Rust f32::acosh

Percentage Accurate: 52.6% → 98.3%

Time: 5.6s

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: 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 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. associate-*r/ 98.8%

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

   2. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Final simplification 98.8%

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

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 54.0%

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

2. Taylor expanded in x around inf 97.6%

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

3. Final simplification 97.6%

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

Alternative 3: 6.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\mathsf{log1p}\left(-1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (- (log1p -1.0)))

C

float code(float x) {
	return -log1pf(-1.0f);
}

Julia

function code(x)
	return Float32(-log1p(Float32(-1.0)))
end

Derivation

1. Initial program 54.0%

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

   1. flip-+ 7.1%

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

   2. div-inv 7.0%

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

   3. log-prod 7.0%

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

   4. add-sqr-sqrt 7.0%

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

   5. fma-neg 7.0%

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

   6. metadata-eval 7.0%

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

   7. fma-neg 7.0%

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

   8. metadata-eval 7.0%

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

3. Applied egg-rr 7.0%

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

   1. fma-def 7.0%

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

   2. associate--r+ 9.6%

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

   3. +-inverses 10.2%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   4. metadata-eval 10.2%

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

   5. metadata-eval 10.2%

      \[\leadsto \color{blue}{0} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   6. +-lft-identity 10.2%

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

   7. log-rec 9.6%

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

5. Simplified 9.6%

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

6. Taylor expanded in x around inf 97.5%

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

   1. log1p-expm1-u 16.4%

      \[\leadsto -\left(\frac{0.25}{x \cdot x} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{0.5}{x}\right)\right)\right)}\right) \]

   2. expm1-udef 16.4%

      \[\leadsto -\left(\frac{0.25}{x \cdot x} + \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{0.5}{x}\right)} - 1}\right)\right) \]

   3. add-exp-log 16.4%

      \[\leadsto -\left(\frac{0.25}{x \cdot x} + \mathsf{log1p}\left(\color{blue}{\frac{0.5}{x}} - 1\right)\right) \]

8. Applied egg-rr 15.4%

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

9. Taylor expanded in x around inf 6.4%

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

10. Final simplification 6.4%

    \[\leadsto -\mathsf{log1p}\left(-1\right) \]

Alternative 4: 5.3% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \frac{-0.25}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (/ (- 0.25) (* x x)))

C

float code(float x) {
	return -0.25f / (x * x);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -0.25e0 / (x * x)
end function

Julia

function code(x)
	return Float32(Float32(-Float32(0.25)) / Float32(x * x))
end

MATLAB

function tmp = code(x)
	tmp = -single(0.25) / (x * x);
end

Derivation

1. Initial program 54.0%

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

   1. flip-+ 7.1%

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

   2. div-inv 7.0%

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

   3. log-prod 7.0%

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

   4. add-sqr-sqrt 7.0%

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

   5. fma-neg 7.0%

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

   6. metadata-eval 7.0%

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

   7. fma-neg 7.0%

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

   8. metadata-eval 7.0%

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

3. Applied egg-rr 7.0%

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

   1. fma-def 7.0%

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

   2. associate--r+ 9.6%

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

   3. +-inverses 10.2%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   4. metadata-eval 10.2%

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

   5. metadata-eval 10.2%

      \[\leadsto \color{blue}{0} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   6. +-lft-identity 10.2%

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

   7. log-rec 9.6%

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

5. Simplified 9.6%

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

6. Taylor expanded in x around inf 97.7%

   \[\leadsto -\color{blue}{\left(\log \left(\frac{1}{x}\right) + \left(\log 0.5 + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative 97.7%

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

   2. +-commutative 97.7%

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

   3. associate-+l+ 97.8%

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

   4. unpow2 97.8%

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

   5. associate-*r/ 97.8%

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

   6. metadata-eval 97.8%

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

   7. log-prod 98.7%

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

   8. associate-*r/ 98.7%

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

   9. metadata-eval 98.7%

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

8. Simplified 98.7%

   \[\leadsto -\color{blue}{\left(\frac{0.25}{x \cdot x} + \log \left(\frac{0.5}{x}\right)\right)} \]
9. Step-by-step derivation

   1. log1p-expm1-u 16.4%

      \[\leadsto -\left(\frac{0.25}{x \cdot x} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{0.5}{x}\right)\right)\right)}\right) \]

   2. expm1-udef 16.4%

      \[\leadsto -\left(\frac{0.25}{x \cdot x} + \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{0.5}{x}\right)} - 1}\right)\right) \]

   3. add-exp-log 16.4%

      \[\leadsto -\left(\frac{0.25}{x \cdot x} + \mathsf{log1p}\left(\color{blue}{\frac{0.5}{x}} - 1\right)\right) \]

10. Applied egg-rr 16.4%

    \[\leadsto -\left(\frac{0.25}{x \cdot x} + \color{blue}{\mathsf{log1p}\left(\frac{0.5}{x} - 1\right)}\right) \]

11. Taylor expanded in x around 0 5.4%

    \[\leadsto -\color{blue}{\frac{0.25}{{x}^{2}}} \]
12. Step-by-step derivation

    1. unpow2 5.4%

       \[\leadsto -\frac{0.25}{\color{blue}{x \cdot x}} \]

13. Simplified 5.4%

    \[\leadsto -\color{blue}{\frac{0.25}{x \cdot x}} \]

14. Final simplification 5.4%

    \[\leadsto \frac{-0.25}{x \cdot x} \]

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 2023238 
(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)))))