Rust f32::acosh

Percentage Accurate: 52.7% → 96.8%

Time: 5.9s

Alternatives: 6

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: 52.7% 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: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log 2 + \log x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (+ (log 2.0) (log x)))

C

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

Fortran

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

Julia

function code(x)
	return Float32(log(Float32(2.0)) + log(x))
end

MATLAB

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

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in x around inf 97.9%

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

   1. mul-1-neg 97.9%

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

   2. log-rec 97.9%

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

   3. remove-double-neg 97.9%

      \[\leadsto \log 2 + \color{blue}{\log x} \]

5. Simplified 97.9%

   \[\leadsto \color{blue}{\log 2 + \log x} \]

6. Final simplification 97.9%

   \[\leadsto \log 2 + \log x \]
7. Add Preprocessing

Alternative 2: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in x around inf 97.6%

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

   1. associate-*r/ 97.6%

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

   2. metadata-eval 97.6%

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

5. Simplified 97.6%

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

6. Final simplification 97.6%

   \[\leadsto \log \left(x + \left(x - \frac{0.5}{x}\right)\right) \]
7. Add Preprocessing

Alternative 3: 97.1% 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 55.8%

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

3. Taylor expanded in x around inf 96.9%

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

4. Final simplification 96.9%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Alternative 4: 21.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log 3.0))

C

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

Fortran

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

Julia

function code(x)
	return log(Float32(3.0))
end

MATLAB

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

Derivation

1. Initial program 55.8%

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

   1. log1p-expm1-u 55.8%

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

   2. expm1-undefine 55.8%

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

   3. add-exp-log 55.7%

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

   4. fma-neg 55.7%

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

   5. metadata-eval 55.7%

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

4. Applied egg-rr 55.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]
5. Step-by-step derivation

   1. associate--l+ 55.7%

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

6. Simplified 55.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} - 1\right)\right)} \]
7. Step-by-step derivation

   1. flip-- 52.7%

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

   2. add-sqr-sqrt 52.7%

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

   3. fma-undefine 52.7%

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

   4. metadata-eval 52.7%

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

   5. associate--l+ 52.7%

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

   6. pow2 52.7%

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

   7. metadata-eval 52.7%

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

8. Applied egg-rr 52.7%

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

10. Simplified 44.4%

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

11. Taylor expanded in x around 0 21.1%

    \[\leadsto \color{blue}{\log 3} \]

12. Final simplification 21.1%

    \[\leadsto \log 3 \]
13. Add Preprocessing

Alternative 5: 44.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log x \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

function code(x)
	return log(x)
end

MATLAB

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

Derivation

1. Initial program 55.8%

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

   1. log1p-expm1-u 55.8%

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

   2. expm1-undefine 55.8%

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

   3. add-exp-log 55.7%

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

   4. fma-neg 55.7%

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

   5. metadata-eval 55.7%

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

4. Applied egg-rr 55.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]
5. Step-by-step derivation

   1. associate--l+ 55.7%

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

6. Simplified 55.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} - 1\right)\right)} \]
7. Step-by-step derivation

   1. flip-- 52.7%

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

   2. add-sqr-sqrt 52.7%

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

   3. fma-undefine 52.7%

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

   4. metadata-eval 52.7%

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

   5. associate--l+ 52.7%

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

   6. pow2 52.7%

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

   7. metadata-eval 52.7%

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

8. Applied egg-rr 52.7%

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

10. Simplified 44.4%

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

11. Taylor expanded in x around inf 44.3%

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

    1. mul-1-neg 44.3%

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

    2. log-rec 44.3%

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

    3. remove-double-neg 44.3%

       \[\leadsto \color{blue}{\log x} \]

13. Simplified 44.3%

    \[\leadsto \color{blue}{\log x} \]

14. Final simplification 44.3%

    \[\leadsto \log x \]
15. Add Preprocessing

Alternative 6: 3.1% 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 55.8%

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

3. Taylor expanded in x around inf 96.9%

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

4. Taylor expanded in x around 0 97.9%

   \[\leadsto \color{blue}{\log 2 + \log x} \]

6. Simplified 3.1%

   \[\leadsto \color{blue}{-1} \]

7. Final simplification 3.1%

   \[\leadsto -1 \]
8. Add Preprocessing

Developer target: 99.3% 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 2024044 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

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

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