bug333 (missed optimization)

Percentage Accurate: 8.6% → 100.0%

Time: 7.9s

Alternatives: 9

Speedup: 207.0×

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: 8.6% 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, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot 2}{\mathsf{hypot}\left(1, \sqrt{x\_m}\right) + \sqrt{1 - x\_m}} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (* x_m 2.0) (+ (hypot 1.0 (sqrt x_m)) (sqrt (- 1.0 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m * 2.0) / (hypot(1.0, sqrt(x_m)) + sqrt((1.0 - x_m))));
}

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 * ((x_m * 2.0) / (Math.hypot(1.0, Math.sqrt(x_m)) + Math.sqrt((1.0 - x_m))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m * 2.0) / (math.hypot(1.0, math.sqrt(x_m)) + math.sqrt((1.0 - x_m))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * 2.0) / Float64(hypot(1.0, sqrt(x_m)) + sqrt(Float64(1.0 - x_m)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m * 2.0) / (hypot(1.0, sqrt(x_m)) + sqrt((1.0 - x_m))));
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 * N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[x$95$m], $MachinePrecision] ^ 2], $MachinePrecision] + N[Sqrt[N[(1.0 - x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. flip-- 7.6%

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

   2. div-inv 7.6%

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

   3. add-sqr-sqrt 7.6%

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

   4. add-sqr-sqrt 7.6%

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

   5. associate--r- 20.5%

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

   6. add-exp-log 20.5%

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

   7. expm1-undefine 20.5%

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

   8. log1p-define 100.0%

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

   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. *-commutative 100.0%

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

   5. metadata-eval 100.0%

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

   6. rem-square-sqrt 52.0%

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

   7. hypot-undefine 52.0%

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

6. Simplified 52.0%

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

7. Final simplification 52.0%

   \[\leadsto \frac{x \cdot 2}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{1 - x}} \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m + x\_m}{\sqrt{1 - x\_m} + \sqrt{x\_m + 1}} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (+ x_m x_m) (+ (sqrt (- 1.0 x_m)) (sqrt (+ x_m 1.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m + x_m) / (sqrt((1.0 - x_m)) + sqrt((x_m + 1.0))));
}

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 * ((x_m + x_m) / (sqrt((1.0d0 - x_m)) + sqrt((x_m + 1.0d0))))
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 * ((x_m + x_m) / (Math.sqrt((1.0 - x_m)) + Math.sqrt((x_m + 1.0))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m + x_m) / (math.sqrt((1.0 - x_m)) + math.sqrt((x_m + 1.0))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m + x_m) / Float64(sqrt(Float64(1.0 - x_m)) + sqrt(Float64(x_m + 1.0)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m + x_m) / (sqrt((1.0 - x_m)) + sqrt((x_m + 1.0))));
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 * N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[Sqrt[N[(1.0 - x$95$m), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.6%

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

   3. add-sqr-sqrt 7.6%

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

   4. associate--r- 20.5%

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

   5. add-exp-log 20.5%

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

   6. expm1-undefine 20.5%

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

   7. log1p-define 100.0%

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

   8. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{x + x}{\sqrt{1 - x} + \sqrt{x + 1}} \]
6. Add Preprocessing

Alternative 3: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m + x\_m}{\sqrt{1 - x\_m} + \left(1 + x\_m \cdot \left(0.5 + x\_m \cdot \left(x\_m \cdot 0.0625 - 0.125\right)\right)\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (+ x_m x_m)
   (+
    (sqrt (- 1.0 x_m))
    (+ 1.0 (* x_m (+ 0.5 (* x_m (- (* x_m 0.0625) 0.125)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m + x_m) / (sqrt((1.0 - x_m)) + (1.0 + (x_m * (0.5 + (x_m * ((x_m * 0.0625) - 0.125)))))));
}

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 * ((x_m + x_m) / (sqrt((1.0d0 - x_m)) + (1.0d0 + (x_m * (0.5d0 + (x_m * ((x_m * 0.0625d0) - 0.125d0)))))))
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 * ((x_m + x_m) / (Math.sqrt((1.0 - x_m)) + (1.0 + (x_m * (0.5 + (x_m * ((x_m * 0.0625) - 0.125)))))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m + x_m) / (math.sqrt((1.0 - x_m)) + (1.0 + (x_m * (0.5 + (x_m * ((x_m * 0.0625) - 0.125)))))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m + x_m) / Float64(sqrt(Float64(1.0 - x_m)) + Float64(1.0 + Float64(x_m * Float64(0.5 + Float64(x_m * Float64(Float64(x_m * 0.0625) - 0.125))))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m + x_m) / (sqrt((1.0 - x_m)) + (1.0 + (x_m * (0.5 + (x_m * ((x_m * 0.0625) - 0.125)))))));
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 * N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[Sqrt[N[(1.0 - x$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x$95$m * N[(0.5 + N[(x$95$m * N[(N[(x$95$m * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.6%

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

   3. add-sqr-sqrt 7.6%

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

   4. associate--r- 20.5%

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

   5. add-exp-log 20.5%

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

   6. expm1-undefine 20.5%

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

   7. log1p-define 100.0%

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

   8. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.9%

   \[\leadsto \frac{x + x}{\color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right)} + \sqrt{1 - x}} \]

6. Final simplification 99.9%

   \[\leadsto \frac{x + x}{\sqrt{1 - x} + \left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right)} \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot 2}{2 + -0.25 \cdot {x\_m}^{2}} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (* x_m 2.0) (+ 2.0 (* -0.25 (pow x_m 2.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m * 2.0) / (2.0 + (-0.25 * pow(x_m, 2.0))));
}

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 * ((x_m * 2.0d0) / (2.0d0 + ((-0.25d0) * (x_m ** 2.0d0))))
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 * ((x_m * 2.0) / (2.0 + (-0.25 * Math.pow(x_m, 2.0))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m * 2.0) / (2.0 + (-0.25 * math.pow(x_m, 2.0))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * 2.0) / Float64(2.0 + Float64(-0.25 * (x_m ^ 2.0)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m * 2.0) / (2.0 + (-0.25 * (x_m ^ 2.0))));
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 * N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(2.0 + N[(-0.25 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. flip-- 7.6%

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

   2. div-inv 7.6%

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

   3. add-sqr-sqrt 7.6%

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

   4. add-sqr-sqrt 7.6%

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

   5. associate--r- 20.5%

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

   6. add-exp-log 20.5%

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

   7. expm1-undefine 20.5%

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

   8. log1p-define 100.0%

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

   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. *-commutative 100.0%

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

   5. metadata-eval 100.0%

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

   6. rem-square-sqrt 52.0%

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

   7. hypot-undefine 52.0%

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

6. Simplified 52.0%

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

7. Taylor expanded in x around 0 99.9%

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

8. Final simplification 99.9%

   \[\leadsto \frac{x \cdot 2}{2 + -0.25 \cdot {x}^{2}} \]
9. Add Preprocessing

Alternative 5: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m + 0.125 \cdot {x\_m}^{3}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (+ 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) {
	return x_s * (x_m + (0.125 * pow(x_m, 3.0)));
}

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 * (x_m + (0.125d0 * (x_m ** 3.0d0)))
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 * (x_m + (0.125 * Math.pow(x_m, 3.0)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m + (0.125 * math.pow(x_m, 3.0)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m + Float64(0.125 * (x_m ^ 3.0))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m + (0.125 * (x_m ^ 3.0)));
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 * N[(x$95$m + N[(0.125 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.125 \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 99.9%

      \[\leadsto \color{blue}{1 \cdot x + \left(0.125 \cdot {x}^{2}\right) \cdot x} \]

   2. *-lft-identity 99.9%

      \[\leadsto \color{blue}{x} + \left(0.125 \cdot {x}^{2}\right) \cdot x \]

   3. associate-*l* 99.9%

      \[\leadsto x + \color{blue}{0.125 \cdot \left({x}^{2} \cdot x\right)} \]

   4. unpow2 99.9%

      \[\leadsto x + 0.125 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

   5. unpow3 99.9%

      \[\leadsto x + 0.125 \cdot \color{blue}{{x}^{3}} \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{x + 0.125 \cdot {x}^{3}} \]

6. Final simplification 99.9%

   \[\leadsto x + 0.125 \cdot {x}^{3} \]
7. Add Preprocessing

Alternative 6: 99.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m + x\_m}{\left(1 + x\_m \cdot \left(0.5 + x\_m \cdot \left(x\_m \cdot 0.0625 - 0.125\right)\right)\right) + \left(1 + x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot -0.0625 - 0.125\right) - 0.5\right)\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (+ x_m x_m)
   (+
    (+ 1.0 (* x_m (+ 0.5 (* x_m (- (* x_m 0.0625) 0.125)))))
    (+ 1.0 (* x_m (- (* x_m (- (* x_m -0.0625) 0.125)) 0.5)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * ((x_m * 0.0625) - 0.125))))) + (1.0 + (x_m * ((x_m * ((x_m * -0.0625) - 0.125)) - 0.5)))));
}

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 * ((x_m + x_m) / ((1.0d0 + (x_m * (0.5d0 + (x_m * ((x_m * 0.0625d0) - 0.125d0))))) + (1.0d0 + (x_m * ((x_m * ((x_m * (-0.0625d0)) - 0.125d0)) - 0.5d0)))))
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 * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * ((x_m * 0.0625) - 0.125))))) + (1.0 + (x_m * ((x_m * ((x_m * -0.0625) - 0.125)) - 0.5)))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * ((x_m * 0.0625) - 0.125))))) + (1.0 + (x_m * ((x_m * ((x_m * -0.0625) - 0.125)) - 0.5)))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m + x_m) / Float64(Float64(1.0 + Float64(x_m * Float64(0.5 + Float64(x_m * Float64(Float64(x_m * 0.0625) - 0.125))))) + Float64(1.0 + Float64(x_m * Float64(Float64(x_m * Float64(Float64(x_m * -0.0625) - 0.125)) - 0.5))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * ((x_m * 0.0625) - 0.125))))) + (1.0 + (x_m * ((x_m * ((x_m * -0.0625) - 0.125)) - 0.5)))));
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 * N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[(1.0 + N[(x$95$m * N[(0.5 + N[(x$95$m * N[(N[(x$95$m * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x$95$m * N[(N[(x$95$m * N[(N[(x$95$m * -0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.6%

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

   3. add-sqr-sqrt 7.6%

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

   4. associate--r- 20.5%

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

   5. add-exp-log 20.5%

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

   6. expm1-undefine 20.5%

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

   7. log1p-define 100.0%

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

   8. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.9%

   \[\leadsto \frac{x + x}{\color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right)} + \sqrt{1 - x}} \]

6. Taylor expanded in x around 0 99.9%

   \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.0625 \cdot x - 0.125\right) - 0.5\right)\right)}} \]

7. Final simplification 99.9%

   \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) + \left(1 + x \cdot \left(x \cdot \left(x \cdot -0.0625 - 0.125\right) - 0.5\right)\right)} \]
8. Add Preprocessing

Alternative 7: 99.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m + x\_m}{\left(1 + x\_m \cdot \left(0.5 + x\_m \cdot -0.125\right)\right) + \left(1 + x\_m \cdot \left(x\_m \cdot -0.125 - 0.5\right)\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (+ x_m x_m)
   (+
    (+ 1.0 (* x_m (+ 0.5 (* x_m -0.125))))
    (+ 1.0 (* x_m (- (* x_m -0.125) 0.5)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * -0.125)))) + (1.0 + (x_m * ((x_m * -0.125) - 0.5)))));
}

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 * ((x_m + x_m) / ((1.0d0 + (x_m * (0.5d0 + (x_m * (-0.125d0))))) + (1.0d0 + (x_m * ((x_m * (-0.125d0)) - 0.5d0)))))
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 * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * -0.125)))) + (1.0 + (x_m * ((x_m * -0.125) - 0.5)))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * -0.125)))) + (1.0 + (x_m * ((x_m * -0.125) - 0.5)))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m + x_m) / Float64(Float64(1.0 + Float64(x_m * Float64(0.5 + Float64(x_m * -0.125)))) + Float64(1.0 + Float64(x_m * Float64(Float64(x_m * -0.125) - 0.5))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * -0.125)))) + (1.0 + (x_m * ((x_m * -0.125) - 0.5)))));
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 * N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[(1.0 + N[(x$95$m * N[(0.5 + N[(x$95$m * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x$95$m * N[(N[(x$95$m * -0.125), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.6%

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

   3. add-sqr-sqrt 7.6%

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

   4. associate--r- 20.5%

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

   5. add-exp-log 20.5%

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

   6. expm1-undefine 20.5%

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

   7. log1p-define 100.0%

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

   8. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.9%

   \[\leadsto \frac{x + x}{\color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right)} + \sqrt{1 - x}} \]

6. Taylor expanded in x around 0 99.8%

   \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \color{blue}{\left(1 + x \cdot \left(-0.125 \cdot x - 0.5\right)\right)}} \]

7. Taylor expanded in x around 0 99.9%

   \[\leadsto \frac{x + x}{\left(1 + \color{blue}{x \cdot \left(0.5 + -0.125 \cdot x\right)}\right) + \left(1 + x \cdot \left(-0.125 \cdot x - 0.5\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \frac{x + x}{\left(1 + x \cdot \color{blue}{\left(-0.125 \cdot x + 0.5\right)}\right) + \left(1 + x \cdot -0.5\right)} \]

   2. *-commutative 99.5%

      \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(\color{blue}{x \cdot -0.125} + 0.5\right)\right) + \left(1 + x \cdot -0.5\right)} \]

9. Simplified 99.9%

   \[\leadsto \frac{x + x}{\left(1 + \color{blue}{x \cdot \left(x \cdot -0.125 + 0.5\right)}\right) + \left(1 + x \cdot \left(-0.125 \cdot x - 0.5\right)\right)} \]

10. Final simplification 99.9%

    \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) + \left(1 + x \cdot \left(x \cdot -0.125 - 0.5\right)\right)} \]
11. Add Preprocessing

Alternative 8: 99.0% accurate, 10.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m + x\_m}{\left(1 + x\_m \cdot \left(0.5 + x\_m \cdot -0.125\right)\right) + \left(1 + x\_m \cdot -0.5\right)} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (+ x_m x_m)
   (+ (+ 1.0 (* x_m (+ 0.5 (* x_m -0.125)))) (+ 1.0 (* x_m -0.5))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * -0.125)))) + (1.0 + (x_m * -0.5))));
}

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 * ((x_m + x_m) / ((1.0d0 + (x_m * (0.5d0 + (x_m * (-0.125d0))))) + (1.0d0 + (x_m * (-0.5d0)))))
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 * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * -0.125)))) + (1.0 + (x_m * -0.5))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * -0.125)))) + (1.0 + (x_m * -0.5))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m + x_m) / Float64(Float64(1.0 + Float64(x_m * Float64(0.5 + Float64(x_m * -0.125)))) + Float64(1.0 + Float64(x_m * -0.5)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m + x_m) / ((1.0 + (x_m * (0.5 + (x_m * -0.125)))) + (1.0 + (x_m * -0.5))));
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 * N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[(1.0 + N[(x$95$m * N[(0.5 + N[(x$95$m * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.6%

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

   3. add-sqr-sqrt 7.6%

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

   4. associate--r- 20.5%

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

   5. add-exp-log 20.5%

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

   6. expm1-undefine 20.5%

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

   7. log1p-define 100.0%

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

   8. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.9%

   \[\leadsto \frac{x + x}{\color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right)} + \sqrt{1 - x}} \]

6. Taylor expanded in x around 0 99.5%

   \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \color{blue}{\left(1 + -0.5 \cdot x\right)}} \]
7. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \left(1 + \color{blue}{x \cdot -0.5}\right)} \]

8. Simplified 99.5%

   \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \color{blue}{\left(1 + x \cdot -0.5\right)}} \]

9. Taylor expanded in x around 0 99.5%

   \[\leadsto \frac{x + x}{\left(1 + \color{blue}{x \cdot \left(0.5 + -0.125 \cdot x\right)}\right) + \left(1 + x \cdot -0.5\right)} \]
10. Step-by-step derivation

    1. +-commutative 99.5%

       \[\leadsto \frac{x + x}{\left(1 + x \cdot \color{blue}{\left(-0.125 \cdot x + 0.5\right)}\right) + \left(1 + x \cdot -0.5\right)} \]

    2. *-commutative 99.5%

       \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(\color{blue}{x \cdot -0.125} + 0.5\right)\right) + \left(1 + x \cdot -0.5\right)} \]

11. Simplified 99.5%

    \[\leadsto \frac{x + x}{\left(1 + \color{blue}{x \cdot \left(x \cdot -0.125 + 0.5\right)}\right) + \left(1 + x \cdot -0.5\right)} \]

12. Final simplification 99.5%

    \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) + \left(1 + x \cdot -0.5\right)} \]
13. Add Preprocessing

Alternative 9: 99.0% accurate, 207.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

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

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 * x_m
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 * x_m;
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
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 * x$95$m), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.4%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 100.0% 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 2024082 
(FPCore (x)
  :name "bug333 (missed optimization)"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :alt
  (/ (* 2.0 x) (+ (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

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