Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 7.0s

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 99.8%

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

2. Final simplification 99.8%

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

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. associate-/l* 99.1%

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

   2. div-sub 99.1%

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

   3. sub-neg 99.1%

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

   4. *-inverses 99.1%

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

   5. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. metadata-eval 99.1%

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

   2. sub-neg 99.1%

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

   3. *-inverses 99.1%

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

   4. div-sub 99.1%

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

   5. associate-/r/ 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 96.9% 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 99.8%

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

   1. associate-/l* 99.1%

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

   2. div-sub 99.1%

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

   3. sub-neg 99.1%

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

   4. *-inverses 99.1%

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

   5. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. flip-+ 59.9%

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

   2. associate-/r/ 59.6%

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

   3. metadata-eval 59.6%

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

   4. sub-neg 59.6%

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

   5. inv-pow 59.6%

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

   6. inv-pow 59.6%

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

   7. pow-prod-up 59.6%

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

   8. metadata-eval 59.6%

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

   9. metadata-eval 59.6%

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

   10. sub-neg 59.6%

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

   11. metadata-eval 59.6%

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

5. Applied egg-rr 59.6%

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

   1. associate-*l/ 59.8%

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

   2. +-commutative 59.8%

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

   3. distribute-lft-in 59.8%

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

   4. metadata-eval 59.8%

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

   5. associate-*r/ 59.8%

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

   6. metadata-eval 59.8%

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

   7. +-commutative 59.8%

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

7. Simplified 59.8%

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

8. Taylor expanded in x around 0 94.9%

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

10. Simplified 94.9%

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

11. Taylor expanded in x around 0 95.4%

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

13. Simplified 95.4%

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

14. Final simplification 95.4%

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

Alternative 4: 7.6% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ 1.5 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.5)

C

float code(float x) {
	return 1.5f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.5e0
end function

Julia

function code(x)
	return Float32(1.5)
end

MATLAB

function tmp = code(x)
	tmp = single(1.5);
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* 99.1%

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

   2. div-sub 99.1%

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

   3. sub-neg 99.1%

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

   4. *-inverses 99.1%

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

   5. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. metadata-eval 99.1%

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

   2. sub-neg 99.1%

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

   3. *-inverses 99.1%

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

   4. div-sub 99.1%

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

   5. associate-/l* 99.8%

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

   6. log1p-udef 23.9%

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

   7. add-cube-cbrt 23.0%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left(\left(\sqrt[3]{1 + \frac{2 \cdot x}{1 - x}} \cdot \sqrt[3]{1 + \frac{2 \cdot x}{1 - x}}\right) \cdot \sqrt[3]{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

   8. log-prod 22.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left(\sqrt[3]{1 + \frac{2 \cdot x}{1 - x}} \cdot \sqrt[3]{1 + \frac{2 \cdot x}{1 - x}}\right) + \log \left(\sqrt[3]{1 + \frac{2 \cdot x}{1 - x}}\right)\right)} \]

5. Applied egg-rr 22.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right)} \]
6. Step-by-step derivation

   1. log-prod 22.9%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right) \]

   2. *-un-lft-identity 22.9%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{1 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right) \]

   3. *-un-lft-identity 22.9%

      \[\leadsto 0.5 \cdot \left(\left(1 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right) + \color{blue}{1 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right) \]

   4. distribute-rgt-out 22.9%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right) \cdot \left(1 + 1\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right) \]

   5. metadata-eval 22.9%

      \[\leadsto 0.5 \cdot \left(\log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right) \cdot \color{blue}{2} + \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right) \]

7. Applied egg-rr 22.9%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right) \cdot 2} + \log \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{x}{1 - x}, 1\right)}\right)\right) \]

8. Taylor expanded in x around inf -0.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{-1}\right) + \log \left(\sqrt[3]{-1}\right)\right)} \]

10. Simplified 8.0%

    \[\leadsto 0.5 \cdot \color{blue}{3} \]

11. Final simplification 8.0%

    \[\leadsto 1.5 \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))