sqrtexp (problem 3.4.4)

Percentage Accurate: 36.5% → 99.9%

Time: 8.6s

Alternatives: 4

Speedup: 3.1×

• could not determine a ground truth

  1. x = 2.0984912935949951e+279

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: 99.9% 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 39.1%

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

   1. *-commutative N/A

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

   2. exp-lft-sqr N/A

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

   3. metadata-eval N/A

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

   4. flip-+ N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

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

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

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

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

   10. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 72.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.9)
   (sqrt 2.0)
   (sqrt (+ 2.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt((2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.9d0)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt((2.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.9:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.9)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.9)
		tmp = sqrt(2.0);
	else
		tmp = sqrt((2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.9], N[Sqrt[2.0], $MachinePrecision], N[Sqrt[N[(2.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8999999999999999

   1. Initial program 100.0%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 20.7%

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

   if -1.8999999999999999 < x

   1. Initial program 7.8%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 93.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 93.6%

      \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0) (sqrt 2.0) (sqrt (+ 2.0 (* x (+ 1.0 (* x 0.5)))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.0d0)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt((2.0d0 + (x * (1.0d0 + (x * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 + (x * (1.0 + (x * 0.5)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sqrt(2.0);
	else
		tmp = sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -2

   1. Initial program 100.0%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 20.7%

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

   if -2 < x

   1. Initial program 7.8%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot x\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 93.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right) \]

   5. Simplified 93.0%

      \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + x \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.1% 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 39.1%

   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{2}\right) \]
4. Step-by-step derivation

5. Simplified 66.7%

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

Reproduce

herbie shell --seed 2024194 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))