Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 4.6s

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 99.8%

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

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 99.8%

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

   1. *-commutative 99.8%

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

   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 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = x + ((x ^ single(3.0)) * single(0.3333333333333333));
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. *-commutative 99.8%

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

   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 99.3%

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

   1. distribute-lft-in 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*r* 99.4%

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

   5. unpow2 99.4%

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

   6. cube-mult 99.4%

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

7. Simplified 99.4%

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

Alternative 4: 98.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* x (+ 1.0 (* 0.3333333333333333 (* x x)))))

C

float code(float x) {
	return x * (1.0f + (0.3333333333333333f * (x * x)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x * (1.0e0 + (0.3333333333333333e0 * (x * x)))
end function

Julia

function code(x)
	return Float32(x * Float32(Float32(1.0) + Float32(Float32(0.3333333333333333) * Float32(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = x * (single(1.0) + (single(0.3333333333333333) * (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. *-commutative 99.8%

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

   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 99.3%

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

   1. *-commutative 99.3%

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

7. Simplified 99.3%

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

   1. unpow2 99.3%

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

9. Applied egg-rr 99.3%

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

10. Final simplification 99.3%

    \[\leadsto x \cdot \left(1 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right) \]
11. Add Preprocessing

Alternative 5: 96.9% 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 99.8%

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

   1. *-commutative 99.8%

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

   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 98.2%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))