Hyperbolic arc-(co)tangent

Percentage Accurate: 8.1% → 99.8%

Time: 9.3s

Alternatives: 6

Speedup: 12.2×

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: 8.1% 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, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.02328042328042328, x \cdot x, -0.044444444444444446\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 0.5\right)} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   x
   (fma
    (fma
     (fma -0.02328042328042328 (* x x) -0.044444444444444446)
     (* x x)
     -0.16666666666666666)
    (* x x)
    0.5))
  0.5))

C

double code(double x) {
	return (x / fma(fma(fma(-0.02328042328042328, (x * x), -0.044444444444444446), (x * x), -0.16666666666666666), (x * x), 0.5)) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(x / fma(fma(fma(-0.02328042328042328, Float64(x * x), -0.044444444444444446), Float64(x * x), -0.16666666666666666), Float64(x * x), 0.5)) * 0.5)
end

Wolfram

code[x_] := N[(N[(x / N[(N[(N[(-0.02328042328042328 * N[(x * x), $MachinePrecision] + -0.044444444444444446), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.4

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right)\right), x \cdot x, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right)\right), x \cdot x, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 99.4%

   \[\leadsto \frac{x}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right)}}} \cdot 0.5 \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{x}{\frac{1}{2} + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-22}{945} \cdot {x}^{2} - \frac{2}{45}\right) - \frac{1}{6}\right)}} \cdot \frac{1}{2} \]
11. Step-by-step derivation

12. Applied rewrites 99.4%

    \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.02328042328042328, x \cdot x, -0.044444444444444446\right), x \cdot x, -0.16666666666666666\right), \color{blue}{x \cdot x}, 0.5\right)} \cdot 0.5 \]
13. Add Preprocessing

Alternative 2: 99.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(-0.044444444444444446, x \cdot x, -0.16666666666666666\right), x \cdot x, 0.5\right)} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   x
   (fma (fma -0.044444444444444446 (* x x) -0.16666666666666666) (* x x) 0.5))
  0.5))

C

double code(double x) {
	return (x / fma(fma(-0.044444444444444446, (x * x), -0.16666666666666666), (x * x), 0.5)) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(x / fma(fma(-0.044444444444444446, Float64(x * x), -0.16666666666666666), Float64(x * x), 0.5)) * 0.5)
end

Wolfram

code[x_] := N[(N[(x / N[(N[(-0.044444444444444446 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.4

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right)\right), x \cdot x, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right)\right), x \cdot x, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 99.4%

   \[\leadsto \frac{x}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right)}}} \cdot 0.5 \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{x}{\frac{1}{2} + \color{blue}{{x}^{2} \cdot \left(\frac{-2}{45} \cdot {x}^{2} - \frac{1}{6}\right)}} \cdot \frac{1}{2} \]
11. Step-by-step derivation

12. Applied rewrites 99.4%

    \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(-0.044444444444444446, x \cdot x, -0.16666666666666666\right), \color{blue}{x \cdot x}, 0.5\right)} \cdot 0.5 \]
13. Add Preprocessing

Alternative 3: 99.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma (fma 0.4 (* x x) 0.6666666666666666) (* x x) 2.0) x) 0.5))

C

double code(double x) {
	return (fma(fma(0.4, (x * x), 0.6666666666666666), (x * x), 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(0.4, Float64(x * x), 0.6666666666666666), Float64(x * x), 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.4 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.4

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. lift-/.f64 N/A

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

   5. metadata-eval 99.4

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{0.5} \]

7. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]
8. Add Preprocessing

Alternative 4: 99.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 0.5\right)} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ x (fma -0.16666666666666666 (* x x) 0.5)) 0.5))

C

double code(double x) {
	return (x / fma(-0.16666666666666666, (x * x), 0.5)) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(x / fma(-0.16666666666666666, Float64(x * x), 0.5)) * 0.5)
end

Wolfram

code[x_] := N[(N[(x / N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.4

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right)\right), x \cdot x, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right)\right), x \cdot x, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 99.4%

   \[\leadsto \frac{x}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right)}}} \cdot 0.5 \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{x}{\frac{1}{2} + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}} \cdot \frac{1}{2} \]
11. Step-by-step derivation

12. Applied rewrites 99.4%

    \[\leadsto \frac{x}{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{x \cdot x}, 0.5\right)} \cdot 0.5 \]
13. Add Preprocessing

Alternative 5: 99.6% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{1}{3} + x \cdot 1 \]

   7. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \frac{1}{3} + \color{blue}{x} \]

   8. lower-fma.f64 N/A

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

   9. lower-pow.f64 99.3

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.3%

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

Alternative 6: 99.2% accurate, 12.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (* x 2.0) 0.5))

C

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

Fortran

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

Java

public static double code(double x) {
	return (x * 2.0) * 0.5;
}

Python

def code(x):
	return (x * 2.0) * 0.5

Julia

function code(x)
	return Float64(Float64(x * 2.0) * 0.5)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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 99.0

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

5. Applied rewrites 99.0%

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot x\right)} \]
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 99.0

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

   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 99.0%

   \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot 0.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))