Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 2.1s

Alternatives: 4

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 99.9%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(1.0) / Float32(Float32(Float32(0.5) / x) + Float32(-0.5)))))
end

Derivation

1. Initial program 99.9%

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

   1. add-log-exp 18.9%

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

   2. *-un-lft-identity 18.9%

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

   3. log-prod 18.9%

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

   4. metadata-eval 18.9%

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

   5. add-log-exp 99.9%

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

   6. associate-/l* 99.9%

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{0 + 0.5 \cdot \mathsf{log1p}\left(2 \cdot \frac{x}{1 - x}\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 99.9%

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

   2. associate-*r/ 99.9%

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

   3. *-lft-identity 99.9%

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

   4. associate-*l/ 99.8%

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

   5. associate-/r/ 99.4%

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

   6. div-sub 99.4%

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

   7. sub-neg 99.4%

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

   8. associate-/r* 99.4%

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

   9. metadata-eval 99.4%

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

   10. *-commutative 99.4%

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

   11. associate-/r* 99.4%

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

   12. *-inverses 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

6. Simplified 99.4%

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

Alternative 3: 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 99.9%

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

3. Taylor expanded in x around 0 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. *-lft-identity 98.8%

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

   3. associate-*l* 98.8%

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

   4. unpow2 98.8%

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

   5. unpow3 98.8%

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

5. Simplified 98.8%

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

Alternative 4: 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 99.9%

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

3. Taylor expanded in x around 0 98.0%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))