bug333 (missed optimization)

Percentage Accurate: 8.3% → 99.8%

Time: 5.5s

Alternatives: 4

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: 8.3% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(0.0546875 \cdot {x}^{5} + 0.125 \cdot {x}^{3}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ x (+ (* 0.0546875 (pow x 5.0)) (* 0.125 (pow x 3.0)))))

C

double code(double x) {
	return x + ((0.0546875 * pow(x, 5.0)) + (0.125 * pow(x, 3.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((0.0546875d0 * (x ** 5.0d0)) + (0.125d0 * (x ** 3.0d0)))
end function

Java

public static double code(double x) {
	return x + ((0.0546875 * Math.pow(x, 5.0)) + (0.125 * Math.pow(x, 3.0)));
}

Python

def code(x):
	return x + ((0.0546875 * math.pow(x, 5.0)) + (0.125 * math.pow(x, 3.0)))

Julia

function code(x)
	return Float64(x + Float64(Float64(0.0546875 * (x ^ 5.0)) + Float64(0.125 * (x ^ 3.0))))
end

MATLAB

function tmp = code(x)
	tmp = x + ((0.0546875 * (x ^ 5.0)) + (0.125 * (x ^ 3.0)));
end

Wolfram

code[x_] := N[(x + N[(N[(0.0546875 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{x + \left(0.0546875 \cdot {x}^{5} + 0.125 \cdot {x}^{3}\right)} \]

4. Final simplification 99.9%

   \[\leadsto x + \left(0.0546875 \cdot {x}^{5} + 0.125 \cdot {x}^{3}\right) \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{\mathsf{fma}\left(-0.25, {x}^{2}, 2\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x 2.0) (fma -0.25 (pow x 2.0) 2.0)))

C

double code(double x) {
	return (x * 2.0) / fma(-0.25, pow(x, 2.0), 2.0);
}

Julia

function code(x)
	return Float64(Float64(x * 2.0) / fma(-0.25, (x ^ 2.0), 2.0))
end

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] / N[(-0.25 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

   1. flip-- 8.6%

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

   2. div-inv 8.6%

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

   3. add-sqr-sqrt 8.6%

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

   4. add-sqr-sqrt 8.7%

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

   5. associate--r- 21.2%

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

   6. add-exp-log 21.2%

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

   7. log1p-udef 21.2%

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

   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* 99.7%

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

   5. remove-double-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. sub-neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 99.5%

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

   1. add-log-exp 8.5%

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

   2. *-un-lft-identity 8.5%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\frac{2}{\frac{2 + -0.25 \cdot {x}^{2}}{x}}}\right)} \]

   3. log-prod 8.5%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{2}{\frac{2 + -0.25 \cdot {x}^{2}}{x}}}\right)} \]

   4. metadata-eval 8.5%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{2}{\frac{2 + -0.25 \cdot {x}^{2}}{x}}}\right) \]

   5. div-inv 8.5%

      \[\leadsto 0 + \log \left(e^{\color{blue}{2 \cdot \frac{1}{\frac{2 + -0.25 \cdot {x}^{2}}{x}}}}\right) \]

   6. add-log-exp 99.5%

      \[\leadsto 0 + \color{blue}{2 \cdot \frac{1}{\frac{2 + -0.25 \cdot {x}^{2}}{x}}} \]

   7. clear-num 99.8%

      \[\leadsto 0 + 2 \cdot \color{blue}{\frac{x}{2 + -0.25 \cdot {x}^{2}}} \]

   8. +-commutative 99.8%

      \[\leadsto 0 + 2 \cdot \frac{x}{\color{blue}{-0.25 \cdot {x}^{2} + 2}} \]

   9. fma-def 99.8%

      \[\leadsto 0 + 2 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(-0.25, {x}^{2}, 2\right)}} \]

9. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{0 + 2 \cdot \frac{x}{\mathsf{fma}\left(-0.25, {x}^{2}, 2\right)}} \]
10. Step-by-step derivation

    1. +-lft-identity 99.8%

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

    2. *-commutative 99.8%

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

    3. associate-*l/ 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \frac{x \cdot 2}{\mathsf{fma}\left(-0.25, {x}^{2}, 2\right)} \]
13. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x + 0.125 \cdot {x}^{3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x (* 0.125 (pow x 3.0))))

C

double code(double x) {
	return x + (0.125 * pow(x, 3.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + (0.125d0 * (x ** 3.0d0))
end function

Java

public static double code(double x) {
	return x + (0.125 * Math.pow(x, 3.0));
}

Python

def code(x):
	return x + (0.125 * math.pow(x, 3.0))

Julia

function code(x)
	return Float64(x + Float64(0.125 * (x ^ 3.0)))
end

MATLAB

function tmp = code(x)
	tmp = x + (0.125 * (x ^ 3.0));
end

Wolfram

code[x_] := N[(x + N[(0.125 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.8%

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

Alternative 4: 99.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto x \]
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 2024020 
(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))))