Rust f32::acosh

Percentage Accurate: 53.1% → 97.9%

Time: 12.6s

Alternatives: 10

Speedup: 2.0×

Specification

\[x \geq 1\]

Math

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

FPCore

(FPCore (x) :precision binary32 (acosh x))

C

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

Julia

function code(x)
	return acosh(x)
end

MATLAB

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

Initial Program: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x \cdot x - 1}\right) \end{array} \]

FPCore

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

C

float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0e0))))
end function

Julia

function code(x)
	return log(Float32(x + sqrt(Float32(Float32(x * x) - Float32(1.0)))))
end

MATLAB

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

Alternative 1: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log x + \left(\log 2 + \frac{-0.25}{{x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (+ (log x) (+ (log 2.0) (/ -0.25 (pow x 2.0)))))

C

float code(float x) {
	return logf(x) + (logf(2.0f) + (-0.25f / powf(x, 2.0f)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(x) + (log(2.0e0) + ((-0.25e0) / (x ** 2.0e0)))
end function

Julia

function code(x)
	return Float32(log(x) + Float32(log(Float32(2.0)) + Float32(Float32(-0.25) / (x ^ Float32(2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = log(x) + (log(single(2.0)) + (single(-0.25) / (x ^ single(2.0))));
end

Derivation

1. Initial program 50.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. +-commutative 98.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. associate-+l+ 98.2%

      \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   4. mul-1-neg 98.2%

      \[\leadsto \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   5. log-rec 98.2%

      \[\leadsto \left(-\color{blue}{\left(-\log x\right)}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   6. remove-double-neg 98.2%

      \[\leadsto \color{blue}{\log x} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) \]

   8. metadata-eval 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right)\right) \]

   9. distribute-neg-frac 98.2%

      \[\leadsto \log x + \left(\log 2 + \color{blue}{\frac{-0.25}{{x}^{2}}}\right) \]

   10. metadata-eval 98.2%

       \[\leadsto \log x + \left(\log 2 + \frac{\color{blue}{-0.25}}{{x}^{2}}\right) \]

5. Simplified 98.2%

   \[\leadsto \color{blue}{\log x + \left(\log 2 + \frac{-0.25}{{x}^{2}}\right)} \]
6. Add Preprocessing

Alternative 2: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 50.3%

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

   1. log1p-expm1-u 50.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

   2. expm1-undefine 50.3%

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

   3. add-exp-log 50.3%

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

   4. fmm-def 50.3%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right) - 1\right) \]

   5. metadata-eval 50.3%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) - 1\right) \]

4. Applied egg-rr 50.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]

5. Taylor expanded in x around inf 97.8%

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

   1. mul-1-neg 97.8%

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

   2. unsub-neg 97.8%

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

   3. associate-*r/ 97.8%

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

   4. metadata-eval 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 3: 97.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{0.5}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (- (log (/ 0.5 x))))

C

float code(float x) {
	return -logf((0.5f / x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -log((0.5e0 / x))
end function

Julia

function code(x)
	return Float32(-log(Float32(Float32(0.5) / x)))
end

MATLAB

function tmp = code(x)
	tmp = -log((single(0.5) / x));
end

Derivation

1. Initial program 50.3%

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

   1. flip-+ 5.9%

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

   2. div-inv 5.9%

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

   3. log-prod 5.9%

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

   4. pow2 5.9%

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

   5. add-sqr-sqrt 5.7%

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

   6. fmm-def 5.7%

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

   7. metadata-eval 5.7%

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

   8. fmm-def 5.7%

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

   9. metadata-eval 5.7%

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

4. Applied egg-rr 5.7%

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

   1. log-rec 5.7%

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

   2. sub-neg 5.7%

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

   3. fma-undefine 5.7%

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

   4. unpow2 5.7%

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

   5. associate--r+ 8.2%

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

   6. +-inverses 8.2%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   7. metadata-eval 8.2%

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

   8. metadata-eval 8.2%

      \[\leadsto \color{blue}{0} - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   9. neg-sub0 8.2%

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

6. Simplified 8.2%

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

7. Taylor expanded in x around inf 97.8%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5}{x}\right)} \]
8. Add Preprocessing

Alternative 4: 96.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x) {
	return logf((x + x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + x))
end function

Julia

function code(x)
	return log(Float32(x + x))
end

MATLAB

function tmp = code(x)
	tmp = log((x + x));
end

Derivation

1. Initial program 50.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 96.7%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
4. Add Preprocessing

Alternative 5: 25.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 8 \end{array} \]

FPCore

(FPCore (x) :precision binary32 8.0)

C

float code(float x) {
	return 8.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 8.0e0
end function

Julia

function code(x)
	return Float32(8.0)
end

MATLAB

function tmp = code(x)
	tmp = single(8.0);
end

Derivation

1. Initial program 50.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. +-commutative 98.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. associate-+l+ 98.2%

      \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   4. mul-1-neg 98.2%

      \[\leadsto \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   5. log-rec 98.2%

      \[\leadsto \left(-\color{blue}{\left(-\log x\right)}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   6. remove-double-neg 98.2%

      \[\leadsto \color{blue}{\log x} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) \]

   8. metadata-eval 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right)\right) \]

   9. distribute-neg-frac 98.2%

      \[\leadsto \log x + \left(\log 2 + \color{blue}{\frac{-0.25}{{x}^{2}}}\right) \]

   10. metadata-eval 98.2%

       \[\leadsto \log x + \left(\log 2 + \frac{\color{blue}{-0.25}}{{x}^{2}}\right) \]

5. Simplified 98.2%

   \[\leadsto \color{blue}{\log x + \left(\log 2 + \frac{-0.25}{{x}^{2}}\right)} \]

7. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{8} \]
8. Add Preprocessing

Alternative 6: 23.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x) :precision binary32 4.0)

C

float code(float x) {
	return 4.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 4.0e0
end function

Julia

function code(x)
	return Float32(4.0)
end

MATLAB

function tmp = code(x)
	tmp = single(4.0);
end

Derivation

1. Initial program 50.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. +-commutative 98.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. associate-+l+ 98.2%

      \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   4. mul-1-neg 98.2%

      \[\leadsto \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   5. log-rec 98.2%

      \[\leadsto \left(-\color{blue}{\left(-\log x\right)}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   6. remove-double-neg 98.2%

      \[\leadsto \color{blue}{\log x} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) \]

   8. metadata-eval 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right)\right) \]

   9. distribute-neg-frac 98.2%

      \[\leadsto \log x + \left(\log 2 + \color{blue}{\frac{-0.25}{{x}^{2}}}\right) \]

   10. metadata-eval 98.2%

       \[\leadsto \log x + \left(\log 2 + \frac{\color{blue}{-0.25}}{{x}^{2}}\right) \]

5. Simplified 98.2%

   \[\leadsto \color{blue}{\log x + \left(\log 2 + \frac{-0.25}{{x}^{2}}\right)} \]

7. Applied egg-rr 23.4%

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

Alternative 7: 22.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 3 \end{array} \]

FPCore

(FPCore (x) :precision binary32 3.0)

C

float code(float x) {
	return 3.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 3.0e0
end function

Julia

function code(x)
	return Float32(3.0)
end

MATLAB

function tmp = code(x)
	tmp = single(3.0);
end

Derivation

1. Initial program 50.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. +-commutative 98.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. associate-+l+ 98.2%

      \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   4. mul-1-neg 98.2%

      \[\leadsto \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   5. log-rec 98.2%

      \[\leadsto \left(-\color{blue}{\left(-\log x\right)}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   6. remove-double-neg 98.2%

      \[\leadsto \color{blue}{\log x} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) \]

   8. metadata-eval 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right)\right) \]

   9. distribute-neg-frac 98.2%

      \[\leadsto \log x + \left(\log 2 + \color{blue}{\frac{-0.25}{{x}^{2}}}\right) \]

   10. metadata-eval 98.2%

       \[\leadsto \log x + \left(\log 2 + \frac{\color{blue}{-0.25}}{{x}^{2}}\right) \]

5. Simplified 98.2%

   \[\leadsto \color{blue}{\log x + \left(\log 2 + \frac{-0.25}{{x}^{2}}\right)} \]

7. Applied egg-rr 22.6%

   \[\leadsto \color{blue}{3} \]
8. Add Preprocessing

Alternative 8: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.0)

C

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

Fortran

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

Julia

function code(x)
	return Float32(2.0)
end

MATLAB

function tmp = code(x)
	tmp = single(2.0);
end

Derivation

1. Initial program 50.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. +-commutative 98.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. associate-+l+ 98.2%

      \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   4. mul-1-neg 98.2%

      \[\leadsto \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   5. log-rec 98.2%

      \[\leadsto \left(-\color{blue}{\left(-\log x\right)}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   6. remove-double-neg 98.2%

      \[\leadsto \color{blue}{\log x} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) \]

   8. metadata-eval 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right)\right) \]

   9. distribute-neg-frac 98.2%

      \[\leadsto \log x + \left(\log 2 + \color{blue}{\frac{-0.25}{{x}^{2}}}\right) \]

   10. metadata-eval 98.2%

       \[\leadsto \log x + \left(\log 2 + \frac{\color{blue}{-0.25}}{{x}^{2}}\right) \]

5. Simplified 98.2%

   \[\leadsto \color{blue}{\log x + \left(\log 2 + \frac{-0.25}{{x}^{2}}\right)} \]

7. Applied egg-rr 22.0%

   \[\leadsto \color{blue}{2} \]
8. Add Preprocessing

Alternative 9: 21.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.0)

C

float code(float x) {
	return 1.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.0e0
end function

Julia

function code(x)
	return Float32(1.0)
end

MATLAB

function tmp = code(x)
	tmp = single(1.0);
end

Derivation

1. Initial program 50.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. +-commutative 98.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. associate-+l+ 98.2%

      \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   4. mul-1-neg 98.2%

      \[\leadsto \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   5. log-rec 98.2%

      \[\leadsto \left(-\color{blue}{\left(-\log x\right)}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   6. remove-double-neg 98.2%

      \[\leadsto \color{blue}{\log x} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) \]

   8. metadata-eval 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right)\right) \]

   9. distribute-neg-frac 98.2%

      \[\leadsto \log x + \left(\log 2 + \color{blue}{\frac{-0.25}{{x}^{2}}}\right) \]

   10. metadata-eval 98.2%

       \[\leadsto \log x + \left(\log 2 + \frac{\color{blue}{-0.25}}{{x}^{2}}\right) \]

5. Simplified 98.2%

   \[\leadsto \color{blue}{\log x + \left(\log 2 + \frac{-0.25}{{x}^{2}}\right)} \]

7. Applied egg-rr 20.9%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Alternative 10: 19.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary32 0.25)

C

float code(float x) {
	return 0.25f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.25e0
end function

Julia

function code(x)
	return Float32(0.25)
end

MATLAB

function tmp = code(x)
	tmp = single(0.25);
end

Derivation

1. Initial program 50.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. +-commutative 98.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. associate-+l+ 98.2%

      \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   4. mul-1-neg 98.2%

      \[\leadsto \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   5. log-rec 98.2%

      \[\leadsto \left(-\color{blue}{\left(-\log x\right)}\right) + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   6. remove-double-neg 98.2%

      \[\leadsto \color{blue}{\log x} + \left(\log 2 + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) \]

   8. metadata-eval 98.2%

      \[\leadsto \log x + \left(\log 2 + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right)\right) \]

   9. distribute-neg-frac 98.2%

      \[\leadsto \log x + \left(\log 2 + \color{blue}{\frac{-0.25}{{x}^{2}}}\right) \]

   10. metadata-eval 98.2%

       \[\leadsto \log x + \left(\log 2 + \frac{\color{blue}{-0.25}}{{x}^{2}}\right) \]

5. Simplified 98.2%

   \[\leadsto \color{blue}{\log x + \left(\log 2 + \frac{-0.25}{{x}^{2}}\right)} \]

7. Applied egg-rr 19.4%

   \[\leadsto \color{blue}{0.25} \]
8. Add Preprocessing

Developer Target 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \end{array} \]

FPCore

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

C

float code(float x) {
	return logf((x + (sqrtf((x - 1.0f)) * sqrtf((x + 1.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + (sqrt((x - 1.0e0)) * sqrt((x + 1.0e0)))))
end function

Julia

function code(x)
	return log(Float32(x + Float32(sqrt(Float32(x - Float32(1.0))) * sqrt(Float32(x + Float32(1.0))))))
end

MATLAB

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

Reproduce

herbie shell --seed 2024188 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :alt
  (! :herbie-platform default (log (+ x (* (sqrt (- x 1)) (sqrt (+ x 1))))))

  (log (+ x (sqrt (- (* x x) 1.0)))))