Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 3

Speedup: 1.0×

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: 99.8% 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: 99.8% 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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 96.8% accurate, 21.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (* 0.5 (* 2.0 x)))

C

float code(float x) {
	return 0.5f * (2.0f * x);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (2.0e0 * x)
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(Float32(2.0) * x))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.7%

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

3. Simplified 99.7%

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

   1. associate-*r/ 99.8%

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

   2. *-commutative 99.8%

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

   3. add-log-exp 23.6%

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

   4. add-sqr-sqrt 22.7%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left(\sqrt{e^{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)}} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)}}\right)} \]

   5. log-prod 22.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left(\sqrt{e^{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)}}\right) + \log \left(\sqrt{e^{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)}}\right)\right)} \]

   6. log1p-undefine 22.7%

      \[\leadsto 0.5 \cdot \left(\log \left(\sqrt{e^{\color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)}}}\right) + \log \left(\sqrt{e^{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)}}\right)\right) \]

   7. rem-exp-log 22.7%

      \[\leadsto 0.5 \cdot \left(\log \left(\sqrt{\color{blue}{1 + \frac{2 \cdot x}{1 - x}}}\right) + \log \left(\sqrt{e^{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)}}\right)\right) \]

   8. +-commutative 22.7%

      \[\leadsto 0.5 \cdot \left(\log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x} + 1}}\right) + \log \left(\sqrt{e^{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)}}\right)\right) \]

   9. associate-/l* 22.7%

      \[\leadsto 0.5 \cdot \left(\log \left(\sqrt{\color{blue}{2 \cdot \frac{x}{1 - x}} + 1}\right) + \log \left(\sqrt{e^{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)}}\right)\right) \]

   10. fma-define 22.7%

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

   11. log1p-undefine 22.7%

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

   12. rem-exp-log 22.7%

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

   13. +-commutative 22.7%

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

6. Applied egg-rr 22.7%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left(\sqrt{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right) + \log \left(\sqrt{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right)} \]
7. Step-by-step derivation

   1. count-2 22.7%

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

   2. fma-undefine 22.7%

      \[\leadsto 0.5 \cdot \left(2 \cdot \log \left(\sqrt{\color{blue}{2 \cdot \frac{x}{1 - x} + 1}}\right)\right) \]

   3. *-commutative 22.7%

      \[\leadsto 0.5 \cdot \left(2 \cdot \log \left(\sqrt{\color{blue}{\frac{x}{1 - x} \cdot 2} + 1}\right)\right) \]

   4. associate-*l/ 22.7%

      \[\leadsto 0.5 \cdot \left(2 \cdot \log \left(\sqrt{\color{blue}{\frac{x \cdot 2}{1 - x}} + 1}\right)\right) \]

   5. associate-*r/ 22.7%

      \[\leadsto 0.5 \cdot \left(2 \cdot \log \left(\sqrt{\color{blue}{x \cdot \frac{2}{1 - x}} + 1}\right)\right) \]

   6. fma-define 22.7%

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

   7. metadata-eval 22.7%

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

   8. associate-/r* 22.7%

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

   9. neg-mul-1 22.7%

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

   10. sub-neg 22.7%

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

   11. +-commutative 22.7%

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

   12. distribute-neg-in 22.7%

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

   13. remove-double-neg 22.7%

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

   14. metadata-eval 22.7%

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

8. Simplified 22.7%

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

9. Taylor expanded in x around 0 97.4%

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

10. Final simplification 97.4%

    \[\leadsto 0.5 \cdot \left(2 \cdot x\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))