Rust f32::acosh

Percentage Accurate: 53.9% → 99.5%

Time: 5.3s

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: 53.9% 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 99.5%

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

      1. mul-1-neg 99.5%

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

      2. log-rec 99.5%

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

      3. remove-double-neg 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.7%

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

Alternative 2: 97.0% accurate, 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 54.3%

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

3. Taylor expanded in x around inf 96.6%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
4. Add Preprocessing

Alternative 3: 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 54.3%

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

3. Taylor expanded in x around inf 96.6%

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

4. Taylor expanded in x around 0 96.2%

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

6. Simplified 44.0%

   \[\leadsto \color{blue}{\log x} \]
7. Add Preprocessing

Alternative 4: 19.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 0.3333333333333333)

C

float code(float x) {
	return 0.3333333333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.3333333333333333e0
end function

Julia

function code(x)
	return Float32(0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(0.3333333333333333);
end

Derivation

1. Initial program 54.3%

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

   1. pow2 54.3%

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

   2. add-sqr-sqrt 54.3%

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

   3. pow2 54.3%

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

   4. pow-pow 54.3%

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

   5. metadata-eval 54.3%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 54.3%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]
5. Step-by-step derivation

   1. +-commutative 54.3%

      \[\leadsto \log \color{blue}{\left(\sqrt{{\left(\sqrt{x}\right)}^{4} - 1} + x\right)} \]

   2. add-sqr-sqrt 54.3%

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

   3. fma-define 54.3%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\sqrt{\sqrt{{\left(\sqrt{x}\right)}^{4} - 1}}, \sqrt{\sqrt{{\left(\sqrt{x}\right)}^{4} - 1}}, x\right)\right)} \]

6. Applied egg-rr 37.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{3}\right)} \]

8. Simplified 19.7%

   \[\leadsto \color{blue}{0.3333333333333333} \]
9. Add Preprocessing

Developer Target 1: 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 2024177 
(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)))))