Rust f32::atanh

Percentage Accurate: 92.9% → 97.0%

Time: 5.4s

Alternatives: 2

Speedup: 125.0×

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: 92.9% 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, 125.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 x)

C

float code(float x) {
	return x;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x
end function

Julia

function code(x)
	return x
end

MATLAB

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

Derivation

1. Initial program 91.4%

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

4. Applied rewrites 62.9%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{fma}\left(1 - x, x, 1\right) \cdot \left(x \cdot 2\right)}{1 - {x}^{3}}}\right) \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-in N/A

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

   6. lift-+.f32 N/A

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

   7. un-div-inv N/A

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

   8. metadata-eval N/A

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

   9. frac-2neg N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. neg-mul-1 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. lift-+.f32 N/A

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

   19. +-commutative N/A

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

   20. lift-pow.f32 N/A

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

   21. sqr-pow N/A

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

6. Applied rewrites 22.2%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{fma}\left(1 - x, x, 1\right)}{{x}^{1.5} + 1} \cdot \frac{-2 \cdot x}{{x}^{1.5} - 1}}\right) \]

7. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{1 \cdot x + \left(\frac{1}{4} \cdot \left(x \cdot \left(4 \cdot \left(1 + \left(\sqrt{x} + -1 \cdot \sqrt{x}\right)\right) - 4\right)\right)\right) \cdot x} \]

   2. *-lft-identity N/A

      \[\leadsto \color{blue}{x} + \left(\frac{1}{4} \cdot \left(x \cdot \left(4 \cdot \left(1 + \left(\sqrt{x} + -1 \cdot \sqrt{x}\right)\right) - 4\right)\right)\right) \cdot x \]

   3. associate-*r* N/A

      \[\leadsto x + \color{blue}{\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(4 \cdot \left(1 + \left(\sqrt{x} + -1 \cdot \sqrt{x}\right)\right) - 4\right)\right)} \cdot x \]

   4. associate-*l* N/A

      \[\leadsto x + \color{blue}{\left(\frac{1}{4} \cdot x\right) \cdot \left(\left(4 \cdot \left(1 + \left(\sqrt{x} + -1 \cdot \sqrt{x}\right)\right) - 4\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x + \color{blue}{\left(x \cdot \frac{1}{4}\right)} \cdot \left(\left(4 \cdot \left(1 + \left(\sqrt{x} + -1 \cdot \sqrt{x}\right)\right) - 4\right) \cdot x\right) \]

   6. *-commutative N/A

      \[\leadsto x + \left(x \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(x \cdot \left(4 \cdot \left(1 + \left(\sqrt{x} + -1 \cdot \sqrt{x}\right)\right) - 4\right)\right)} \]

   7. associate-*r* N/A

      \[\leadsto x + \color{blue}{x \cdot \left(\frac{1}{4} \cdot \left(x \cdot \left(4 \cdot \left(1 + \left(\sqrt{x} + -1 \cdot \sqrt{x}\right)\right) - 4\right)\right)\right)} \]

   8. associate-*r* N/A

      \[\leadsto x + x \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(4 \cdot \left(1 + \left(\sqrt{x} + -1 \cdot \sqrt{x}\right)\right) - 4\right)\right)} \]

   9. associate-*r* N/A

      \[\leadsto x + \color{blue}{\left(x \cdot \left(\frac{1}{4} \cdot x\right)\right) \cdot \left(4 \cdot \left(1 + \left(\sqrt{x} + -1 \cdot \sqrt{x}\right)\right) - 4\right)} \]

9. Applied rewrites 97.1%

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

Alternative 2: 7.7% accurate, 125.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.0)

C

float code(float x) {
	return 1.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.0e0
end function

Julia

function code(x)
	return Float32(1.0)
end

MATLAB

function tmp = code(x)
	tmp = single(1.0);
end

Derivation

1. Initial program 92.2%

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

4. Applied rewrites 63.3%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{fma}\left(1 - x, x, 1\right) \cdot \left(x \cdot 2\right)}{1 - {x}^{3}}}\right) \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-in N/A

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

   6. lift-+.f32 N/A

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

   7. un-div-inv N/A

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

   8. metadata-eval N/A

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

   9. frac-2neg N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. neg-mul-1 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. lift-+.f32 N/A

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

   19. +-commutative N/A

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

   20. lift-pow.f32 N/A

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

   21. sqr-pow N/A

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

6. Applied rewrites 21.2%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{fma}\left(1 - x, x, 1\right)}{{x}^{1.5} + 1} \cdot \frac{-2 \cdot x}{{x}^{1.5} - 1}}\right) \]

7. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{\frac{-1}{{\left(\sqrt{-1}\right)}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}} \]

   2. rem-square-sqrt N/A

      \[\leadsto \frac{-1}{\color{blue}{-1}} \]

   3. metadata-eval 7.5

      \[\leadsto \color{blue}{1} \]

9. Applied rewrites 7.5%

   \[\leadsto \color{blue}{1} \]
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)))))