Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 4.8s

Alternatives: 6

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: 99.7% 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. Step-by-step derivation

   1. associate-/l* 99.4%

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

   2. associate-/r/ 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 3: 98.7% 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. Taylor expanded in x around 0 98.5%

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

   1. unpow3 98.5%

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 4: 98.6% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 98.5%

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

   1. unpow3 98.5%

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

4. Applied egg-rr 98.5%

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

   1. associate-*r* 98.5%

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

   2. distribute-rgt-out 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 5: 97.1% accurate, 21.8× 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 99.7%

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

2. Taylor expanded in x around 0 96.6%

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

   1. distribute-lft-out 96.6%

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

   2. unpow2 96.6%

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

4. Simplified 96.6%

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

5. Taylor expanded in x around 0 97.0%

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

   1. count-2 97.0%

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

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 6: 10.8% accurate, 109.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 0.0)

C

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

Fortran

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

Julia

function code(x)
	return Float32(0.0)
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 92.9%

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

   1. count-2 92.9%

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

4. Simplified 92.9%

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

   1. log1p-udef 19.4%

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

   2. flip-+ 19.2%

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

   3. metadata-eval 19.2%

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

   4. flip-+ -0.0%

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

   5. +-inverses -0.0%

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

   6. +-inverses -0.0%

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

   7. associate-*r/ -0.0%

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

   8. +-inverses -0.0%

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

   9. +-inverses -0.0%

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

   10. difference-of-squares -0.0%

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

   11. +-inverses -0.0%

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

   12. +-inverses -0.0%

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

   13. flip-+ 10.8%

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

   14. log-div 10.8%

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

6. Applied egg-rr 10.8%

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

   1. +-inverses 10.8%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]

8. Simplified 10.8%

   \[\leadsto 0.5 \cdot \color{blue}{0} \]

9. Final simplification 10.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023174 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))