Hyperbolic arcsine

Percentage Accurate: 17.8% → 99.8%

Time: 7.0s

Alternatives: 4

Speedup: 7.2×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asinh x))

C

double code(double x) {
	return asinh(x);
}

Python

def code(x):
	return math.asinh(x)

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

code[x_] := N[ArcSinh[x], $MachinePrecision]

Derivation

1. Initial program 17.2%

   \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. asinh-def-rev N/A

      \[\leadsto \color{blue}{\sinh^{-1} x} \]

   7. lower-asinh.f64 99.8

      \[\leadsto \color{blue}{\sinh^{-1} x} \]

4. Applied rewrites 99.8%

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

Alternative 2: 51.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right) \cdot x, x \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (fma 0.075 (* x x) -0.16666666666666666) x) (* x x) x))

C

double code(double x) {
	return fma((fma(0.075, (x * x), -0.16666666666666666) * x), (x * x), x);
}

Julia

function code(x)
	return fma(Float64(fma(0.075, Float64(x * x), -0.16666666666666666) * x), Float64(x * x), x)
end

Wolfram

code[x_] := N[(N[(N[(0.075 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2}\right)} \cdot x + 1 \cdot x \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right)} + 1 \cdot x \]

   5. *-lft-identity N/A

      \[\leadsto \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right) + \color{blue}{x} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot x\right) \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot x, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right)} \]

   8. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   11. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}}, x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2}} - \frac{1}{6}, x\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{6}, x\right) \]

   14. lower-*.f64 48.6

       \[\leadsto \mathsf{fma}\left({x}^{3}, 0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666, x\right) \]

5. Applied rewrites 48.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 48.6%

   \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right) \cdot x, \color{blue}{x \cdot x}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x \cdot x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 48.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right) \cdot x, x \cdot x, x\right) \]
11. Add Preprocessing

Alternative 3: 50.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.075\right) \cdot x, x \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* (* (* x x) 0.075) x) (* x x) x))

C

double code(double x) {
	return fma((((x * x) * 0.075) * x), (x * x), x);
}

Julia

function code(x)
	return fma(Float64(Float64(Float64(x * x) * 0.075) * x), Float64(x * x), x)
end

Wolfram

code[x_] := N[(N[(N[(N[(x * x), $MachinePrecision] * 0.075), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2}\right)} \cdot x + 1 \cdot x \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right)} + 1 \cdot x \]

   5. *-lft-identity N/A

      \[\leadsto \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right) + \color{blue}{x} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot x\right) \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot x, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right)} \]

   8. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   11. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}}, x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2}} - \frac{1}{6}, x\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{6}, x\right) \]

   14. lower-*.f64 48.6

       \[\leadsto \mathsf{fma}\left({x}^{3}, 0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666, x\right) \]

5. Applied rewrites 48.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 48.6%

   \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right) \cdot x, \color{blue}{x \cdot x}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x \cdot x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 48.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right) \cdot x, x \cdot x, x\right) \]

11. Taylor expanded in x around inf

    \[\leadsto \mathsf{fma}\left(\left(\frac{3}{40} \cdot {x}^{2}\right) \cdot x, x \cdot x, x\right) \]
12. Step-by-step derivation

13. Applied rewrites 48.3%

    \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.075\right) \cdot x, x \cdot x, x\right) \]
14. Add Preprocessing

Alternative 4: 49.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fma (* -0.16666666666666666 x) x 1.0) x))

C

double code(double x) {
	return fma((-0.16666666666666666 * x), x, 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2}\right)} \cdot x + 1 \cdot x \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right)} + 1 \cdot x \]

   5. *-lft-identity N/A

      \[\leadsto \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right) + \color{blue}{x} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot x\right) \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot x, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right)} \]

   8. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   11. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}}, x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2}} - \frac{1}{6}, x\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{6}, x\right) \]

   14. lower-*.f64 48.6

       \[\leadsto \mathsf{fma}\left({x}^{3}, 0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666, x\right) \]

5. Applied rewrites 48.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 48.6%

   \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right) \cdot x, \color{blue}{x \cdot x}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot x, x \cdot x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 47.0%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x \cdot x, x\right) \]
11. Step-by-step derivation

12. Applied rewrites 47.0%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot \color{blue}{x} \]
13. Add Preprocessing

Developer Target 1: 29.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot x + 1}\\ \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* x x) 1.0))))
   (if (< x 0.0) (log (/ -1.0 (- x t_0))) (log (+ x t_0)))))

C

double code(double x) {
	double t_0 = sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = log((-1.0 / (x - t_0)));
	} else {
		tmp = log((x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((x * x) + 1.0d0))
    if (x < 0.0d0) then
        tmp = log(((-1.0d0) / (x - t_0)))
    else
        tmp = log((x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = Math.log((-1.0 / (x - t_0)));
	} else {
		tmp = Math.log((x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt(((x * x) + 1.0))
	tmp = 0
	if x < 0.0:
		tmp = math.log((-1.0 / (x - t_0)))
	else:
		tmp = math.log((x + t_0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(Float64(x * x) + 1.0))
	tmp = 0.0
	if (x < 0.0)
		tmp = log(Float64(-1.0 / Float64(x - t_0)));
	else
		tmp = log(Float64(x + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt(((x * x) + 1.0));
	tmp = 0.0;
	if (x < 0.0)
		tmp = log((-1.0 / (x - t_0)));
	else
		tmp = log((x + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, If[Less[x, 0.0], N[Log[N[(-1.0 / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[N[(x + t$95$0), $MachinePrecision]], $MachinePrecision]]]

Reproduce

herbie shell --seed 2024337 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1))))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))