Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 6.3s

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. Final simplification 99.7%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 98.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (* 0.5 (+ (* 2.0 x) (* 0.6666666666666666 (* x (* x x))))))

C

float code(float x) {
	return 0.5f * ((2.0f * x) + (0.6666666666666666f * (x * (x * x))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * ((2.0e0 * x) + (0.6666666666666666e0 * (x * (x * x))))
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(Float32(Float32(2.0) * x) + Float32(Float32(0.6666666666666666) * Float32(x * Float32(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * ((single(2.0) * x) + (single(0.6666666666666666) * (x * (x * x))));
end

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 97.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]
5. Step-by-step derivation

   1. unpow3 97.0%

      \[\leadsto 0.5 \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right) \]

6. Applied egg-rr 97.0%

   \[\leadsto 0.5 \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right) \]

7. Final simplification 97.0%

   \[\leadsto 0.5 \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \]

Alternative 4: 96.7% 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. Step-by-step derivation

   1. associate-/l* 99.2%

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

3. Simplified 99.2%

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

   1. log1p-udef 24.8%

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

   2. associate-/l* 24.9%

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

   3. add-cube-cbrt 23.9%

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

   4. log-prod 23.9%

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

5. Applied egg-rr 23.9%

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

   1. pow1/3 23.9%

      \[\leadsto 0.5 \cdot \left(\log \left(\color{blue}{{\left(\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)\right)}^{0.3333333333333333}} \cdot \sqrt[3]{\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)}\right)\right) \]

   2. pow1/3 23.9%

      \[\leadsto 0.5 \cdot \left(\log \left({\left(\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)\right)}^{0.3333333333333333} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)\right)}^{0.3333333333333333}}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)}\right)\right) \]

   3. pow-prod-up 24.7%

      \[\leadsto 0.5 \cdot \left(\log \color{blue}{\left({\left(\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)}\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)}\right)\right) \]

   4. metadata-eval 24.7%

      \[\leadsto 0.5 \cdot \left(\log \left({\left(\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)\right)}^{\color{blue}{0.6666666666666666}}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)}\right)\right) \]

7. Applied egg-rr 24.7%

   \[\leadsto 0.5 \cdot \left(\log \color{blue}{\left({\left(\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)\right)}^{0.6666666666666666}\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(x, \frac{2}{1 - x}, 1\right)}\right)\right) \]

8. Taylor expanded in x around 0 95.5%

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

   1. *-commutative 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

    \[\leadsto 0.5 \cdot \left(2 \cdot x\right) \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))