bug333 (missed optimization)

Percentage Accurate: 95.4% → 100.0%

Time: 4.4s

Alternatives: 5

Speedup: 34.4×

Specification

\[-1 \leq x \land x \leq 1\]

Math

\[\begin{array}{l} \\ \sqrt{1 + x} - \sqrt{1 - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

C

double code(double x) {
	return sqrt((1.0 + x)) - sqrt((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 + x)) - sqrt((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 + x)) - Math.sqrt((1.0 - x));
}

Python

def code(x):
	return math.sqrt((1.0 + x)) - math.sqrt((1.0 - x))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 + x)) - sqrt((1.0 - x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 + x} - \sqrt{1 - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

C

double code(double x) {
	return sqrt((1.0 + x)) - sqrt((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 + x)) - sqrt((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 + x)) - Math.sqrt((1.0 - x));
}

Python

def code(x):
	return math.sqrt((1.0 + x)) - math.sqrt((1.0 - x))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 + x)) - sqrt((1.0 - x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m + x\_m}{\sqrt{x\_m + 1} + \sqrt{1 - x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.6e-26)
    0.0
    (/ (+ x_m x_m) (+ (sqrt (+ x_m 1.0)) (sqrt (- 1.0 x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.6e-26) {
		tmp = 0.0;
	} else {
		tmp = (x_m + x_m) / (sqrt((x_m + 1.0)) + sqrt((1.0 - x_m)));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.6d-26) then
        tmp = 0.0d0
    else
        tmp = (x_m + x_m) / (sqrt((x_m + 1.0d0)) + sqrt((1.0d0 - x_m)))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.6e-26) {
		tmp = 0.0;
	} else {
		tmp = (x_m + x_m) / (Math.sqrt((x_m + 1.0)) + Math.sqrt((1.0 - x_m)));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 2.6e-26:
		tmp = 0.0
	else:
		tmp = (x_m + x_m) / (math.sqrt((x_m + 1.0)) + math.sqrt((1.0 - x_m)))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.6e-26)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x_m + x_m) / Float64(sqrt(Float64(x_m + 1.0)) + sqrt(Float64(1.0 - x_m))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 2.6e-26)
		tmp = 0.0;
	else
		tmp = (x_m + x_m) / (sqrt((x_m + 1.0)) + sqrt((1.0 - x_m)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 2.6e-26], 0.0, N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[Sqrt[N[(x$95$m + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 - x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e-26

   1. Initial program 97.8%

      \[\sqrt{1 + x} - \sqrt{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 95.6%

      \[\leadsto \color{blue}{0} \]

   if 2.6000000000000001e-26 < x

   1. Initial program 33.4%

      \[\sqrt{1 + x} - \sqrt{1 - x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 33.4%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 - x} \cdot \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}}} \]

      2. add-sqr-sqrt 33.5%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{1 - x} \cdot \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      3. add-sqr-sqrt 34.0%

         \[\leadsto \frac{\left(1 + x\right) - \color{blue}{\left(1 - x\right)}}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      4. associate--r- 41.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + x\right) - 1\right) + x}}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      5. add-exp-log 41.8%

         \[\leadsto \frac{\left(\color{blue}{e^{\log \left(1 + x\right)}} - 1\right) + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      6. expm1-undefine 41.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + x\right)\right)} + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      7. log1p-define 99.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right) + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      8. expm1-log1p-u 99.8%

         \[\leadsto \frac{\color{blue}{x} + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{x + x}{\sqrt{1 + x} + \sqrt{1 - x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x}{\sqrt{x + 1} + \sqrt{1 - x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\_m + \left(0.0546875 \cdot {x\_m}^{5} + 0.125 \cdot {x\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.6e-26)
    0.0
    (+ x_m (+ (* 0.0546875 (pow x_m 5.0)) (* 0.125 (pow x_m 3.0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.6e-26) {
		tmp = 0.0;
	} else {
		tmp = x_m + ((0.0546875 * pow(x_m, 5.0)) + (0.125 * pow(x_m, 3.0)));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.6d-26) then
        tmp = 0.0d0
    else
        tmp = x_m + ((0.0546875d0 * (x_m ** 5.0d0)) + (0.125d0 * (x_m ** 3.0d0)))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.6e-26) {
		tmp = 0.0;
	} else {
		tmp = x_m + ((0.0546875 * Math.pow(x_m, 5.0)) + (0.125 * Math.pow(x_m, 3.0)));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 2.6e-26:
		tmp = 0.0
	else:
		tmp = x_m + ((0.0546875 * math.pow(x_m, 5.0)) + (0.125 * math.pow(x_m, 3.0)))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.6e-26)
		tmp = 0.0;
	else
		tmp = Float64(x_m + Float64(Float64(0.0546875 * (x_m ^ 5.0)) + Float64(0.125 * (x_m ^ 3.0))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 2.6e-26)
		tmp = 0.0;
	else
		tmp = x_m + ((0.0546875 * (x_m ^ 5.0)) + (0.125 * (x_m ^ 3.0)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 2.6e-26], 0.0, N[(x$95$m + N[(N[(0.0546875 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e-26

   1. Initial program 97.8%

      \[\sqrt{1 + x} - \sqrt{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 95.6%

      \[\leadsto \color{blue}{0} \]

   if 2.6000000000000001e-26 < x

   1. Initial program 33.4%

      \[\sqrt{1 + x} - \sqrt{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 95.9%

      \[\leadsto \color{blue}{x + \left(0.0546875 \cdot {x}^{5} + 0.125 \cdot {x}^{3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.0546875 \cdot {x}^{5} + 0.125 \cdot {x}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m + x\_m}{2 + -0.25 \cdot \left(x\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.6e-26) 0.0 (/ (+ x_m x_m) (+ 2.0 (* -0.25 (* x_m x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.6e-26) {
		tmp = 0.0;
	} else {
		tmp = (x_m + x_m) / (2.0 + (-0.25 * (x_m * x_m)));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.6d-26) then
        tmp = 0.0d0
    else
        tmp = (x_m + x_m) / (2.0d0 + ((-0.25d0) * (x_m * x_m)))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.6e-26) {
		tmp = 0.0;
	} else {
		tmp = (x_m + x_m) / (2.0 + (-0.25 * (x_m * x_m)));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 2.6e-26:
		tmp = 0.0
	else:
		tmp = (x_m + x_m) / (2.0 + (-0.25 * (x_m * x_m)))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.6e-26)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x_m + x_m) / Float64(2.0 + Float64(-0.25 * Float64(x_m * x_m))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 2.6e-26)
		tmp = 0.0;
	else
		tmp = (x_m + x_m) / (2.0 + (-0.25 * (x_m * x_m)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 2.6e-26], 0.0, N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(2.0 + N[(-0.25 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e-26

   1. Initial program 97.8%

      \[\sqrt{1 + x} - \sqrt{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 95.6%

      \[\leadsto \color{blue}{0} \]

   if 2.6000000000000001e-26 < x

   1. Initial program 33.4%

      \[\sqrt{1 + x} - \sqrt{1 - x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 33.4%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 - x} \cdot \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}}} \]

      2. add-sqr-sqrt 33.5%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right)} - \sqrt{1 - x} \cdot \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      3. add-sqr-sqrt 34.0%

         \[\leadsto \frac{\left(1 + x\right) - \color{blue}{\left(1 - x\right)}}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      4. associate--r- 41.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + x\right) - 1\right) + x}}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      5. add-exp-log 41.8%

         \[\leadsto \frac{\left(\color{blue}{e^{\log \left(1 + x\right)}} - 1\right) + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      6. expm1-undefine 41.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + x\right)\right)} + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      7. log1p-define 99.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right) + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      8. expm1-log1p-u 99.8%

         \[\leadsto \frac{\color{blue}{x} + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{x + x}{\sqrt{1 + x} + \sqrt{1 - x}}} \]

   5. Taylor expanded in x around 0 93.9%

      \[\leadsto \frac{x + x}{\color{blue}{2 + -0.25 \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 93.9%

         \[\leadsto \frac{x + x}{2 + -0.25 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   7. Applied egg-rr 93.9%

      \[\leadsto \frac{x + x}{2 + -0.25 \cdot \color{blue}{\left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x}{2 + -0.25 \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 34.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (if (<= x_m 2.6e-26) 0.0 x_m)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.6e-26) {
		tmp = 0.0;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.6d-26) then
        tmp = 0.0d0
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.6e-26) {
		tmp = 0.0;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 2.6e-26:
		tmp = 0.0
	else:
		tmp = x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.6e-26)
		tmp = 0.0;
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 2.6e-26)
		tmp = 0.0;
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 2.6e-26], 0.0, x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e-26

   1. Initial program 97.8%

      \[\sqrt{1 + x} - \sqrt{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 95.6%

      \[\leadsto \color{blue}{0} \]

   if 2.6000000000000001e-26 < x

   1. Initial program 33.4%

      \[\sqrt{1 + x} - \sqrt{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 87.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.0))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 0.0;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * 0.0d0
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * 0.0;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 0.0

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 0.0)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 90.5%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 90.5%

   \[\leadsto 0 \]
5. Add Preprocessing

Developer target: 12.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x}{\sqrt{1 + x} + \sqrt{1 - x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 2.0 x) (+ (sqrt (+ 1.0 x)) (sqrt (- 1.0 x)))))

C

double code(double x) {
	return (2.0 * x) / (sqrt((1.0 + x)) + sqrt((1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 * x) / (sqrt((1.0d0 + x)) + sqrt((1.0d0 - x)))
end function

Java

public static double code(double x) {
	return (2.0 * x) / (Math.sqrt((1.0 + x)) + Math.sqrt((1.0 - x)));
}

Python

def code(x):
	return (2.0 * x) / (math.sqrt((1.0 + x)) + math.sqrt((1.0 - x)))

Julia

function code(x)
	return Float64(Float64(2.0 * x) / Float64(sqrt(Float64(1.0 + x)) + sqrt(Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 * x) / (sqrt((1.0 + x)) + sqrt((1.0 - x)));
end

Wolfram

code[x_] := N[(N[(2.0 * x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024046 
(FPCore (x)
  :name "bug333 (missed optimization)"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :herbie-target
  (/ (* 2.0 x) (+ (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

  (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))