Rust f32::acosh

Percentage Accurate: 53.4% → 98.2%

Time: 5.6s

Alternatives: 8

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: 53.4% 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

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

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 51.7%

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

2. Taylor expanded in x around inf 97.9%

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

   1. associate-*r/ 97.9%

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

   2. metadata-eval 97.9%

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

4. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 2: 44.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + 0.875\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log (+ x 0.875)))

C

float code(float x) {
	return logf((x + 0.875f));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + 0.875e0))
end function

Julia

function code(x)
	return log(Float32(x + Float32(0.875)))
end

MATLAB

function tmp = code(x)
	tmp = log((x + single(0.875)));
end

Derivation

1. Initial program 51.7%

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

   1. difference-of-sqr-1 51.7%

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

   2. sub-neg 51.7%

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

   3. metadata-eval 51.7%

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

3. Applied egg-rr 51.7%

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

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto \log \left(x + \color{blue}{\left(-0.125 \cdot \frac{{x}^{4}}{{\left(\sqrt{-1}\right)}^{3}} + \left(\sqrt{-1} + 0.5 \cdot \frac{{x}^{2}}{\sqrt{-1}}\right)\right)}\right) \]

6. Simplified 44.2%

   \[\leadsto \log \left(x + \color{blue}{0.875}\right) \]

7. Final simplification 44.2%

   \[\leadsto \log \left(x + 0.875\right) \]

Alternative 3: 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 51.7%

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

2. Taylor expanded in x around inf 96.6%

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

3. Final simplification 96.6%

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

Alternative 4: 44.1% 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 51.7%

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

   1. difference-of-sqr-1 51.7%

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

   2. sub-neg 51.7%

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

   3. metadata-eval 51.7%

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

3. Applied egg-rr 51.7%

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

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto \log \left(x + \color{blue}{\left(-0.125 \cdot \frac{{x}^{4}}{{\left(\sqrt{-1}\right)}^{3}} + \left(\sqrt{-1} + 0.5 \cdot \frac{{x}^{2}}{\sqrt{-1}}\right)\right)}\right) \]

6. Simplified 44.2%

   \[\leadsto \log \left(x + \color{blue}{0.875}\right) \]

7. Taylor expanded in x around inf 44.1%

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

   1. mul-1-neg 44.1%

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

   2. log-rec 44.1%

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

   3. remove-double-neg 44.1%

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

9. Simplified 44.1%

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

10. Final simplification 44.1%

    \[\leadsto \log x \]

Alternative 5: 20.8% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 \end{array} \]

FPCore

(FPCore (x) :precision binary32 0.8333333333333334)

C

float code(float x) {
	return 0.8333333333333334f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.8333333333333334e0
end function

Julia

function code(x)
	return Float32(0.8333333333333334)
end

MATLAB

function tmp = code(x)
	tmp = single(0.8333333333333334);
end

Derivation

1. Initial program 51.7%

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

   1. difference-of-sqr-1 51.7%

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

   2. sub-neg 51.7%

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

   3. metadata-eval 51.7%

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

3. Applied egg-rr 51.7%

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

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{x}^{3}}{{\left(\sqrt{-1}\right)}^{3}} + \left(\frac{x}{\sqrt{-1}} + \log \left(\sqrt{-1}\right)\right)} \]

6. Simplified 20.8%

   \[\leadsto \color{blue}{0.8333333333333334} \]

7. Final simplification 20.8%

   \[\leadsto 0.8333333333333334 \]

Alternative 6: 22.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.9583333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.9583333333333333)

C

float code(float x) {
	return 1.9583333333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.9583333333333333e0
end function

Julia

function code(x)
	return Float32(1.9583333333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(1.9583333333333333);
end

Derivation

1. Initial program 51.7%

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

   1. difference-of-sqr-1 51.7%

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

   2. sub-neg 51.7%

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

   3. metadata-eval 51.7%

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

3. Applied egg-rr 51.7%

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

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

   \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{-1}{x}\right) + \left(\log \left({\left(\sqrt{-1}\right)}^{2} - 1\right) + \left(-0.041666666666666664 \cdot \frac{3 \cdot \frac{1}{{\left({\left(\sqrt{-1}\right)}^{2} - 1\right)}^{2}} + 3 \cdot \frac{1}{\left({\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}{{x}^{4}} + 0.5 \cdot \frac{1}{\left({\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot {x}^{2}}\right)\right)} \]

6. Simplified 22.2%

   \[\leadsto \color{blue}{1.9583333333333333} \]

7. Final simplification 22.2%

   \[\leadsto 1.9583333333333333 \]

Alternative 7: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2.09375 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.09375)

C

float code(float x) {
	return 2.09375f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.09375e0
end function

Julia

function code(x)
	return Float32(2.09375)
end

MATLAB

function tmp = code(x)
	tmp = single(2.09375);
end

Derivation

1. Initial program 51.7%

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

   1. log1p-expm1-u 51.7%

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

   2. expm1-udef 51.7%

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

   3. add-exp-log 51.7%

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

   4. fma-neg 51.8%

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

   5. metadata-eval 51.8%

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

3. Applied egg-rr 51.8%

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

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

   \[\leadsto \color{blue}{\log \left(\frac{-1}{x}\right) + \left(\log -0.5 + \left(0.09375 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

6. Simplified 22.3%

   \[\leadsto \color{blue}{2.09375} \]

7. Final simplification 22.3%

   \[\leadsto 2.09375 \]

Alternative 8: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2.1458333333333335 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.1458333333333335)

C

float code(float x) {
	return 2.1458333333333335f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.1458333333333335e0
end function

Julia

function code(x)
	return Float32(2.1458333333333335)
end

MATLAB

function tmp = code(x)
	tmp = single(2.1458333333333335);
end

Derivation

1. Initial program 51.7%

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

   1. log1p-expm1-u 51.7%

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

   2. expm1-udef 51.7%

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

   3. add-exp-log 51.7%

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

   4. fma-neg 51.8%

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

   5. metadata-eval 51.8%

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

3. Applied egg-rr 51.8%

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

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

   \[\leadsto \color{blue}{\log \left(\frac{-1}{x}\right) + \left(\log -0.5 + \left(0.09375 \cdot \frac{1}{{x}^{4}} + \left(0.052083333333333336 \cdot \frac{1}{{x}^{6}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

6. Simplified 22.3%

   \[\leadsto \color{blue}{2.1458333333333335} \]

7. Final simplification 22.3%

   \[\leadsto 2.1458333333333335 \]

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 2023196 
(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)))))