Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 6.1s

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: 99.8% 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: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(2.0) * Float32(x + Float32(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* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 96.6%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{2 \cdot {x}^{2} + 2 \cdot x}\right) \]
5. 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%

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

6. Simplified 96.6%

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

7. Final simplification 96.6%

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

Alternative 3: 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%

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

3. Simplified 99.4%

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

   1. associate-/r/ 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 93.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (* 0.5 (log1p (+ x x))))

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(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* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 93.2%

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

   1. count-2 93.2%

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

6. Simplified 93.2%

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

7. Final simplification 93.2%

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

Alternative 5: 7.8% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (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. Step-by-step derivation

   1. associate-/l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 92.9%

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

   1. associate-/r/ 93.2%

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

   2. metadata-eval 93.2%

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

   3. count-2 93.2%

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

   4. flip-+ -0.0%

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

   5. clear-num -0.0%

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

   6. +-inverses -0.0%

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

   7. +-inverses -0.0%

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

   8. +-inverses -0.0%

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

   9. +-inverses -0.0%

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

   10. flip-+ 5.3%

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

   11. count-2 5.3%

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

   12. metadata-eval 5.3%

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

   13. associate-/r/ 5.3%

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

   14. clear-num 5.3%

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

   15. div-inv 5.3%

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

   16. metadata-eval 5.3%

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

6. Applied egg-rr 5.3%

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

8. Simplified 5.3%

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

9. Taylor expanded in x around inf 7.8%

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

10. Final simplification 7.8%

    \[\leadsto 0.5 \cdot \frac{0.5}{x} \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))