Rust f32::acosh

Percentage Accurate: 53.4% → 99.2%

Time: 8.8s

Alternatives: 5

Speedup: 1.2×

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: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x + -1}, x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log (fma (sqrt (+ x 1.0)) (sqrt (+ x -1.0)) x)))

C

float code(float x) {
	return logf(fmaf(sqrtf((x + 1.0f)), sqrtf((x + -1.0f)), x));
}

Julia

function code(x)
	return log(fma(sqrt(Float32(x + Float32(1.0))), sqrt(Float32(x + Float32(-1.0))), x))
end

Derivation

1. Initial program 51.7%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. pow1/2 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. difference-of-sqr-1 N/A

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

   8. unpow-prod-down N/A

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

   9. lower-fma.f32 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-+.f32 N/A

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

   13. pow1/2 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. sub-neg N/A

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

   16. lower-+.f32 N/A

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

   17. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. pow1/2 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. difference-of-sqr-1 N/A

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

   8. unpow-prod-down N/A

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

   9. lower-fma.f32 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-+.f32 N/A

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

   13. pow1/2 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. sub-neg N/A

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

   16. lower-+.f32 N/A

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

   17. metadata-eval 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 44.3%

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

8. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. associate--l+ N/A

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

   9. lower-*.f32 N/A

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

   10. associate--l+ N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. unpow2 N/A

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

   14. rem-square-sqrt N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

10. Applied rewrites 98.0%

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

Developer Target 1: 99.2% accurate, 0.9× 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 2024228 
(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)))))