Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 7.0s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(2.0) * fma(x, x, x)) / Float32(Float32(1.0) - 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* 99.3%

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

3. Simplified 99.3%

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

   1. associate-/l* 99.7%

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

   2. flip-- 99.8%

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

   3. associate-/r/ 99.8%

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

   4. add-log-exp 23.5%

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

   5. *-commutative 23.5%

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

   6. exp-lft-sqr 22.7%

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

   7. log-prod 22.7%

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

   8. add-log-exp 36.5%

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

   9. add-log-exp 99.8%

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

   10. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. count-2 99.8%

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

   3. associate-*l* 99.8%

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

   4. +-commutative 99.8%

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

   5. distribute-rgt-in 99.9%

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

   6. unpow2 99.9%

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

   7. *-lft-identity 99.9%

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

   8. unpow2 99.9%

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

   9. fma-def 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(x + x) / Float32(Float32(1.0) - Float32(x * x))) * 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. Step-by-step derivation

   1. associate-/l* 99.3%

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

3. Simplified 99.3%

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

   1. associate-/l* 99.7%

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

   2. flip-- 99.8%

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

   3. associate-/r/ 99.8%

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

   4. add-log-exp 23.5%

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

   5. *-commutative 23.5%

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

   6. exp-lft-sqr 22.7%

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

   7. log-prod 22.7%

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

   8. add-log-exp 36.5%

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

   9. add-log-exp 99.8%

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

   10. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.2% 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 99.7%

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

   1. associate-/l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 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 5: 96.7% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (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* 99.3%

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

3. Simplified 99.3%

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

   1. log1p-udef 23.4%

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

   2. associate-/l* 23.5%

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

   3. +-commutative 23.5%

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

   4. associate-*l/ 23.4%

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

   5. *-commutative 23.4%

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

   6. fma-def 23.5%

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

5. Applied egg-rr 23.5%

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

6. Taylor expanded in x around 0 19.6%

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

7. Taylor expanded in x around 0 97.1%

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

   1. count-2 97.1%

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

9. Simplified 97.1%

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

10. Final simplification 97.1%

    \[\leadsto 0.5 \cdot \left(x + x\right) \]

Reproduce

herbie shell --seed 2023172 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))