sqrtexp (problem 3.4.4)

Percentage Accurate: 36.5% → 100.0%

Time: 7.8s

Alternatives: 3

Speedup: 3.1×

• could not determine a ground truth

  1. x = 1.2398057374039986e+188

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{-1}{-1 - e^{x}}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (/ -1.0 (- -1.0 (exp x))) -0.5))

C

double code(double x) {
	return pow((-1.0 / (-1.0 - exp(x))), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-1.0d0) / ((-1.0d0) - exp(x))) ** (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow((-1.0 / (-1.0 - Math.exp(x))), -0.5);
}

Python

def code(x):
	return math.pow((-1.0 / (-1.0 - math.exp(x))), -0.5)

Julia

function code(x)
	return Float64(-1.0 / Float64(-1.0 - exp(x))) ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 / (-1.0 - exp(x))) ^ -0.5;
end

Wolfram

code[x_] := N[Power[N[(-1.0 / N[(-1.0 - N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{e^{x \cdot 2} - 1}{e^{x} - 1}} \]

   2. exp-lft-sqr N/A

      \[\leadsto \sqrt{\frac{e^{x} \cdot e^{x} - 1}{e^{x} - 1}} \]

   3. difference-of-sqr-1 N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

      \[\leadsto \sqrt{\left(e^{x} + 1\right) \cdot 1} \]

   6. *-rgt-identity N/A

      \[\leadsto \sqrt{e^{x} + 1} \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(e^{x} + 1\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(1 + e^{x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(e^{x}\right)\right)\right) \]

   10. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \sqrt{\frac{1 \cdot 1 - e^{x} \cdot e^{x}}{1 - e^{x}}} \]

   2. clear-num N/A

      \[\leadsto \sqrt{\frac{1}{\frac{1 - e^{x}}{1 \cdot 1 - e^{x} \cdot e^{x}}}} \]

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1 - e^{x}}{1 \cdot 1 - e^{x} \cdot e^{x}}}\right)}\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1 - e^{x}}{1 \cdot 1 - e^{x} \cdot e^{x}}\right)\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{1 \cdot 1 - e^{x} \cdot e^{x}}{1 - e^{x}}}\right)\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{1 + e^{x}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{x}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{x}\right)\right)\right)\right)\right) \]

   11. exp-lowering-exp.f64 99.1%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{1}{1 + e^{x}}}}} \]
7. Step-by-step derivation

   1. inv-pow N/A

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

   2. sqrt-pow2 N/A

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

   3. metadata-eval N/A

      \[\leadsto {\left(\frac{1}{1 + e^{x}}\right)}^{\frac{-1}{2}} \]

   4. metadata-eval N/A

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

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{x}}\right), \color{blue}{\left(-1 \cdot \frac{1}{2}\right)}\right) \]

   6. frac-2neg N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(1 + e^{x}\right)\right)}\right), \left(\color{blue}{-1} \cdot \frac{1}{2}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\left(1 + e^{x}\right)\right)}\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(1 + e^{x}\right)\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{1}{2}\right)\right) \]

   9. distribute-neg-in N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(e^{x}\right)\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \left(-1 + \left(\mathsf{neg}\left(e^{x}\right)\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   11. unsub-neg N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \left(-1 - e^{x}\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(-1, \left(e^{x}\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   13. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(-1, \mathsf{exp.f64}\left(x\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   14. metadata-eval 100.0%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(-1, \mathsf{exp.f64}\left(x\right)\right)\right), \frac{-1}{2}\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\frac{-1}{-1 - e^{x}}\right)}^{-0.5}} \]
9. Add Preprocessing

Alternative 2: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{e^{x \cdot 2} - 1}{e^{x} - 1}} \]

   2. exp-lft-sqr N/A

      \[\leadsto \sqrt{\frac{e^{x} \cdot e^{x} - 1}{e^{x} - 1}} \]

   3. difference-of-sqr-1 N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

      \[\leadsto \sqrt{\left(e^{x} + 1\right) \cdot 1} \]

   6. *-rgt-identity N/A

      \[\leadsto \sqrt{e^{x} + 1} \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(e^{x} + 1\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(1 + e^{x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(e^{x}\right)\right)\right) \]

   10. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \sqrt{e^{x} + 1} \]
6. Add Preprocessing

Alternative 3: 71.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt 2.0))

C

double code(double x) {
	return sqrt(2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(2.0d0)
end function

Java

public static double code(double x) {
	return Math.sqrt(2.0);
}

Python

def code(x):
	return math.sqrt(2.0)

Julia

function code(x)
	return sqrt(2.0)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(2.0);
end

Wolfram

code[x_] := N[Sqrt[2.0], $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{e^{x \cdot 2} - 1}{e^{x} - 1}} \]

   2. exp-lft-sqr N/A

      \[\leadsto \sqrt{\frac{e^{x} \cdot e^{x} - 1}{e^{x} - 1}} \]

   3. difference-of-sqr-1 N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

      \[\leadsto \sqrt{\left(e^{x} + 1\right) \cdot 1} \]

   6. *-rgt-identity N/A

      \[\leadsto \sqrt{e^{x} + 1} \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(e^{x} + 1\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(1 + e^{x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(e^{x}\right)\right)\right) \]

   10. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{2}} \]
6. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 68.1%

      \[\leadsto \mathsf{sqrt.f64}\left(2\right) \]

7. Simplified 68.1%

   \[\leadsto \color{blue}{\sqrt{2}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))