Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 5.8s

Alternatives: 3

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. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(2.0) / Float32(Float32(Float32(1.0) / x) + Float32(-1.0)))))
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* 99.0%

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

   2. div-sub 99.0%

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

   3. sub-neg 99.0%

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

   4. *-inverses 99.0%

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

   5. metadata-eval 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 96.6% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (2.0e0 * x)
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(Float32(2.0) * x))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (single(2.0) * 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. associate-/l* 99.0%

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

   2. div-sub 99.0%

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

   3. sub-neg 99.0%

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

   4. *-inverses 99.0%

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

   5. metadata-eval 99.0%

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

3. Simplified 99.0%

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

   1. flip-+ 55.3%

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

   2. associate-/r/ 55.2%

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

   3. metadata-eval 55.2%

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

   4. sub-neg 55.2%

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

   5. inv-pow 55.2%

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

   6. inv-pow 55.2%

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

   7. pow-prod-up 55.0%

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

   8. metadata-eval 55.0%

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

   9. metadata-eval 55.0%

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

   10. sub-neg 55.0%

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

   11. metadata-eval 55.0%

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

6. Applied egg-rr 55.0%

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

   1. associate-*l/ 55.2%

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

   2. +-commutative 55.2%

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

   3. distribute-lft-in 55.2%

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

   4. metadata-eval 55.2%

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

   5. associate-*r/ 55.2%

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

   6. metadata-eval 55.2%

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

   7. +-commutative 55.2%

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

8. Simplified 55.2%

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

9. Taylor expanded in x around 0 96.3%

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

    1. distribute-lft-out 96.3%

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

    2. unpow2 96.3%

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

    3. distribute-rgt1-in 96.2%

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

11. Simplified 96.2%

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

12. Taylor expanded in x around 0 96.7%

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

    1. *-commutative 96.7%

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

14. Simplified 96.7%

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

15. Final simplification 96.7%

    \[\leadsto 0.5 \cdot \left(2 \cdot x\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))