

   1. Initial program 63.4%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if -1.25e6 < a < 3.6e21

   1. Initial program 99.1%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   if 3.6e21 < a

   1. Initial program 26.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.015:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 2.45:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot a\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.015)
   (* (* a a) (* a a))
   (if (<= a 2.45) -1.0 (* (* a (* a a)) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.015) {
		tmp = (a * a) * (a * a);
	} else if (a <= 2.45) {
		tmp = -1.0;
	} else {
		tmp = (a * (a * a)) * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.015d0)) then
        tmp = (a * a) * (a * a)
    else if (a <= 2.45d0) then
        tmp = -1.0d0
    else
        tmp = (a * (a * a)) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.015) {
		tmp = (a * a) * (a * a);
	} else if (a <= 2.45) {
		tmp = -1.0;
	} else {
		tmp = (a * (a * a)) * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.015:
		tmp = (a * a) * (a * a)
	elif a <= 2.45:
		tmp = -1.0
	else:
		tmp = (a * (a * a)) * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.015)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 2.45)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * Float64(a * a)) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.015)
		tmp = (a * a) * (a * a);
	elseif (a <= 2.45)
		tmp = -1.0;
	else
		tmp = (a * (a * a)) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.015], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.45], -1.0, N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.014999999999999999

   1. Initial program 63.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if -0.014999999999999999 < a < 2.4500000000000002

   1. Initial program 99.8%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 2.4500000000000002 < a

   1. Initial program 31.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 68.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -0.015:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.45:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -0.015) t_0 (if (<= a 2.45) -1.0 t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -0.015) {
		tmp = t_0;
	} else if (a <= 2.45) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (a * a)
    if (a <= (-0.015d0)) then
        tmp = t_0
    else if (a <= 2.45d0) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -0.015) {
		tmp = t_0;
	} else if (a <= 2.45) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * a)
	tmp = 0
	if a <= -0.015:
		tmp = t_0
	elif a <= 2.45:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -0.015)
		tmp = t_0;
	elseif (a <= 2.45)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * (a * a);
	tmp = 0.0;
	if (a <= -0.015)
		tmp = t_0;
	elseif (a <= 2.45)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.015], t$95$0, If[LessEqual[a, 2.45], -1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.014999999999999999 or 2.4500000000000002 < a

   1. Initial program 45.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if -0.014999999999999999 < a < 2.4500000000000002

   1. Initial program 99.8%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 37.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0009:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.0009) -1.0 (* b (* b 12.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.0009) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.0009d0) then
        tmp = -1.0d0
    else
        tmp = b * (b * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.0009) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.0009:
		tmp = -1.0
	else:
		tmp = b * (b * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.0009)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.0009)
		tmp = -1.0;
	else
		tmp = b * (b * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.0009], -1.0, N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.9999999999999998e-4

   1. Initial program 76.6%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 8.9999999999999998e-4 < b

   1. Initial program 60.9%

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

   3. Taylor expanded in b around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in b around 0 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   12. Taylor expanded in b around inf 0

       \[\leadsto expr\]

   14. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 24.6% accurate, 128.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 72.9%

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

3. Taylor expanded in b around inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in b around 0 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))