Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 8.2s

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 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. add-log-exp 21.6%

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

   2. *-un-lft-identity 21.6%

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

   3. log-prod 21.6%

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

   4. metadata-eval 21.6%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{0} + \log \left(e^{\frac{2 \cdot x}{1 - x}}\right)\right) \]

   5. add-log-exp 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(0 + \color{blue}{\frac{2 \cdot x}{1 - x}}\right) \]

   6. *-commutative 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(0 + \frac{\color{blue}{x \cdot 2}}{1 - x}\right) \]

   7. associate-/l* 99.6%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(0 + \color{blue}{x \cdot \frac{2}{1 - x}}\right) \]

4. Applied egg-rr 99.6%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{0 + x \cdot \frac{2}{1 - x}}\right) \]
5. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. associate-*r/ 99.7%

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

   3. *-commutative 99.7%

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

   4. *-lft-identity 99.7%

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

   5. associate-*l/ 99.6%

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

   6. associate-/r/ 99.1%

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

   7. div-sub 99.1%

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

   8. sub-neg 99.1%

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

   9. associate-/r* 99.1%

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

   10. metadata-eval 99.1%

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

   11. *-commutative 99.1%

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

   12. associate-/r* 99.1%

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

   13. *-inverses 99.1%

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

   14. metadata-eval 99.1%

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

   15. metadata-eval 99.1%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 3: 97.1% 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 99.7%

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

   1. add-cube-cbrt 97.5%

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

   2. pow3 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-/l* 97.4%

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

4. Applied egg-rr 97.4%

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

5. Taylor expanded in x around 0 97.2%

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

   1. *-commutative 97.2%

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Reproduce

herbie shell --seed 2024079 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))