Rust f32::acosh

Percentage Accurate: 52.5% → 99.5%

Time: 4.1s

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: 52.5% 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: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sqrt{x \cdot x + -1}\\ \mathbf{if}\;t\_0 \leq 5.000000100204387 \cdot 10^{+19}:\\ \;\;\;\;\log t\_0\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ x (sqrt (+ (* x x) -1.0)))))
   (if (<= t_0 5.000000100204387e+19) (log t_0) (+ (log 2.0) (log x)))))

C

float code(float x) {
	float t_0 = x + sqrtf(((x * x) + -1.0f));
	float tmp;
	if (t_0 <= 5.000000100204387e+19f) {
		tmp = logf(t_0);
	} else {
		tmp = logf(2.0f) + logf(x);
	}
	return tmp;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x + sqrt(((x * x) + (-1.0e0)))
    if (t_0 <= 5.000000100204387e+19) then
        tmp = log(t_0)
    else
        tmp = log(2.0e0) + log(x)
    end if
    code = tmp
end function

Julia

function code(x)
	t_0 = Float32(x + sqrt(Float32(Float32(x * x) + Float32(-1.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(5.000000100204387e+19))
		tmp = log(t_0);
	else
		tmp = Float32(log(Float32(2.0)) + log(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x + sqrt(((x * x) + single(-1.0)));
	tmp = single(0.0);
	if (t_0 <= single(5.000000100204387e+19))
		tmp = log(t_0);
	else
		tmp = log(single(2.0)) + log(x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 x (sqrt.f32 (-.f32 (*.f32 x x) #s(literal 1 binary32)))) < 5.0000001e19

   1. Initial program 100.0%

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

   if 5.0000001e19 < (+.f32 x (sqrt.f32 (-.f32 (*.f32 x x) #s(literal 1 binary32))))

   1. Initial program 6.5%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. log-rec 99.2%

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

      3. remove-double-neg 99.2%

         \[\leadsto \log 2 + \color{blue}{\log x} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\log 2 + \log x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 5.000000100204387 \cdot 10^{+19}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{fma}\left(x, 2, -1\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log1p (fma x 2.0 -1.0)))

C

float code(float x) {
	return log1pf(fmaf(x, 2.0f, -1.0f));
}

Julia

function code(x)
	return log1p(fma(x, Float32(2.0), Float32(-1.0)))
end

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in x around inf 96.1%

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

   1. log1p-expm1-u 96.1%

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

   2. expm1-undefine 96.1%

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

   3. count-2 96.1%

      \[\leadsto \mathsf{log1p}\left(e^{\log \color{blue}{\left(2 \cdot x\right)}} - 1\right) \]

   4. add-exp-log 96.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{2 \cdot x} - 1\right) \]

   5. *-commutative 96.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot 2} - 1\right) \]

   6. fmm-def 96.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, 2, -1\right)}\right) \]

   7. metadata-eval 96.1%

      \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left(x, 2, \color{blue}{-1}\right)\right) \]

5. Applied egg-rr 96.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, 2, -1\right)\right)} \]

6. Final simplification 96.1%

   \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left(x, 2, -1\right)\right) \]
7. Add Preprocessing

Alternative 3: 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 58.0%

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

3. Taylor expanded in x around inf 96.1%

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

4. Final simplification 96.1%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Alternative 4: 6.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary32 0.0)

C

float code(float x) {
	return 0.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.0e0
end function

Julia

function code(x)
	return Float32(0.0)
end

MATLAB

function tmp = code(x)
	tmp = single(0.0);
end

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in x around inf 96.1%

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

   1. add-sqr-sqrt 96.0%

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

   2. log-prod 96.0%

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

   3. flip-+ -0.0%

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

   4. difference-of-squares -0.0%

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

   5. associate-*r/ -0.0%

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

   6. +-inverses -0.0%

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

   7. +-inverses -0.0%

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

   8. flip-+ 19.5%

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

   9. sqrt-unprod 30.3%

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

   10. add-sqr-sqrt 30.3%

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

   11. flip-+ -0.0%

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

   12. +-inverses -0.0%

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

   13. +-inverses -0.0%

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

   14. flip-+ -0.0%

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

   15. difference-of-squares -0.0%

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

   16. associate-*r/ -0.0%

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

   17. +-inverses -0.0%

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

   18. +-inverses -0.0%

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

   19. flip-+ -0.0%

       \[\leadsto \log \left(\frac{0}{0}\right) + \log \left(\sqrt{\left(x + x\right) \cdot \color{blue}{\left(x + x\right)}}\right) \]

5. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\log \left(\frac{0}{0}\right) + \log \left(\frac{0}{0}\right)} \]

7. Simplified 6.1%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 6.1%

   \[\leadsto 0 \]
9. Add Preprocessing

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 2024095 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

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

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