Rust f32::asinh

Percentage Accurate: 38.6% → 58.1%

Time: 9.2s

Alternatives: 6

Speedup: 4.0×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (asinh x))

C

float code(float x) {
	return asinhf(x);
}

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Initial Program: 38.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

Alternative 1: 58.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (log1p x) x))

C

float code(float x) {
	return copysignf(log1pf(x), x);
}

Julia

function code(x)
	return copysign(log1p(x), x)
end

Derivation

1. Initial program 38.2%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 16.3%

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

   1. log1p-def 52.9%

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

5. Simplified 52.9%

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

   1. unpow2 52.9%

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

   2. *-un-lft-identity 52.9%

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

   3. times-frac 53.2%

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

   4. +-commutative 53.2%

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

   5. add-sqr-sqrt 28.0%

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

   6. fabs-sqr 28.0%

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

   7. add-sqr-sqrt 52.9%

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

7. Applied egg-rr 52.9%

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

8. Taylor expanded in x around 0 31.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
9. Step-by-step derivation

   1. log1p-def 67.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   2. unpow1 67.4%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{{x}^{1}}\right|\right), x\right) \]

   3. sqr-pow 35.5%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right), x\right) \]

   4. fabs-sqr 35.5%

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

   5. sqr-pow 57.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{{x}^{1}}\right), x\right) \]

   6. unpow1 57.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right), x\right) \]

10. Simplified 57.6%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]

11. Final simplification 57.6%

    \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right) \]
12. Add Preprocessing

Alternative 2: 40.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (log (+ x (hypot 1.0 x))) x))

C

float code(float x) {
	return copysignf(logf((x + hypotf(1.0f, x))), x);
}

Julia

function code(x)
	return copysign(log(Float32(x + hypot(Float32(1.0), x))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
end

Derivation

1. Initial program 38.2%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   2. *-commutative 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

   3. log-prod 38.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

   4. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)} + \log 1, x\right) \]

   5. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} + \log 1, x\right) \]

   6. add-sqr-sqrt 22.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   7. fabs-sqr 22.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   8. add-sqr-sqrt 30.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   9. +-commutative 30.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

   10. hypot-1-def 43.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

   11. metadata-eval 43.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

4. Applied egg-rr 43.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
5. Step-by-step derivation

   1. +-rgt-identity 43.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

6. Simplified 43.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

7. Final simplification 43.7%

   \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
8. Add Preprocessing

Alternative 3: 27.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))) x))

C

float code(float x) {
	return copysignf(logf(((x * 2.0f) + (0.5f * (1.0f / x)))), x);
}

Julia

function code(x)
	return copysign(log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log(((x * single(2.0)) + (single(0.5) * (single(1.0) / x)))));
end

Derivation

1. Initial program 38.2%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   2. *-commutative 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

   3. log-prod 38.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

   4. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)} + \log 1, x\right) \]

   5. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} + \log 1, x\right) \]

   6. add-sqr-sqrt 22.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   7. fabs-sqr 22.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   8. add-sqr-sqrt 30.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   9. +-commutative 30.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

   10. hypot-1-def 43.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

   11. metadata-eval 43.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

4. Applied egg-rr 43.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
5. Step-by-step derivation

   1. +-rgt-identity 43.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

6. Simplified 43.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

7. Taylor expanded in x around inf 31.4%

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

8. Final simplification 31.4%

   \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right) \]
9. Add Preprocessing

Alternative 4: 27.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (log (* x 2.0)) x))

C

float code(float x) {
	return copysignf(logf((x * 2.0f)), x);
}

Julia

function code(x)
	return copysign(log(Float32(x * Float32(2.0))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((x * single(2.0))));
end

Derivation

1. Initial program 38.2%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   2. *-commutative 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

   3. log-prod 38.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

   4. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)} + \log 1, x\right) \]

   5. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} + \log 1, x\right) \]

   6. add-sqr-sqrt 22.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   7. fabs-sqr 22.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   8. add-sqr-sqrt 30.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   9. +-commutative 30.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

   10. hypot-1-def 43.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

   11. metadata-eval 43.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

4. Applied egg-rr 43.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
5. Step-by-step derivation

   1. +-rgt-identity 43.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

6. Simplified 43.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

7. Taylor expanded in x around inf 30.7%

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

   1. *-commutative 30.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)}, x\right) \]

9. Simplified 30.7%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)}, x\right) \]

10. Final simplification 30.7%

    \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right) \]
11. Add Preprocessing

Alternative 5: 13.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (log x) x))

C

float code(float x) {
	return copysignf(logf(x), x);
}

Julia

function code(x)
	return copysign(log(x), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log(x));
end

Derivation

1. Initial program 38.2%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 15.6%

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

   1. mul-1-neg 15.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{1}{x}\right)}, x\right) \]

   2. log-rec 15.6%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(-\log x\right)}, x\right) \]

   3. remove-double-neg 15.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]

5. Simplified 15.6%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]

6. Final simplification 15.6%

   \[\leadsto \mathsf{copysign}\left(\log x, x\right) \]
7. Add Preprocessing

Alternative 6: 54.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign x x))

C

float code(float x) {
	return copysignf(x, x);
}

Julia

function code(x)
	return copysign(x, x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(x);
end

Derivation

1. Initial program 38.2%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   2. *-commutative 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

   3. log-prod 38.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

   4. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)} + \log 1, x\right) \]

   5. *-un-lft-identity 38.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} + \log 1, x\right) \]

   6. add-sqr-sqrt 22.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   7. fabs-sqr 22.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   8. add-sqr-sqrt 30.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   9. +-commutative 30.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

   10. hypot-1-def 43.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

   11. metadata-eval 43.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

4. Applied egg-rr 43.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
5. Step-by-step derivation

   1. +-rgt-identity 43.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

6. Simplified 43.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

7. Taylor expanded in x around 0 52.6%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

8. Final simplification 52.6%

   \[\leadsto \mathsf{copysign}\left(x, x\right) \]
9. Add Preprocessing

Developer target: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2024033 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))