Rust f32::acosh

Percentage Accurate: 52.8% → 99.1%

Time: 4.9s

Alternatives: 7

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: 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \mathsf{hypot}\left(1, \sqrt{x}\right) \cdot \sqrt{x + -1}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log (+ x (* (hypot 1.0 (sqrt x)) (sqrt (+ x -1.0))))))

C

float code(float x) {
	return logf((x + (hypotf(1.0f, sqrtf(x)) * sqrtf((x + -1.0f)))));
}

Julia

function code(x)
	return log(Float32(x + Float32(hypot(Float32(1.0), sqrt(x)) * sqrt(Float32(x + Float32(-1.0))))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + (hypot(single(1.0), sqrt(x)) * sqrt((x + single(-1.0))))));
end

Derivation

1. Initial program 54.6%

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

   1. difference-of-sqr-1 54.6%

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

   2. sqrt-prod 98.8%

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

   3. +-commutative 98.8%

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

   4. add-sqr-sqrt 98.8%

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

   5. hypot-1-def 98.8%

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

   6. sub-neg 98.8%

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

   7. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 98.5% 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 54.6%

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

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 3: 98.0% 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 54.6%

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

2. Taylor expanded in x around inf 97.7%

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

   1. *-commutative 97.7%

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

   2. associate-*r/ 97.7%

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

   3. metadata-eval 97.7%

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

4. Simplified 97.7%

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

5. Final simplification 97.7%

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

Alternative 4: 97.6% accurate, 2.0× 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 54.6%

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

   1. flip-+ 6.3%

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

   2. div-inv 6.3%

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

   3. log-prod 6.3%

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

   4. add-sqr-sqrt 6.0%

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

   5. fma-neg 6.0%

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

   6. metadata-eval 6.0%

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

   7. fma-neg 6.0%

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

   8. metadata-eval 6.0%

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

3. Applied egg-rr 6.0%

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

   1. fma-def 6.0%

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

   2. associate--r+ 8.8%

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

   3. +-inverses 9.4%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   4. metadata-eval 9.4%

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

   5. metadata-eval 9.4%

      \[\leadsto \color{blue}{0} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   6. +-lft-identity 9.4%

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

   7. log-rec 8.8%

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

5. Simplified 8.8%

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

6. Taylor expanded in x around inf 97.7%

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

7. Final simplification 97.7%

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

Alternative 5: 96.7% 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 54.6%

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

2. Taylor expanded in x around inf 96.6%

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

3. Final simplification 96.6%

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

Alternative 6: 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 54.6%

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

   1. difference-of-sqr-1 54.6%

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

   2. sqrt-prod 98.8%

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

   3. +-commutative 98.8%

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

   4. add-sqr-sqrt 98.8%

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

   5. hypot-1-def 98.8%

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

   6. sub-neg 98.8%

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

   7. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

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

4. Taylor expanded in x around inf 44.4%

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

5. Final simplification 44.4%

   \[\leadsto \log x \]

Alternative 7: 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 54.6%

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

2. Taylor expanded in x around inf 97.7%

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

   1. *-commutative 97.7%

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

   2. associate-*r/ 97.7%

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

   3. metadata-eval 97.7%

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

4. Simplified 97.7%

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

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%

   \[\leadsto -0.03125 \]

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 2023172 
(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)))))