sqrtexp (problem 3.4.4)

Percentage Accurate: 35.3% → 100.0%

Time: 6.1s

Alternatives: 4

Speedup: 3.0×

• could not determine a ground truth

  1. x = 8.393955272903564e+280

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: 35.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. *-commutative 38.6%

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

   2. exp-lft-sqr 39.6%

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

   3. difference-of-sqr-1 40.2%

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

   4. associate-*r/ 40.6%

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

   5. *-inverses 100.0%

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

   6. *-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 67.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ 2.0 (* x (+ 1.0 (* x 0.5))))))

C

double code(double x) {
	return sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 + (x * (1.0d0 + (x * 0.5d0)))))
end function

Java

public static double code(double x) {
	return Math.sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
}

Python

def code(x):
	return math.sqrt((2.0 + (x * (1.0 + (x * 0.5)))))

Julia

function code(x)
	return sqrt(Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * 0.5)))))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
end

Wolfram

code[x_] := N[Sqrt[N[(2.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. *-commutative 38.6%

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

   2. exp-lft-sqr 39.6%

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

   3. difference-of-sqr-1 40.2%

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

   4. associate-*r/ 40.6%

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

   5. *-inverses 100.0%

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

   6. *-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 65.2%

   \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + 0.5 \cdot x\right)}} \]

6. Final simplification 65.2%

   \[\leadsto \sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)} \]
7. Add Preprocessing

Alternative 3: 67.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{0.5 + x \cdot -0.25}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (sqrt (+ 0.5 (* x -0.25)))))

C

double code(double x) {
	return 1.0 / sqrt((0.5 + (x * -0.25)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / sqrt((0.5d0 + (x * (-0.25d0))))
end function

Java

public static double code(double x) {
	return 1.0 / Math.sqrt((0.5 + (x * -0.25)));
}

Python

def code(x):
	return 1.0 / math.sqrt((0.5 + (x * -0.25)))

Julia

function code(x)
	return Float64(1.0 / sqrt(Float64(0.5 + Float64(x * -0.25))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / sqrt((0.5 + (x * -0.25)));
end

Wolfram

code[x_] := N[(1.0 / N[Sqrt[N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. *-commutative 38.6%

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

   2. exp-lft-sqr 39.6%

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

   3. difference-of-sqr-1 40.2%

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

   4. associate-*r/ 40.6%

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

   5. *-inverses 100.0%

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

   6. *-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

3. Simplified 100.0%

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

   1. +-commutative 100.0%

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

   2. flip-+ 39.6%

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

   3. metadata-eval 39.6%

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

   4. exp-lft-sqr 38.6%

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

   5. *-commutative 38.6%

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

   6. expm1-undefine 38.5%

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}} \]

   7. expm1-undefine 98.8%

      \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

   8. clear-num 98.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

   9. sqrt-div 97.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

   10. metadata-eval 97.9%

       \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}} \]

   11. clear-num 97.9%

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}}}}} \]

   12. expm1-undefine 38.6%

       \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{2 \cdot x} - 1}}{\mathsf{expm1}\left(x\right)}}}} \]

   13. *-commutative 38.6%

       \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{\color{blue}{x \cdot 2}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

   14. exp-lft-sqr 38.5%

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

   15. metadata-eval 38.5%

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

   16. expm1-undefine 39.6%

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

   17. flip-+ 99.1%

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

   18. +-commutative 99.1%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in x around 0 65.0%

   \[\leadsto \frac{1}{\sqrt{\color{blue}{0.5 + -0.25 \cdot x}}} \]
8. Step-by-step derivation

   1. *-commutative 65.0%

      \[\leadsto \frac{1}{\sqrt{0.5 + \color{blue}{x \cdot -0.25}}} \]

9. Simplified 65.0%

   \[\leadsto \frac{1}{\sqrt{\color{blue}{0.5 + x \cdot -0.25}}} \]
10. Add Preprocessing

Alternative 4: 66.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt(Float64(x + 2.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. *-commutative 38.6%

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

   2. exp-lft-sqr 39.6%

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

   3. difference-of-sqr-1 40.2%

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

   4. associate-*r/ 40.6%

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

   5. *-inverses 100.0%

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

   6. *-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 63.1%

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

6. Final simplification 63.1%

   \[\leadsto \sqrt{x + 2} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024179 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))