Rust f32::acosh

Percentage Accurate: 53.4% → 98.7%

Time: 4.9s

Alternatives: 5

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: 53.4% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log (- (* x 2.0) (+ (/ 0.5 x) (/ 0.125 (pow x 3.0))))))

C

float code(float x) {
	return logf(((x * 2.0f) - ((0.5f / x) + (0.125f / powf(x, 3.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(((x * 2.0e0) - ((0.5e0 / x) + (0.125e0 / (x ** 3.0e0)))))
end function

Julia

function code(x)
	return log(Float32(Float32(x * Float32(2.0)) - Float32(Float32(Float32(0.5) / x) + Float32(Float32(0.125) / (x ^ Float32(3.0))))))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * single(2.0)) - ((single(0.5) / x) + (single(0.125) / (x ^ single(3.0))))));
end

Derivation

1. Initial program 56.9%

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

2. Taylor expanded in x around inf 98.0%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative 98.0%

      \[\leadsto \log \left(\color{blue}{x \cdot 2} - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   2. associate-*r/ 98.0%

      \[\leadsto \log \left(x \cdot 2 - \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   3. metadata-eval 98.0%

      \[\leadsto \log \left(x \cdot 2 - \left(\frac{\color{blue}{0.5}}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   4. associate-*r/ 98.0%

      \[\leadsto \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right)\right) \]

   5. metadata-eval 98.0%

      \[\leadsto \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{\color{blue}{0.125}}{{x}^{3}}\right)\right) \]

4. Simplified 98.0%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)} \]

5. Final simplification 98.0%

   \[\leadsto \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right) \]

Alternative 2: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

2. Taylor expanded in x around inf 97.5%

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

   1. *-commutative 97.5%

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

   2. associate-*r/ 97.5%

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

   3. metadata-eval 97.5%

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

4. Simplified 97.5%

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

5. Final simplification 97.5%

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

Alternative 3: 96.8% 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 56.9%

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

2. Taylor expanded in x around inf 96.2%

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

3. Final simplification 96.2%

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

Alternative 4: 3.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -0.03125 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -0.03125)

C

float code(float x) {
	return -0.03125f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -0.03125e0
end function

Julia

function code(x)
	return Float32(-0.03125)
end

MATLAB

function tmp = code(x)
	tmp = single(-0.03125);
end

Derivation

1. Initial program 56.9%

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

2. Taylor expanded in x around inf 97.5%

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

   1. *-commutative 97.5%

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

   2. associate-*r/ 97.5%

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

   3. metadata-eval 97.5%

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

4. Simplified 97.5%

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

5. Taylor expanded in x around inf 98.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - \left(0.03125 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

7. Simplified 3.2%

   \[\leadsto \color{blue}{-0.03125} \]

8. Final simplification 3.2%

   \[\leadsto -0.03125 \]

Alternative 5: 21.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.0)

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.0e0
end function

Julia

function code(x)
	return Float32(1.0)
end

MATLAB

function tmp = code(x)
	tmp = single(1.0);
end

Derivation

1. Initial program 56.9%

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

   1. add-cube-cbrt 56.8%

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

   2. pow3 56.8%

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

   3. log-pow 56.3%

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

   4. fma-neg 56.3%

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

   5. metadata-eval 56.3%

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

3. Applied egg-rr 56.3%

   \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right)} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 56.3%

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

   2. log-prod 56.3%

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

   3. pow1/3 56.3%

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

   4. sqrt-pow1 56.3%

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

   5. metadata-eval 56.3%

      \[\leadsto 3 \cdot \left(\log \left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{0.16666666666666666}}\right) + \log \left(\sqrt{\sqrt[3]{x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}}}\right)\right) \]

   6. pow1/3 56.2%

      \[\leadsto 3 \cdot \left(\log \left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{0.16666666666666666}\right) + \log \left(\sqrt{\color{blue}{{\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{0.3333333333333333}}}\right)\right) \]

   7. sqrt-pow1 56.2%

      \[\leadsto 3 \cdot \left(\log \left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{0.16666666666666666}\right) + \log \color{blue}{\left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}\right) \]

   8. metadata-eval 56.2%

      \[\leadsto 3 \cdot \left(\log \left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{0.16666666666666666}\right) + \log \left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{0.16666666666666666}}\right)\right) \]

5. Applied egg-rr 56.2%

   \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{0.16666666666666666}\right) + \log \left({\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}^{0.16666666666666666}\right)\right)} \]

7. Simplified 21.1%

   \[\leadsto 3 \cdot \color{blue}{0.3333333333333333} \]

8. Final simplification 21.1%

   \[\leadsto 1 \]

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 2023217 
(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)))))