bug333 (missed optimization)

Percentage Accurate: 65.2% → 99.7%

Time: 7.2s

Alternatives: 6

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: 65.2% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\_m + \left(0.0546875 \cdot {x\_m}^{5} + 0.125 \cdot {x\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 6e-123)
    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 <= 6e-123) {
		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 <= 6d-123) 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 <= 6e-123) {
		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 <= 6e-123:
		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 <= 6e-123)
		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 <= 6e-123)
		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, 6e-123], 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 < 5.99999999999999968e-123

   1. Initial program 80.6%

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

   3. Taylor expanded in x around inf 78.3%

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

   if 5.99999999999999968e-123 < x

   1. Initial program 9.5%

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

   3. Taylor expanded in x around 0 99.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.0546875 \cdot {x}^{5} + 0.125 \cdot {x}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x\_m}{\mathsf{fma}\left({x\_m}^{2}, -0.25, 2\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 6e-123) 0.0 (* 2.0 (/ x_m (fma (pow x_m 2.0) -0.25 2.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 6e-123) {
		tmp = 0.0;
	} else {
		tmp = 2.0 * (x_m / fma(pow(x_m, 2.0), -0.25, 2.0));
	}
	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 <= 6e-123)
		tmp = 0.0;
	else
		tmp = Float64(2.0 * Float64(x_m / fma((x_m ^ 2.0), -0.25, 2.0)));
	end
	return Float64(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, 6e-123], 0.0, N[(2.0 * N[(x$95$m / N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.25 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999968e-123

   1. Initial program 80.6%

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

   3. Taylor expanded in x around inf 78.3%

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

   if 5.99999999999999968e-123 < x

   1. Initial program 9.5%

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

      1. flip-- 9.4%

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

      2. div-inv 9.4%

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

      3. add-sqr-sqrt 9.5%

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

      4. add-sqr-sqrt 9.6%

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

      5. associate--r- 22.9%

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

      6. add-exp-log 22.9%

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

      7. log1p-udef 22.9%

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

      8. expm1-udef 100.0%

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

      9. expm1-log1p-u 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. count-2 100.0%

         \[\leadsto \frac{\color{blue}{2 \cdot x}}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      4. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{2}{\frac{\sqrt{1 + x} + \sqrt{1 - x}}{x}}} \]

      5. remove-double-neg 99.9%

         \[\leadsto \frac{2}{\frac{\sqrt{1 + x} + \color{blue}{\left(-\left(-\sqrt{1 - x}\right)\right)}}{x}} \]

      6. sub-neg 99.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{1 + x} - \left(-\sqrt{1 - x}\right)}}{x}} \]

      7. sub-neg 99.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{1 + x} + \left(-\left(-\sqrt{1 - x}\right)\right)}}{x}} \]

      8. remove-double-neg 99.9%

         \[\leadsto \frac{2}{\frac{\sqrt{1 + x} + \color{blue}{\sqrt{1 - x}}}{x}} \]

      9. +-commutative 99.9%

         \[\leadsto \frac{2}{\frac{\sqrt{\color{blue}{x + 1}} + \sqrt{1 - x}}{x}} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{2}{\frac{\sqrt{x + 1} + \sqrt{1 - x}}{x}}} \]

   7. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

         \[\leadsto \frac{2}{\frac{2 + \color{blue}{{x}^{2} \cdot -0.25}}{x}} \]

   9. Simplified 99.0%

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

       1. clear-num 99.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{2 + {x}^{2} \cdot -0.25}{x}}{2}}} \]

       2. associate-/r/ 99.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{2 + {x}^{2} \cdot -0.25}{x}} \cdot 2} \]

       3. clear-num 99.2%

          \[\leadsto \color{blue}{\frac{x}{2 + {x}^{2} \cdot -0.25}} \cdot 2 \]

       4. +-commutative 99.2%

          \[\leadsto \frac{x}{\color{blue}{{x}^{2} \cdot -0.25 + 2}} \cdot 2 \]

       5. fma-def 99.2%

          \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.25, 2\right)}} \cdot 2 \]

   11. Applied egg-rr 99.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.25, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\_m + 0.125 \cdot {x\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 6e-123) 0.0 (+ x_m (* 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 <= 6e-123) {
		tmp = 0.0;
	} else {
		tmp = x_m + (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 <= 6d-123) then
        tmp = 0.0d0
    else
        tmp = x_m + (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 <= 6e-123) {
		tmp = 0.0;
	} else {
		tmp = x_m + (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 <= 6e-123:
		tmp = 0.0
	else:
		tmp = x_m + (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 <= 6e-123)
		tmp = 0.0;
	else
		tmp = Float64(x_m + 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 <= 6e-123)
		tmp = 0.0;
	else
		tmp = x_m + (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, 6e-123], 0.0, N[(x$95$m + N[(0.125 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999968e-123

   1. Initial program 80.6%

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

   3. Taylor expanded in x around inf 78.3%

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

   if 5.99999999999999968e-123 < x

   1. Initial program 9.5%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{x + 0.125 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x + 0.125 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x\_m \cdot -0.25 + 2 \cdot \frac{1}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 6e-123) 0.0 (/ 2.0 (+ (* x_m -0.25) (* 2.0 (/ 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 <= 6e-123) {
		tmp = 0.0;
	} else {
		tmp = 2.0 / ((x_m * -0.25) + (2.0 * (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 <= 6d-123) then
        tmp = 0.0d0
    else
        tmp = 2.0d0 / ((x_m * (-0.25d0)) + (2.0d0 * (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 <= 6e-123) {
		tmp = 0.0;
	} else {
		tmp = 2.0 / ((x_m * -0.25) + (2.0 * (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 <= 6e-123:
		tmp = 0.0
	else:
		tmp = 2.0 / ((x_m * -0.25) + (2.0 * (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 <= 6e-123)
		tmp = 0.0;
	else
		tmp = Float64(2.0 / Float64(Float64(x_m * -0.25) + Float64(2.0 * 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 <= 6e-123)
		tmp = 0.0;
	else
		tmp = 2.0 / ((x_m * -0.25) + (2.0 * (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, 6e-123], 0.0, N[(2.0 / N[(N[(x$95$m * -0.25), $MachinePrecision] + N[(2.0 * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999968e-123

   1. Initial program 80.6%

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

   3. Taylor expanded in x around inf 78.3%

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

   if 5.99999999999999968e-123 < x

   1. Initial program 9.5%

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

      1. flip-- 9.4%

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

      2. div-inv 9.4%

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

      3. add-sqr-sqrt 9.5%

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

      4. add-sqr-sqrt 9.6%

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

      5. associate--r- 22.9%

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

      6. add-exp-log 22.9%

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

      7. log1p-udef 22.9%

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

      8. expm1-udef 100.0%

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

      9. expm1-log1p-u 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. count-2 100.0%

         \[\leadsto \frac{\color{blue}{2 \cdot x}}{\sqrt{1 + x} + \sqrt{1 - x}} \]

      4. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{2}{\frac{\sqrt{1 + x} + \sqrt{1 - x}}{x}}} \]

      5. remove-double-neg 99.9%

         \[\leadsto \frac{2}{\frac{\sqrt{1 + x} + \color{blue}{\left(-\left(-\sqrt{1 - x}\right)\right)}}{x}} \]

      6. sub-neg 99.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{1 + x} - \left(-\sqrt{1 - x}\right)}}{x}} \]

      7. sub-neg 99.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{1 + x} + \left(-\left(-\sqrt{1 - x}\right)\right)}}{x}} \]

      8. remove-double-neg 99.9%

         \[\leadsto \frac{2}{\frac{\sqrt{1 + x} + \color{blue}{\sqrt{1 - x}}}{x}} \]

      9. +-commutative 99.9%

         \[\leadsto \frac{2}{\frac{\sqrt{\color{blue}{x + 1}} + \sqrt{1 - x}}{x}} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{2}{\frac{\sqrt{x + 1} + \sqrt{1 - x}}{x}}} \]

   7. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

         \[\leadsto \frac{2}{\frac{2 + \color{blue}{{x}^{2} \cdot -0.25}}{x}} \]

   9. Simplified 99.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{2 + {x}^{2} \cdot -0.25}}{x}} \]

   10. Taylor expanded in x around 0 99.1%

       \[\leadsto \frac{2}{\color{blue}{-0.25 \cdot x + 2 \cdot \frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot -0.25 + 2 \cdot \frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 34.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (if (<= x_m 6e-123) 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 <= 6e-123) {
		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 <= 6d-123) 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 <= 6e-123) {
		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 <= 6e-123:
		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 <= 6e-123)
		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 <= 6e-123)
		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, 6e-123], 0.0, x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999968e-123

   1. Initial program 80.6%

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

   3. Taylor expanded in x around inf 78.3%

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

   if 5.99999999999999968e-123 < x

   1. Initial program 9.5%

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

   3. Taylor expanded in x around 0 98.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.2% 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 66.1%

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

3. Taylor expanded in x around inf 63.1%

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

4. Final simplification 63.1%

   \[\leadsto 0 \]
5. Add Preprocessing

Developer target: 43.2% 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 2024031 
(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))))