Rust f32::acosh

Percentage Accurate: 52.8% → 98.6%

Time: 6.1s

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: 52.8% 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: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(\log 2 - \frac{0.25}{x \cdot x}\right) - \left(\frac{0.09375 + \frac{0.052083333333333336}{x \cdot x}}{{x}^{4}} - \log x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (-
  (- (log 2.0) (/ 0.25 (* x x)))
  (- (/ (+ 0.09375 (/ 0.052083333333333336 (* x x))) (pow x 4.0)) (log x))))

C

float code(float x) {
	return (logf(2.0f) - (0.25f / (x * x))) - (((0.09375f + (0.052083333333333336f / (x * x))) / powf(x, 4.0f)) - logf(x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (log(2.0e0) - (0.25e0 / (x * x))) - (((0.09375e0 + (0.052083333333333336e0 / (x * x))) / (x ** 4.0e0)) - log(x))
end function

Julia

function code(x)
	return Float32(Float32(log(Float32(2.0)) - Float32(Float32(0.25) / Float32(x * x))) - Float32(Float32(Float32(Float32(0.09375) + Float32(Float32(0.052083333333333336) / Float32(x * x))) / (x ^ Float32(4.0))) - log(x)))
end

MATLAB

function tmp = code(x)
	tmp = (log(single(2.0)) - (single(0.25) / (x * x))) - (((single(0.09375) + (single(0.052083333333333336) / (x * x))) / (x ^ single(4.0))) - log(x));
end

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left(\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) + -1 \cdot \frac{\frac{3}{32} + \frac{5}{96} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + -1 \cdot \frac{\frac{3}{32} + \frac{5}{96} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) + \log 2\right)} - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + -1 \cdot \frac{\frac{3}{32} + \frac{5}{96} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) + \left(\log 2 - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\log 2 - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + -1 \cdot \frac{\frac{3}{32} + \frac{5}{96} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)} \]

   4. distribute-lft-out N/A

      \[\leadsto \left(\log 2 - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) + \color{blue}{-1 \cdot \left(\log \left(\frac{1}{x}\right) + \frac{\frac{3}{32} + \frac{5}{96} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)} \]

   5. mul-1-neg N/A

      \[\leadsto \left(\log 2 - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\log \left(\frac{1}{x}\right) + \frac{\frac{3}{32} + \frac{5}{96} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right)\right)} \]

   6. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\log 2 - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) - \left(\log \left(\frac{1}{x}\right) + \frac{\frac{3}{32} + \frac{5}{96} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)} \]

   7. lower--.f32 N/A

      \[\leadsto \color{blue}{\left(\log 2 - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) - \left(\log \left(\frac{1}{x}\right) + \frac{\frac{3}{32} + \frac{5}{96} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)} \]

5. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\left(\log 2 - \frac{0.25}{x \cdot x}\right) - \left(\frac{\frac{0.052083333333333336}{x \cdot x} + 0.09375}{{x}^{4}} - \log x\right)} \]

6. Final simplification 98.9%

   \[\leadsto \left(\log 2 - \frac{0.25}{x \cdot x}\right) - \left(\frac{0.09375 + \frac{0.052083333333333336}{x \cdot x}}{{x}^{4}} - \log x\right) \]
7. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\left(\left(2 - \frac{0.5}{x \cdot x}\right) - \frac{0.125 + \frac{0.0625}{x \cdot x}}{{x}^{4}}\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (*
   (- (- 2.0 (/ 0.5 (* x x))) (/ (+ 0.125 (/ 0.0625 (* x x))) (pow x 4.0)))
   x)))

C

float code(float x) {
	return logf((((2.0f - (0.5f / (x * x))) - ((0.125f + (0.0625f / (x * x))) / powf(x, 4.0f))) * x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((((2.0e0 - (0.5e0 / (x * x))) - ((0.125e0 + (0.0625e0 / (x * x))) / (x ** 4.0e0))) * x))
end function

Julia

function code(x)
	return log(Float32(Float32(Float32(Float32(2.0) - Float32(Float32(0.5) / Float32(x * x))) - Float32(Float32(Float32(0.125) + Float32(Float32(0.0625) / Float32(x * x))) / (x ^ Float32(4.0)))) * x))
end

MATLAB

function tmp = code(x)
	tmp = log((((single(2.0) - (single(0.5) / (x * x))) - ((single(0.125) + (single(0.0625) / (x * x))) / (x ^ single(4.0)))) * x));
end

Derivation

1. Initial program 50.6%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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. lower-+.f32 50.6

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 28.2

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

4. Applied rewrites 27.8%

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

5. Taylor expanded in x around inf

   \[\leadsto \log \color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log \color{blue}{\left(\left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \log \color{blue}{\left(\left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)} \]

7. Applied rewrites 98.5%

   \[\leadsto \log \color{blue}{\left(\left(\left(2 - \frac{0.5}{x \cdot x}\right) - \frac{\frac{0.0625}{x \cdot x} + 0.125}{{x}^{4}}\right) \cdot x\right)} \]

8. Final simplification 98.5%

   \[\leadsto \log \left(\left(\left(2 - \frac{0.5}{x \cdot x}\right) - \frac{0.125 + \frac{0.0625}{x \cdot x}}{{x}^{4}}\right) \cdot x\right) \]
9. Add Preprocessing

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\log x + \log 2\right) - \frac{0.25}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (- (+ (log x) (log 2.0)) (/ 0.25 (* x x))))

C

float code(float x) {
	return (logf(x) + logf(2.0f)) - (0.25f / (x * x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (log(x) + log(2.0e0)) - (0.25e0 / (x * x))
end function

Julia

function code(x)
	return Float32(Float32(log(x) + log(Float32(2.0))) - Float32(Float32(0.25) / Float32(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (log(x) + log(single(2.0))) - (single(0.25) / (x * x));
end

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in x around inf

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. remove-double-neg N/A

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

   7. lower-log.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\left(\log x + \log 2\right) - \frac{0.25}{x \cdot x}} \]
6. Add Preprocessing

Alternative 4: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 50.6%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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. lower-+.f32 50.6

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 27.7

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

4. Applied rewrites 27.8%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. log-rec N/A

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

   5. remove-double-neg N/A

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

   6. lower-log.f32 N/A

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

   7. lower-log.f32 97.2

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

7. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\log x + \log 2} \]
8. Step-by-step derivation

9. Applied rewrites 98.2%

   \[\leadsto -\log \left(\frac{0.5}{x}\right) \]
10. Add Preprocessing

Alternative 5: 97.2% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log (* 2.0 x)))

C

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

Fortran

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

Julia

function code(x)
	return log(Float32(Float32(2.0) * x))
end

MATLAB

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

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f32 96.7

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

5. Applied rewrites 96.7%

   \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
6. Add Preprocessing

Developer Target 1: 99.3% 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 2024266 
(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)))))