Rust f32::atanh

Percentage Accurate: 92.7% → 97.0%

Time: 6.0s

Alternatives: 3

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: 92.7% 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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 94.5%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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 97.8

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 97.8%

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

Alternative 2: 11.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(2 \cdot \frac{\frac{-1}{x}}{x}, \mathsf{fma}\left(x, x, x\right), 2 \cdot \frac{x}{1 - {x}^{3}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  0.5
  (log1p
   (fma
    (* 2.0 (/ (/ -1.0 x) x))
    (fma x x x)
    (* 2.0 (/ x (- 1.0 (pow x 3.0))))))))

C

float code(float x) {
	return 0.5f * log1pf(fmaf((2.0f * ((-1.0f / x) / x)), fmaf(x, x, x), (2.0f * (x / (1.0f - powf(x, 3.0f))))));
}

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(fma(Float32(Float32(2.0) * Float32(Float32(Float32(-1.0) / x) / x)), fma(x, x, x), Float32(Float32(2.0) * Float32(x / Float32(Float32(1.0) - (x ^ Float32(3.0))))))))
end

Derivation

1. Initial program 85.2%

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

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

   3. flip3-- N/A

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

   4. associate-/r/ N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. lower-fma.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. metadata-eval N/A

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

   14. lower--.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. *-lft-identity N/A

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

   17. lower-fma.f32 N/A

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

4. Applied rewrites 11.9%

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

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 11.4

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

7. Applied rewrites 11.7%

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

8. Final simplification 11.5%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(2 \cdot \frac{\frac{-1}{x}}{x}, \mathsf{fma}\left(x, x, x\right), 2 \cdot \frac{x}{1 - {x}^{3}}\right)\right) \]
9. Add Preprocessing

Alternative 3: 10.8% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ 0 \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* 0.0 0.5))

C

float code(float x) {
	return 0.0f * 0.5f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.0e0 * 0.5e0
end function

Julia

function code(x)
	return Float32(Float32(0.0) * Float32(0.5))
end

MATLAB

function tmp = code(x)
	tmp = single(0.0) * single(0.5);
end

Derivation

1. Initial program 90.7%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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 97.8

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 97.8%

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

9. Applied rewrites 7.1%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f32 7.4

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

11. Applied rewrites 11.0%

    \[\leadsto \color{blue}{0 \cdot 0.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))