Rust f32::atanh

Percentage Accurate: 93.0% → 97.0%

Time: 5.8s

Alternatives: 2

Speedup: 13.9×

Specification

Math

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

FPCore

(FPCore (x) :precision binary32 (atanh x))

C

float code(float x) {
	return atanhf(x);
}

Julia

function code(x)
	return atanh(x)
end

MATLAB

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

Initial Program: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

float code(float x) {
	return 0.5f * log1pf(((2.0f * x) / (1.0f - x)));
}

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(2.0) * x) / Float32(Float32(1.0) - x))))
end

Alternative 1: 97.0% accurate, 13.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (* (+ x x) 0.5))

C

float code(float x) {
	return (x + x) * 0.5f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (x + x) * 0.5e0
end function

Julia

function code(x)
	return Float32(Float32(x + x) * Float32(0.5))
end

MATLAB

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

Derivation

1. Initial program 92.6%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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-*.f32 97.1

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 97.1%

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

8. Final simplification 97.1%

   \[\leadsto \left(x + x\right) \cdot 0.5 \]
9. Add Preprocessing

Alternative 2: 7.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ -2 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* -2.0 x))

C

float code(float x) {
	return -2.0f * x;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (-2.0e0) * x
end function

Julia

function code(x)
	return Float32(Float32(-2.0) * x)
end

MATLAB

function tmp = code(x)
	tmp = single(-2.0) * x;
end

Derivation

1. Initial program 92.6%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. flip-+ N/A

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

   5. difference-of-squares N/A

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

   6. count-2-rev N/A

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

   7. lift-*.f32 N/A

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

   8. +-inverses N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\frac{\left(2 \cdot x\right) \cdot \color{blue}{0}}{x - x}}{1 - x}\right) \]

   9. +-inverses N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\frac{\left(2 \cdot x\right) \cdot 0}{\color{blue}{0}}}{1 - x}\right) \]

   10. associate-*r/ N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(2 \cdot x\right) \cdot \frac{0}{0}}}{1 - x}\right) \]

   11. +-inverses N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\left(2 \cdot x\right) \cdot \frac{\color{blue}{x \cdot x - x \cdot x}}{0}}{1 - x}\right) \]

   12. +-inverses N/A

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

   13. flip-+ N/A

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

   14. count-2-rev N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*l/ N/A

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

   17. lift--.f32 N/A

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

   18. flip-- N/A

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

   19. associate-/r/ N/A

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

   20. associate-*l* N/A

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

   21. lower-*.f32 N/A

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

4. Applied rewrites 11.9%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\left(2 \cdot \frac{x}{1 - x \cdot x}\right) \cdot \left(\mathsf{fma}\left(x, x, x\right) \cdot 2\right)}\right) \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot 2} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot 2} \]

   3. unpow2 N/A

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

   4. lower-*.f32 11.8

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

7. Applied rewrites 11.8%

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

9. Applied rewrites 7.5%

   \[\leadsto \color{blue}{-2 \cdot x} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))