Rust f32::acosh

Percentage Accurate: 53.2% → 98.2%

Time: 7.0s

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: 53.2% 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: 98.2% 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 48.8%

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

   1. log1p-expm1-u 48.8%

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

   2. expm1-undefine 48.8%

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

   3. add-exp-log 48.8%

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

   4. fmm-def 48.8%

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

   5. metadata-eval 48.8%

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

4. Applied egg-rr 48.8%

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

5. Taylor expanded in x around inf 98.6%

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

   1. mul-1-neg 98.6%

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

   2. unsub-neg 98.6%

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

   3. *-un-lft-identity 98.6%

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

   4. *-un-lft-identity 98.6%

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

   5. un-div-inv 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 2: 96.9% 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 48.8%

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

3. Taylor expanded in x around inf 97.2%

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

Alternative 3: 50.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot 1.6875\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log (* x 1.6875)))

C

float code(float x) {
	return logf((x * 1.6875f));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x * 1.6875e0))
end function

Julia

function code(x)
	return log(Float32(x * Float32(1.6875)))
end

MATLAB

function tmp = code(x)
	tmp = log((x * single(1.6875)));
end

Derivation

1. Initial program 48.8%

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

   1. log1p-expm1-u 48.8%

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

   2. expm1-undefine 48.8%

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

   3. add-exp-log 48.8%

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

   4. fmm-def 48.8%

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

   5. metadata-eval 48.8%

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

4. Applied egg-rr 48.8%

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

5. Taylor expanded in x around inf 99.1%

   \[\leadsto \mathsf{log1p}\left(\left(x + \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - 0.5 \cdot \frac{1}{{x}^{2}}\right)}\right) - 1\right) \]

7. Simplified 50.7%

   \[\leadsto \mathsf{log1p}\left(\left(x + \color{blue}{x \cdot 0.6875}\right) - 1\right) \]
8. Step-by-step derivation

   1. add-exp-log 50.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(x + x \cdot 0.6875\right)}} - 1\right) \]

   2. expm1-define 50.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + x \cdot 0.6875\right)\right)}\right) \]

   3. log1p-expm1-u 50.7%

      \[\leadsto \color{blue}{\log \left(x + x \cdot 0.6875\right)} \]

   4. *-un-lft-identity 50.7%

      \[\leadsto \log \color{blue}{\left(1 \cdot \left(x + x \cdot 0.6875\right)\right)} \]

   5. log-prod 50.7%

      \[\leadsto \color{blue}{\log 1 + \log \left(x + x \cdot 0.6875\right)} \]

   6. metadata-eval 50.7%

      \[\leadsto \color{blue}{0} + \log \left(x + x \cdot 0.6875\right) \]

   7. *-un-lft-identity 50.7%

      \[\leadsto 0 + \log \left(\color{blue}{1 \cdot x} + x \cdot 0.6875\right) \]

   8. *-commutative 50.7%

      \[\leadsto 0 + \log \left(1 \cdot x + \color{blue}{0.6875 \cdot x}\right) \]

   9. distribute-rgt-out 50.7%

      \[\leadsto 0 + \log \color{blue}{\left(x \cdot \left(1 + 0.6875\right)\right)} \]

   10. metadata-eval 50.7%

       \[\leadsto 0 + \log \left(x \cdot \color{blue}{1.6875}\right) \]

9. Applied egg-rr 50.7%

   \[\leadsto \color{blue}{0 + \log \left(x \cdot 1.6875\right)} \]
10. Step-by-step derivation

    1. +-lft-identity 50.7%

       \[\leadsto \color{blue}{\log \left(x \cdot 1.6875\right)} \]

11. Simplified 50.7%

    \[\leadsto \color{blue}{\log \left(x \cdot 1.6875\right)} \]
12. Add Preprocessing

Alternative 4: 44.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return log1p(x)
end

Derivation

1. Initial program 48.8%

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

   1. log1p-expm1-u 48.8%

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

   2. expm1-undefine 48.8%

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

   3. add-exp-log 48.8%

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

   4. fmm-def 48.8%

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

   5. metadata-eval 48.8%

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

4. Applied egg-rr 48.8%

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

5. Taylor expanded in x around -inf 1.4%

   \[\leadsto \mathsf{log1p}\left(\left(x + \color{blue}{-1 \cdot \left(x \cdot \left(\left(1 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}\right) - 1\right) \]

7. Simplified 44.6%

   \[\leadsto \mathsf{log1p}\left(\left(x + \color{blue}{0.6875}\right) - 1\right) \]

8. Taylor expanded in x around inf 44.5%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x}\right) \]
9. 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 48.8%

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

3. Taylor expanded in x around inf 97.2%

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

4. Taylor expanded in x around 0 96.8%

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

6. Simplified 44.4%

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

Alternative 6: 20.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary32 0.5)

C

float code(float x) {
	return 0.5f;
}

Fortran

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

Julia

function code(x)
	return Float32(0.5)
end

MATLAB

function tmp = code(x)
	tmp = single(0.5);
end

Derivation

1. Initial program 48.8%

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

   1. add-cube-cbrt 48.7%

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

   2. pow3 48.7%

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

   3. log-pow 48.4%

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

   4. fmm-def 48.4%

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

   5. metadata-eval 48.4%

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

4. Applied egg-rr 48.4%

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

   1. pow1/3 48.3%

      \[\leadsto 3 \cdot \log \color{blue}{\left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{0.3333333333333333}\right)} \]

   2. log-pow 48.5%

      \[\leadsto 3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

6. Applied egg-rr 48.5%

   \[\leadsto 3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

8. Simplified 20.2%

   \[\leadsto 3 \cdot \color{blue}{0.16666666666666666} \]
9. Step-by-step derivation

   1. metadata-eval 20.2%

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

10. Applied egg-rr 20.2%

    \[\leadsto \color{blue}{0.5} \]
11. 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 2024131 
(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)))))