Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 4.5s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(x * Float32(2.0)) / Float32(Float32(1.0) - (x ^ Float32(2.0)))) * Float32(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. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 99.7%

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

   2. associate-/r/ 99.8%

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

   3. *-commutative 99.8%

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

   4. metadata-eval 99.8%

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

   5. pow2 99.8%

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

   6. +-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(x + Float32(1.0)) * Float32(Float32(x * Float32(-2.0)) / fma(x, 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. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 99.7%

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

   2. associate-/r/ 99.8%

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

   3. *-commutative 99.8%

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

   4. metadata-eval 99.8%

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

   5. pow2 99.8%

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

   6. +-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/l* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.8%

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

   2. associate-*l/ 99.8%

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

   3. *-commutative 99.8%

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

   4. metadata-eval 99.8%

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

   5. times-frac 99.8%

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

   6. *-commutative 99.8%

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

   7. neg-mul-1 99.8%

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

   8. sub-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. distribute-neg-in 99.8%

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

   11. remove-double-neg 99.8%

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

   12. unpow2 99.8%

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

   13. metadata-eval 99.8%

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

   14. fma-undefine 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

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 99.7%

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

3. Final simplification 99.7%

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

Alternative 4: 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. *-commutative 99.7%

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

   2. associate-/l* 99.7%

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

3. Simplified 99.7%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + 0.3333333333333333 \cdot {x}^{3} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (+ x (* 0.3333333333333333 (pow x 3.0))))

C

float code(float x) {
	return x + (0.3333333333333333f * powf(x, 3.0f));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x + (0.3333333333333333e0 * (x ** 3.0e0))
end function

Julia

function code(x)
	return Float32(x + Float32(Float32(0.3333333333333333) * (x ^ Float32(3.0))))
end

MATLAB

function tmp = code(x)
	tmp = x + (single(0.3333333333333333) * (x ^ single(3.0)));
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. *-commutative 99.7%

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

   2. associate-/l* 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 97.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in 97.9%

      \[\leadsto \color{blue}{1 \cdot x + \left(0.3333333333333333 \cdot {x}^{2}\right) \cdot x} \]

   2. *-lft-identity 97.9%

      \[\leadsto \color{blue}{x} + \left(0.3333333333333333 \cdot {x}^{2}\right) \cdot x \]

   3. associate-*l* 97.9%

      \[\leadsto x + \color{blue}{0.3333333333333333 \cdot \left({x}^{2} \cdot x\right)} \]

   4. unpow2 97.9%

      \[\leadsto x + 0.3333333333333333 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

   5. unpow3 97.9%

      \[\leadsto x + 0.3333333333333333 \cdot \color{blue}{{x}^{3}} \]

7. Simplified 97.9%

   \[\leadsto \color{blue}{x + 0.3333333333333333 \cdot {x}^{3}} \]
8. Add Preprocessing

Alternative 6: 97.0% accurate, 109.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 x)

C

float code(float x) {
	return x;
}

Fortran

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

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = 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. *-commutative 99.7%

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

   2. associate-/l* 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 96.0%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))