Rust f32::atanh

Percentage Accurate: 99.8% → 99.7%

Time: 7.7s

Alternatives: 5

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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(x + x) * Float32(x + Float32(1.0))) / Float32(Float32(1.0) - Float32(x * 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. associate-/l* 99.3%

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

3. Simplified 99.3%

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

   1. associate-/l* 99.8%

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

   2. flip-- 99.8%

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

   3. associate-/r/ 99.8%

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

   4. add-log-exp 19.9%

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

   5. *-commutative 19.9%

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

   6. exp-lft-sqr 19.4%

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

   7. log-prod 19.4%

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

   8. add-log-exp 33.9%

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

   9. add-log-exp 99.8%

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

   10. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(2.0) / Float32(Float32(Float32(1.0) - x) / 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. associate-/l* 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

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(Float32(x * 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. Final simplification 99.8%

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

Alternative 4: 96.6% 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 99.8%

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

   1. associate-/l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 95.0%

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

   1. count-2 95.0%

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

6. Simplified 95.0%

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

   1. log1p-udef 17.2%

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

8. Applied egg-rr 17.2%

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

9. Taylor expanded in x around 0 98.2%

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

    1. count-2 98.2%

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

11. Simplified 98.2%

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

12. Final simplification 98.2%

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

Alternative 5: 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 99.8%

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

   1. associate-/l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 95.0%

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

   1. count-2 95.0%

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

6. Simplified 95.0%

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

   1. log1p-udef 17.2%

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

8. Applied egg-rr 17.2%

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

   1. flip-+ 17.2%

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

   2. div-inv 17.1%

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

   3. metadata-eval 17.1%

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

   4. flip-+ -0.0%

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

   5. +-inverses -0.0%

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

   6. metadata-eval -0.0%

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

   7. +-inverses -0.0%

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

   8. metadata-eval -0.0%

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

   9. associate-*r/ -0.0%

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

   10. metadata-eval -0.0%

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

   11. +-inverses -0.0%

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

   12. difference-of-squares -0.0%

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

   13. metadata-eval -0.0%

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

   14. +-inverses -0.0%

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

   15. flip-+ 11.0%

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

   16. log-prod 11.7%

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

10. Applied egg-rr 9.2%

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

    1. log-div 9.1%

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

    2. metadata-eval 9.1%

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

    3. log1p-def 11.0%

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

    4. associate-+r- 11.0%

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

    5. +-rgt-identity 11.0%

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

    6. +-inverses 11.0%

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

12. Simplified 11.0%

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

13. Final simplification 11.0%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023193 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))