Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 8.6s

Alternatives: 4

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: 99.7% 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 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/l* 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 98.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{-1}{0.5 \cdot \frac{-1}{x} - x \cdot -0.16666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (* 0.5 (/ -1.0 (- (* 0.5 (/ -1.0 x)) (* x -0.16666666666666666)))))

C

float code(float x) {
	return 0.5f * (-1.0f / ((0.5f * (-1.0f / x)) - (x * -0.16666666666666666f)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * ((-1.0e0) / ((0.5e0 * ((-1.0e0) / x)) - (x * (-0.16666666666666666e0))))
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(Float32(-1.0) / Float32(Float32(Float32(0.5) * Float32(Float32(-1.0) / x)) - Float32(x * Float32(-0.16666666666666666)))))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (single(-1.0) / ((single(0.5) * (single(-1.0) / x)) - (x * single(-0.16666666666666666))));
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 97.6%

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

   2. pow3 97.7%

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

   3. associate-/l* 97.7%

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

4. Applied egg-rr 97.7%

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

   1. rem-cube-cbrt 99.7%

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

   2. pow1 99.7%

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

   3. metadata-eval 99.7%

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

   4. pow-div 66.2%

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

   5. pow1 66.2%

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

   6. clear-num 65.9%

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

   7. pow1 65.9%

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

   8. pow-div 99.0%

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

   9. metadata-eval 99.0%

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

   10. inv-pow 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in x around 0 97.6%

   \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{-0.16666666666666666 \cdot x + 0.5 \cdot \frac{1}{x}}} \]

8. Final simplification 97.6%

   \[\leadsto 0.5 \cdot \frac{-1}{0.5 \cdot \frac{-1}{x} - x \cdot -0.16666666666666666} \]
9. Add Preprocessing

Alternative 4: 96.5% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{1}{\frac{0.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* 0.5 (/ 1.0 (/ 0.5 x))))

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (1.0e0 / (0.5e0 / x))
end function

Julia

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

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (single(1.0) / (single(0.5) / 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 97.6%

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

   2. pow3 97.7%

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

   3. associate-/l* 97.7%

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

4. Applied egg-rr 97.7%

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

   1. rem-cube-cbrt 99.7%

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

   2. pow1 99.7%

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

   3. metadata-eval 99.7%

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

   4. pow-div 66.2%

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

   5. pow1 66.2%

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

   6. clear-num 65.9%

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

   7. pow1 65.9%

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

   8. pow-div 99.0%

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

   9. metadata-eval 99.0%

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

   10. inv-pow 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in x around 0 96.5%

   \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{0.5}{x}}} \]

8. Final simplification 96.5%

   \[\leadsto 0.5 \cdot \frac{1}{\frac{0.5}{x}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))