Rust f32::acosh

Percentage Accurate: 52.6% → 98.3%

Time: 4.4s

Alternatives: 7

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: 52.6% 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, 1.9× 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 58.3%

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

2. Taylor expanded in x around inf 96.8%

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

   1. associate-*r/ 96.8%

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

   2. metadata-eval 96.8%

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

4. Simplified 96.8%

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

5. Final simplification 96.8%

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

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 58.3%

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

2. Taylor expanded in x around inf 94.7%

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

3. Final simplification 94.7%

   \[\leadsto \log \left(x + x\right) \]

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 58.3%

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

2. Taylor expanded in x around inf 96.8%

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

   1. associate-*r/ 96.8%

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

   2. metadata-eval 96.8%

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

4. Simplified 96.8%

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

   1. *-un-lft-identity 96.8%

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

   2. fma-def 96.8%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(1, x, x - \frac{0.5}{x}\right)\right)} \]

   3. sub-neg 96.8%

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

   4. distribute-neg-frac 96.8%

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

   5. metadata-eval 96.8%

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

6. Applied egg-rr 96.8%

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

8. Simplified 43.0%

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

9. Taylor expanded in x around inf 43.7%

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

10. Final simplification 43.7%

    \[\leadsto \log x \]

Alternative 4: 21.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.1068793402777777 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.1068793402777777)

C

float code(float x) {
	return 1.1068793402777777f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.1068793402777777e0
end function

Julia

function code(x)
	return Float32(1.1068793402777777)
end

MATLAB

function tmp = code(x)
	tmp = single(1.1068793402777777);
end

Derivation

1. Initial program 58.3%

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

   1. add-sqr-sqrt 57.4%

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

   2. pow2 57.4%

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

   3. fma-neg 57.4%

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

   4. metadata-eval 57.4%

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

3. Applied egg-rr 57.4%

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

4. Taylor expanded in x around -inf -0.0%

   \[\leadsto {\color{blue}{\left(0.5 \cdot \left(\frac{0.052083333333333336 - 0.125 \cdot \frac{0.09375 - {\left(0.125 \cdot \sqrt{\frac{1}{\log -0.5 + \log \left(\frac{-1}{x}\right)}}\right)}^{2}}{\log -0.5 + \log \left(\frac{-1}{x}\right)}}{{x}^{6}} \cdot \sqrt{\frac{1}{\log -0.5 + \log \left(\frac{-1}{x}\right)}}\right) + \left(0.5 \cdot \left(\frac{0.09375 - {\left(0.125 \cdot \sqrt{\frac{1}{\log -0.5 + \log \left(\frac{-1}{x}\right)}}\right)}^{2}}{{x}^{4}} \cdot \sqrt{\frac{1}{\log -0.5 + \log \left(\frac{-1}{x}\right)}}\right) + \left(\sqrt{\log -0.5 + \log \left(\frac{-1}{x}\right)} + 0.125 \cdot \left(\sqrt{\frac{1}{\log -0.5 + \log \left(\frac{-1}{x}\right)}} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}}^{2} \]

6. Simplified 21.4%

   \[\leadsto {\color{blue}{1.0520833333333333}}^{2} \]
7. Step-by-step derivation

   1. metadata-eval 21.4%

      \[\leadsto \color{blue}{1.1068793402777777} \]

8. Applied egg-rr 21.4%

   \[\leadsto \color{blue}{1.1068793402777777} \]

9. Final simplification 21.4%

   \[\leadsto 1.1068793402777777 \]

Alternative 5: 21.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.1614583333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.1614583333333333)

C

float code(float x) {
	return 1.1614583333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.1614583333333333e0
end function

Julia

function code(x)
	return Float32(1.1614583333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(1.1614583333333333);
end

Derivation

1. Initial program 58.3%

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

   1. add-sqr-sqrt 57.4%

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

   2. pow2 57.4%

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

   3. fma-neg 57.4%

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

   4. metadata-eval 57.4%

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

3. Applied egg-rr 57.4%

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

4. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\left(0.125 \cdot \frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{{x}^{6} \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right)} + \left(-1 \cdot \frac{0.052083333333333336 + 0.125 \cdot \frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{6}} + \left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(0.015625 \cdot \frac{1}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) \cdot {x}^{4}} + -1 \cdot \frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{{x}^{4}}\right)\right)\right)\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]

6. Simplified 21.5%

   \[\leadsto \color{blue}{1.1614583333333333} \]

7. Final simplification 21.5%

   \[\leadsto 1.1614583333333333 \]

Alternative 6: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2.109375 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.109375)

C

float code(float x) {
	return 2.109375f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.109375e0
end function

Julia

function code(x)
	return Float32(2.109375)
end

MATLAB

function tmp = code(x)
	tmp = single(2.109375);
end

Derivation

1. Initial program 58.3%

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

   1. add-sqr-sqrt 57.4%

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

   2. pow2 57.4%

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

   3. fma-neg 57.4%

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

   4. metadata-eval 57.4%

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

3. Applied egg-rr 57.4%

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

4. Taylor expanded in x around inf 98.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(0.015625 \cdot \frac{1}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) \cdot {x}^{4}} + -1 \cdot \frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{{x}^{4}}\right)\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]

6. Simplified 22.5%

   \[\leadsto \color{blue}{2.109375} \]

7. Final simplification 22.5%

   \[\leadsto 2.109375 \]

Alternative 7: 23.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 4.211046006944445 \end{array} \]

FPCore

(FPCore (x) :precision binary32 4.211046006944445)

C

float code(float x) {
	return 4.211046006944445f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 4.211046006944445e0
end function

Julia

function code(x)
	return Float32(4.211046006944445)
end

MATLAB

function tmp = code(x)
	tmp = single(4.211046006944445);
end

Derivation

1. Initial program 58.3%

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

   1. add-sqr-sqrt 57.4%

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

   2. pow2 57.4%

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

   3. fma-neg 57.4%

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

   4. metadata-eval 57.4%

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

3. Applied egg-rr 57.4%

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

4. Taylor expanded in x around inf 97.1%

   \[\leadsto {\color{blue}{\left(-0.125 \cdot \left(\sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}} \cdot \frac{1}{{x}^{2}}\right) + \left(-0.5 \cdot \left(\frac{0.052083333333333336 + 0.125 \cdot \frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{6}} \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right) + \left(-0.5 \cdot \left(\frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{{x}^{4}} \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right) + \sqrt{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}\right)\right)\right)}}^{2} \]

6. Simplified 24.3%

   \[\leadsto {\color{blue}{2.0520833333333335}}^{2} \]
7. Step-by-step derivation

   1. metadata-eval 24.3%

      \[\leadsto \color{blue}{4.211046006944445} \]

8. Applied egg-rr 24.3%

   \[\leadsto \color{blue}{4.211046006944445} \]

9. Final simplification 24.3%

   \[\leadsto 4.211046006944445 \]

Developer target: 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 2023224 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

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