Hyperbolic arc-(co)tangent

Percentage Accurate: 8.3% → 99.8%

Time: 8.4s

Alternatives: 5

Speedup: 14.9×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))

C

double code(double x) {
	return (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
}

Fortran

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

Java

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

Python

def code(x):
	return (1.0 / 2.0) * math.log(((1.0 + x) / (1.0 - x)))

Julia

function code(x)
	return Float64(Float64(1.0 / 2.0) * log(Float64(Float64(1.0 + x) / Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
end

Wolfram

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

Initial Program: 8.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))

C

double code(double x) {
	return (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
}

Fortran

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

Java

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

Python

def code(x):
	return (1.0 / 2.0) * math.log(((1.0 + x) / (1.0 - x)))

Julia

function code(x)
	return Float64(Float64(1.0 / 2.0) * log(Float64(Float64(1.0 + x) / Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
end

Wolfram

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

Alternative 1: 99.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (*
   (fma (fma (* x x) 0.14285714285714285 0.2) (* x x) 0.3333333333333333)
   (* x x))
  x
  x))

C

double code(double x) {
	return fma((fma(fma((x * x), 0.14285714285714285, 0.2), (x * x), 0.3333333333333333) * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(fma(fma(Float64(x * x), 0.14285714285714285, 0.2), Float64(x * x), 0.3333333333333333) * Float64(x * x)), x, x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.14285714285714285 + 0.2), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto \color{blue}{x} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) + x} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)} + x \]

   5. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + x \]

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) + \frac{1}{3}}, x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{3}, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, {x}^{2}, \frac{1}{3}\right)}, x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\frac{1}{7} \cdot {x}^{2} + \frac{1}{5}}, {x}^{2}, \frac{1}{3}\right), x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{7}, {x}^{2}, \frac{1}{5}\right)}, {x}^{2}, \frac{1}{3}\right), x\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, \color{blue}{x \cdot x}, \frac{1}{5}\right), {x}^{2}, \frac{1}{3}\right), x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, \color{blue}{x \cdot x}, \frac{1}{5}\right), {x}^{2}, \frac{1}{3}\right), x\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, x \cdot x, \frac{1}{5}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), x\right) \]

   17. lower-*.f64 100.0

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), \color{blue}{x \cdot x}, 0.3333333333333333\right), x\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
8. Add Preprocessing

Alternative 2: 99.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (fma 0.2 (* x x) 0.3333333333333333) (* x x)) x x))

C

double code(double x) {
	return fma((fma(0.2, (x * x), 0.3333333333333333) * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(fma(0.2, Float64(x * x), 0.3333333333333333) * Float64(x * x)), x, x)
end

Wolfram

code[x_] := N[(N[(N[(0.2 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto \color{blue}{x} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) + x} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)} + x \]

   5. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + x \]

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) + \frac{1}{3}}, x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{3}, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, {x}^{2}, \frac{1}{3}\right)}, x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\frac{1}{7} \cdot {x}^{2} + \frac{1}{5}}, {x}^{2}, \frac{1}{3}\right), x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{7}, {x}^{2}, \frac{1}{5}\right)}, {x}^{2}, \frac{1}{3}\right), x\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, \color{blue}{x \cdot x}, \frac{1}{5}\right), {x}^{2}, \frac{1}{3}\right), x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, \color{blue}{x \cdot x}, \frac{1}{5}\right), {x}^{2}, \frac{1}{3}\right), x\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, x \cdot x, \frac{1}{5}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), x\right) \]

   17. lower-*.f64 100.0

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), \color{blue}{x \cdot x}, 0.3333333333333333\right), x\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right) \cdot \left(x \cdot x\right), x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right) \]
11. Add Preprocessing

Alternative 3: 99.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (fma 0.2 (* x x) 0.3333333333333333) (* x x) 1.0) x))

C

double code(double x) {
	return fma(fma(0.2, (x * x), 0.3333333333333333), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(fma(0.2, Float64(x * x), 0.3333333333333333), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(0.2 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto \color{blue}{x} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) + x} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)} + x \]

   5. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + x \]

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) + \frac{1}{3}}, x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{3}, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, {x}^{2}, \frac{1}{3}\right)}, x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\frac{1}{7} \cdot {x}^{2} + \frac{1}{5}}, {x}^{2}, \frac{1}{3}\right), x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{7}, {x}^{2}, \frac{1}{5}\right)}, {x}^{2}, \frac{1}{3}\right), x\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, \color{blue}{x \cdot x}, \frac{1}{5}\right), {x}^{2}, \frac{1}{3}\right), x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, \color{blue}{x \cdot x}, \frac{1}{5}\right), {x}^{2}, \frac{1}{3}\right), x\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, x \cdot x, \frac{1}{5}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), x\right) \]

   17. lower-*.f64 100.0

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), \color{blue}{x \cdot x}, 0.3333333333333333\right), x\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right) \cdot \left(x \cdot x\right), x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot \color{blue}{x} \]
13. Add Preprocessing

Alternative 4: 99.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* 0.3333333333333333 (* x x)) x x))

C

double code(double x) {
	return fma((0.3333333333333333 * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(0.3333333333333333 * Float64(x * x)), x, x)
end

Wolfram

code[x_] := N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto \color{blue}{x} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) + x} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)} + x \]

   5. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + x \]

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) + \frac{1}{3}}, x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{3}, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, {x}^{2}, \frac{1}{3}\right)}, x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\frac{1}{7} \cdot {x}^{2} + \frac{1}{5}}, {x}^{2}, \frac{1}{3}\right), x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{7}, {x}^{2}, \frac{1}{5}\right)}, {x}^{2}, \frac{1}{3}\right), x\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, \color{blue}{x \cdot x}, \frac{1}{5}\right), {x}^{2}, \frac{1}{3}\right), x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, \color{blue}{x \cdot x}, \frac{1}{5}\right), {x}^{2}, \frac{1}{3}\right), x\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, x \cdot x, \frac{1}{5}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), x\right) \]

   17. lower-*.f64 100.0

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), \color{blue}{x \cdot x}, 0.3333333333333333\right), x\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]
11. Add Preprocessing

Alternative 5: 99.0% accurate, 14.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (+ x x) 0.5))

C

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

Fortran

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

Java

public static double code(double x) {
	return (x + x) * 0.5;
}

Python

def code(x):
	return (x + x) * 0.5

Julia

function code(x)
	return Float64(Float64(x + x) * 0.5)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.3

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

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(x \cdot 2\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(x \cdot 2\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 99.3%

   \[\leadsto \left(x + \color{blue}{x}\right) \cdot 0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))