Rust f32::acosh

Percentage Accurate: 53.2% → 98.0%

Time: 6.0s

Alternatives: 5

Speedup: 2.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 98.0% 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 49.9%

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

2. Taylor expanded in x around inf 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

4. Simplified 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 96.6% 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 49.9%

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

2. Taylor expanded in x around inf 97.3%

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

3. Final simplification 97.3%

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

Alternative 3: 22.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log 9 \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log 9.0))

C

float code(float x) {
	return logf(9.0f);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(9.0e0)
end function

Julia

function code(x)
	return log(Float32(9.0))
end

MATLAB

function tmp = code(x)
	tmp = log(single(9.0));
end

Derivation

1. Initial program 49.9%

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

2. Taylor expanded in x around inf 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

4. Simplified 98.5%

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

   1. associate-+r- 98.5%

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

   2. flip-+ -0.0%

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

   3. +-inverses -0.0%

      \[\leadsto \log \left(\frac{\color{blue}{0}}{x - x} - \frac{0.5}{x}\right) \]

   4. metadata-eval -0.0%

      \[\leadsto \log \left(\frac{\color{blue}{{0}^{3}}}{x - x} - \frac{0.5}{x}\right) \]

   5. +-inverses -0.0%

      \[\leadsto \log \left(\frac{{\color{blue}{\left(x \cdot x - x \cdot x\right)}}^{3}}{x - x} - \frac{0.5}{x}\right) \]

   6. +-inverses -0.0%

      \[\leadsto \log \left(\frac{{\left(x \cdot x - x \cdot x\right)}^{3}}{\color{blue}{0}} - \frac{0.5}{x}\right) \]

   7. metadata-eval -0.0%

      \[\leadsto \log \left(\frac{{\left(x \cdot x - x \cdot x\right)}^{3}}{\color{blue}{{0}^{3}}} - \frac{0.5}{x}\right) \]

   8. +-inverses -0.0%

      \[\leadsto \log \left(\frac{{\left(x \cdot x - x \cdot x\right)}^{3}}{{\color{blue}{\left(x - x\right)}}^{3}} - \frac{0.5}{x}\right) \]

   9. cube-div -0.0%

      \[\leadsto \log \left(\color{blue}{{\left(\frac{x \cdot x - x \cdot x}{x - x}\right)}^{3}} - \frac{0.5}{x}\right) \]

   10. flip-+ 12.0%

       \[\leadsto \log \left({\color{blue}{\left(x + x\right)}}^{3} - \frac{0.5}{x}\right) \]

   11. count-2 12.0%

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

   12. unpow-prod-down 12.0%

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

   13. metadata-eval 12.0%

       \[\leadsto \log \left(\color{blue}{8} \cdot {x}^{3} - \frac{0.5}{x}\right) \]

   14. metadata-eval 12.0%

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

   15. metadata-eval 12.0%

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

   16. clear-num 12.0%

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

   17. div-inv 12.0%

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

   18. metadata-eval 12.0%

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

   19. *-commutative 12.0%

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

   20. count-2 12.0%

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

   21. flip-+ -0.0%

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

   22. +-inverses -0.0%

       \[\leadsto \log \left(\left(\left(2 \cdot 2\right) \cdot 2\right) \cdot {x}^{3} - \frac{1}{\frac{\color{blue}{0}}{x - x}}\right) \]

   23. +-inverses -0.0%

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

   24. +-inverses -0.0%

       \[\leadsto \log \left(\left(\left(2 \cdot 2\right) \cdot 2\right) \cdot {x}^{3} - \frac{1}{\frac{x - x}{\color{blue}{0}}}\right) \]

   25. +-inverses -0.0%

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

   26. clear-num -0.0%

       \[\leadsto \log \left(\left(\left(2 \cdot 2\right) \cdot 2\right) \cdot {x}^{3} - \color{blue}{\frac{x \cdot x - x \cdot x}{x - x}}\right) \]

   27. flip-+ 12.0%

       \[\leadsto \log \left(\left(\left(2 \cdot 2\right) \cdot 2\right) \cdot {x}^{3} - \color{blue}{\left(x + x\right)}\right) \]

   28. count-2 12.0%

       \[\leadsto \log \left(\left(\left(2 \cdot 2\right) \cdot 2\right) \cdot {x}^{3} - \color{blue}{2 \cdot x}\right) \]

   29. *-commutative 12.0%

       \[\leadsto \log \left(\left(\left(2 \cdot 2\right) \cdot 2\right) \cdot {x}^{3} - \color{blue}{x \cdot 2}\right) \]

6. Applied egg-rr 12.0%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(8, {x}^{3}, \frac{-0.5}{x}\right)\right)} \]

8. Simplified 22.2%

   \[\leadsto \log \color{blue}{9} \]

9. Final simplification 22.2%

   \[\leadsto \log 9 \]

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 49.9%

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

   1. pow2 49.9%

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

   2. pow-to-exp 49.7%

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

   3. expm1-def 49.7%

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\mathsf{expm1}\left(\log x \cdot 2\right)}}\right) \]

3. Applied egg-rr 49.7%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{\mathsf{expm1}\left(\log x \cdot 2\right)}}\right) \]

4. Taylor expanded in x around inf 44.3%

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

5. Final simplification 44.3%

   \[\leadsto \log x \]

Alternative 5: 21.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.3333333333333333)

C

float code(float x) {
	return 1.3333333333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.3333333333333333e0
end function

Julia

function code(x)
	return Float32(1.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(1.3333333333333333);
end

Derivation

1. Initial program 49.9%

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

   1. pow2 49.9%

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

   2. pow-to-exp 49.7%

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

   3. expm1-def 49.7%

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\mathsf{expm1}\left(\log x \cdot 2\right)}}\right) \]

3. Applied egg-rr 49.7%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{\mathsf{expm1}\left(\log x \cdot 2\right)}}\right) \]

4. Taylor expanded in x around 0 13.4%

   \[\leadsto \color{blue}{\log \left(\sqrt{{x}^{2} - 1}\right) + \left(0.3333333333333333 \cdot \left(\sqrt{\frac{1}{{\left({x}^{2} - 1\right)}^{3}}} \cdot {x}^{3}\right) + \left(\sqrt{\frac{1}{{x}^{2} - 1}} \cdot x + -0.5 \cdot \frac{{x}^{2}}{{x}^{2} - 1}\right)\right)} \]

6. Simplified 21.3%

   \[\leadsto \color{blue}{1.3333333333333333} \]

7. Final simplification 21.3%

   \[\leadsto 1.3333333333333333 \]

Developer target: 99.1% 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 2023171 
(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)))))