Hyperbolic arc-(co)tangent

Percentage Accurate: 8.3% → 100.0%

Time: 5.2s

Alternatives: 5

Speedup: 134.0×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (- (log1p x) (log1p (- x))) 0.5))

C

double code(double x) {
	return (log1p(x) - log1p(-x)) * 0.5;
}

Java

public static double code(double x) {
	return (Math.log1p(x) - Math.log1p(-x)) * 0.5;
}

Python

def code(x):
	return (math.log1p(x) - math.log1p(-x)) * 0.5

Julia

function code(x)
	return Float64(Float64(log1p(x) - log1p(Float64(-x))) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 8.3

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

   4. lift-log.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. log-div N/A

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

   7. lower--.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lower-log1p.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 - x\right)\right) \cdot \frac{1}{2} \]

   10. lift--.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 - x\right)}\right) \cdot \frac{1}{2} \]

   11. sub-negate1 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \cdot \frac{1}{2} \]

   12. lower-log1p.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(x\right)\right)}\right) \cdot \frac{1}{2} \]

   13. lower-neg.f64 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\color{blue}{-x}\right)\right) \cdot \frac{1}{2} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   15. metadata-eval 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{0.5} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5} \]
4. Add Preprocessing

Alternative 2: 99.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (fma (fma 0.14285714285714285 (* x x) 0.2) (* x x) 0.3333333333333333)
   (* x x)
   1.0)
  x))

C

double code(double x) {
	return fma(fma(fma(0.14285714285714285, (x * x), 0.2), (x * x), 0.3333333333333333), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(fma(fma(0.14285714285714285, Float64(x * x), 0.2), Float64(x * x), 0.3333333333333333), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.14285714285714285 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision] * 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 8.3

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

   4. lift-log.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. log-div N/A

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

   7. lower--.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lower-log1p.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 - x\right)\right) \cdot \frac{1}{2} \]

   10. lift--.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 - x\right)}\right) \cdot \frac{1}{2} \]

   11. sub-negate1 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \cdot \frac{1}{2} \]

   12. lower-log1p.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(x\right)\right)}\right) \cdot \frac{1}{2} \]

   13. lower-neg.f64 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\color{blue}{-x}\right)\right) \cdot \frac{1}{2} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   15. metadata-eval 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{0.5} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5} \]
4. Step-by-step derivation

   1. lift-log1p.f64 N/A

      \[\leadsto \left(\color{blue}{\log \left(1 + x\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   2. flip3-+ N/A

      \[\leadsto \left(\log \color{blue}{\left(\frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   3. log-div N/A

      \[\leadsto \left(\color{blue}{\left(\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\log \left(\color{blue}{1} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   6. lower-log1p.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\mathsf{log1p}\left({x}^{3}\right)} - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   7. lower-pow.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{{x}^{3}}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \log \left(\color{blue}{1} + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   9. lower-log1p.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \color{blue}{\mathsf{log1p}\left(x \cdot x - 1 \cdot x\right)}\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x} - 1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   11. fp-cancel-sub-sign-inv N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x + \left(\mathsf{neg}\left(1\right)\right) \cdot x}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(1\right)\right) \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(x \cdot x + \color{blue}{-1} \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   14. sqr-neg-rev N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   15. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   16. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \color{blue}{\left(-x\right)} + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   17. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{x \cdot -1}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   18. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   19. distribute-rgt-neg-out N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot 1\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   20. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \left(\mathsf{neg}\left(\color{blue}{1 \cdot x}\right)\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   21. distribute-rgt-neg-out N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{1 \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   22. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \color{blue}{\left(-x\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\color{blue}{\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{{x}^{3}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   2. unpow3 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   4. lower-*.f64 100.0

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]

7. Applied rewrites 100.0%

   \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 8.3

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

   4. lift-log.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. log-div N/A

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

   7. lower--.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lower-log1p.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 - x\right)\right) \cdot \frac{1}{2} \]

   10. lift--.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 - x\right)}\right) \cdot \frac{1}{2} \]

   11. sub-negate1 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \cdot \frac{1}{2} \]

   12. lower-log1p.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(x\right)\right)}\right) \cdot \frac{1}{2} \]

   13. lower-neg.f64 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\color{blue}{-x}\right)\right) \cdot \frac{1}{2} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   15. metadata-eval 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{0.5} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5} \]
4. Step-by-step derivation

   1. lift-log1p.f64 N/A

      \[\leadsto \left(\color{blue}{\log \left(1 + x\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   2. flip3-+ N/A

      \[\leadsto \left(\log \color{blue}{\left(\frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   3. log-div N/A

      \[\leadsto \left(\color{blue}{\left(\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\log \left(\color{blue}{1} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   6. lower-log1p.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\mathsf{log1p}\left({x}^{3}\right)} - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   7. lower-pow.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{{x}^{3}}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \log \left(\color{blue}{1} + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   9. lower-log1p.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \color{blue}{\mathsf{log1p}\left(x \cdot x - 1 \cdot x\right)}\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x} - 1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   11. fp-cancel-sub-sign-inv N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x + \left(\mathsf{neg}\left(1\right)\right) \cdot x}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(1\right)\right) \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(x \cdot x + \color{blue}{-1} \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   14. sqr-neg-rev N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   15. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   16. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \color{blue}{\left(-x\right)} + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   17. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{x \cdot -1}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   18. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   19. distribute-rgt-neg-out N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot 1\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   20. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \left(\mathsf{neg}\left(\color{blue}{1 \cdot x}\right)\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   21. distribute-rgt-neg-out N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{1 \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   22. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \color{blue}{\left(-x\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\color{blue}{\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{{x}^{3}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   2. unpow3 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   4. lower-*.f64 100.0

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]

7. Applied rewrites 100.0%

   \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.7

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \]

10. Applied rewrites 99.7%

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

Alternative 4: 99.6% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 8.3

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

   4. lift-log.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. log-div N/A

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

   7. lower--.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lower-log1p.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 - x\right)\right) \cdot \frac{1}{2} \]

   10. lift--.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 - x\right)}\right) \cdot \frac{1}{2} \]

   11. sub-negate1 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \cdot \frac{1}{2} \]

   12. lower-log1p.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(x\right)\right)}\right) \cdot \frac{1}{2} \]

   13. lower-neg.f64 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\color{blue}{-x}\right)\right) \cdot \frac{1}{2} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   15. metadata-eval 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{0.5} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5} \]
4. Step-by-step derivation

   1. lift-log1p.f64 N/A

      \[\leadsto \left(\color{blue}{\log \left(1 + x\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   2. flip3-+ N/A

      \[\leadsto \left(\log \color{blue}{\left(\frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   3. log-div N/A

      \[\leadsto \left(\color{blue}{\left(\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\log \left(\color{blue}{1} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   6. lower-log1p.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\mathsf{log1p}\left({x}^{3}\right)} - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   7. lower-pow.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{{x}^{3}}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \log \left(\color{blue}{1} + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   9. lower-log1p.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \color{blue}{\mathsf{log1p}\left(x \cdot x - 1 \cdot x\right)}\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x} - 1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   11. fp-cancel-sub-sign-inv N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x + \left(\mathsf{neg}\left(1\right)\right) \cdot x}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(1\right)\right) \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(x \cdot x + \color{blue}{-1} \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   14. sqr-neg-rev N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   15. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   16. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \color{blue}{\left(-x\right)} + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   17. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{x \cdot -1}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   18. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   19. distribute-rgt-neg-out N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot 1\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   20. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \left(\mathsf{neg}\left(\color{blue}{1 \cdot x}\right)\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   21. distribute-rgt-neg-out N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{1 \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   22. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \color{blue}{\left(-x\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\color{blue}{\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{{x}^{3}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   2. unpow3 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   4. lower-*.f64 100.0

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]

7. Applied rewrites 100.0%

   \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 99.6

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x \]

10. Applied rewrites 99.6%

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

Alternative 5: 99.1% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 8.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 8.3

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

   4. lift-log.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. log-div N/A

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

   7. lower--.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lower-log1p.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 - x\right)\right) \cdot \frac{1}{2} \]

   10. lift--.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 - x\right)}\right) \cdot \frac{1}{2} \]

   11. sub-negate1 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \cdot \frac{1}{2} \]

   12. lower-log1p.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(x\right)\right)}\right) \cdot \frac{1}{2} \]

   13. lower-neg.f64 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\color{blue}{-x}\right)\right) \cdot \frac{1}{2} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   15. metadata-eval 100.0

       \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \color{blue}{0.5} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5} \]
4. Step-by-step derivation

   1. lift-log1p.f64 N/A

      \[\leadsto \left(\color{blue}{\log \left(1 + x\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   2. flip3-+ N/A

      \[\leadsto \left(\log \color{blue}{\left(\frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   3. log-div N/A

      \[\leadsto \left(\color{blue}{\left(\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\log \left(\color{blue}{1} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   6. lower-log1p.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\mathsf{log1p}\left({x}^{3}\right)} - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   7. lower-pow.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{{x}^{3}}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \log \left(\color{blue}{1} + \left(x \cdot x - 1 \cdot x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   9. lower-log1p.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \color{blue}{\mathsf{log1p}\left(x \cdot x - 1 \cdot x\right)}\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x} - 1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   11. fp-cancel-sub-sign-inv N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x + \left(\mathsf{neg}\left(1\right)\right) \cdot x}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(1\right)\right) \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(x \cdot x + \color{blue}{-1} \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   14. sqr-neg-rev N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   15. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   16. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \color{blue}{\left(-x\right)} + -1 \cdot x\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   17. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{x \cdot -1}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   18. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   19. distribute-rgt-neg-out N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot 1\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   20. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \left(\mathsf{neg}\left(\color{blue}{1 \cdot x}\right)\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   21. distribute-rgt-neg-out N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + \color{blue}{1 \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   22. lift-neg.f64 N/A

       \[\leadsto \left(\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\left(-x\right) \cdot \left(-x\right) + 1 \cdot \color{blue}{\left(-x\right)}\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\color{blue}{\left(\mathsf{log1p}\left({x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right)} - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{{x}^{3}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   2. unpow3 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2} \]

   4. lower-*.f64 100.0

      \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]

7. Applied rewrites 100.0%

   \[\leadsto \left(\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot x}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, -x\right)\right)\right) - \mathsf{log1p}\left(-x\right)\right) \cdot 0.5 \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
9. Step-by-step derivation

10. Applied rewrites 99.1%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025107 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))