Rust f32::acosh

Percentage Accurate: 53.3% → 98.3%

Time: 3.9s

Alternatives: 4

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: 53.3% 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: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot \left(2 - \frac{0.5}{{x}^{2}}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log (* x (- 2.0 (/ 0.5 (pow x 2.0))))))

C

float code(float x) {
	return logf((x * (2.0f - (0.5f / powf(x, 2.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x * (2.0e0 - (0.5e0 / (x ** 2.0e0)))))
end function

Julia

function code(x)
	return log(Float32(x * Float32(Float32(2.0) - Float32(Float32(0.5) / (x ^ Float32(2.0))))))
end

MATLAB

function tmp = code(x)
	tmp = log((x * (single(2.0) - (single(0.5) / (x ^ single(2.0))))));
end

Derivation

1. Initial program 45.9%

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

3. Taylor expanded in x around inf 98.8%

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

   1. associate-*r/ 98.8%

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

   2. metadata-eval 98.8%

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

5. Simplified 98.8%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 - \frac{0.5}{{x}^{2}}\right)\right)} \]
6. Add Preprocessing

Alternative 2: 97.0% 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 45.9%

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

3. Taylor expanded in x around inf 97.8%

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

Alternative 3: 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 45.9%

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

3. Taylor expanded in x around inf 98.8%

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

   1. associate-*r/ 98.8%

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

   2. metadata-eval 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in x around inf 97.3%

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

8. Simplified 44.9%

   \[\leadsto \color{blue}{\log x} \]
9. Add Preprocessing

Alternative 4: 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 45.9%

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

3. Taylor expanded in x around inf 97.8%

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

   1. count-2 97.8%

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

   2. sum-log 97.3%

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

   3. flip-+ 97.1%

      \[\leadsto \color{blue}{\frac{\log 2 \cdot \log 2 - \log x \cdot \log x}{\log 2 - \log x}} \]

   4. div-sub 96.8%

      \[\leadsto \color{blue}{\frac{\log 2 \cdot \log 2}{\log 2 - \log x} - \frac{\log x \cdot \log x}{\log 2 - \log x}} \]

   5. pow2 96.8%

      \[\leadsto \frac{\color{blue}{{\log 2}^{2}}}{\log 2 - \log x} - \frac{\log x \cdot \log x}{\log 2 - \log x} \]

   6. log1p-expm1-u 96.8%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log 2\right)\right)\right)}}^{2}}{\log 2 - \log x} - \frac{\log x \cdot \log x}{\log 2 - \log x} \]

   7. expm1-undefine 96.8%

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

   8. rem-exp-log 96.8%

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

   9. metadata-eval 96.8%

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

   10. diff-log 96.8%

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

   11. pow2 96.8%

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

   12. diff-log 96.1%

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

5. Applied egg-rr 96.1%

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

7. Simplified 20.9%

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

8. Taylor expanded in x around 0 20.9%

   \[\leadsto \color{blue}{1} \]
9. 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 2024172 
(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)))))