Rust f32::acosh

Percentage Accurate: 52.7% → 98.6%

Time: 5.8s

Alternatives: 7

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: 52.7% 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, 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 48.1%

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

2. Taylor expanded in x around inf 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.5%

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

   3. metadata-eval 98.5%

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

   4. associate-*r/ 98.5%

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

   5. metadata-eval 98.5%

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

4. Simplified 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 48.1%

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

2. Taylor expanded in x around inf 98.0%

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

   1. associate-*r/ 98.0%

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

   2. metadata-eval 98.0%

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

4. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 3: 96.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log1p (+ (+ x x) -1.0)))

C

float code(float x) {
	return log1pf(((x + x) + -1.0f));
}

Julia

function code(x)
	return log1p(Float32(Float32(x + x) + Float32(-1.0)))
end

Derivation

1. Initial program 48.1%

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

2. Taylor expanded in x around inf 96.6%

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

   1. count-2 96.6%

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

   2. sum-log 96.4%

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

   3. log1p-expm1-u 95.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log 2 + \log x\right)\right)} \]

   4. expm1-udef 95.7%

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

   5. exp-sum 95.6%

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

   6. add-exp-log 95.6%

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

   7. add-exp-log 96.6%

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

   8. count-2 96.6%

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

4. Applied egg-rr 96.6%

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

5. Final simplification 96.6%

   \[\leadsto \mathsf{log1p}\left(\left(x + x\right) + -1\right) \]

Alternative 4: 96.9% 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 48.1%

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

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 5: 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 48.1%

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

   1. pow1/2 48.1%

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

   2. add-cube-cbrt 48.1%

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

   3. pow3 48.1%

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

   4. pow-pow 48.1%

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

   5. fma-neg 48.1%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   6. metadata-eval 48.1%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   7. metadata-eval 48.1%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{1.5}}\right) \]

3. Applied egg-rr 48.1%

   \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}}\right) \]

4. Taylor expanded in x around inf 44.2%

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

   1. mul-1-neg 44.2%

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

   2. log-rec 44.2%

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

   3. remove-double-neg 44.2%

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

6. Simplified 44.2%

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

7. Final simplification 44.2%

   \[\leadsto \log x \]

Alternative 6: 3.1% 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 48.1%

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

2. Taylor expanded in x around inf 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.5%

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

   3. metadata-eval 98.5%

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

   4. associate-*r/ 98.5%

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

   5. metadata-eval 98.5%

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

4. Simplified 98.5%

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

5. Taylor expanded in x around inf 96.5%

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

7. Simplified 3.1%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 3.1%

   \[\leadsto -1 \]

Alternative 7: 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 48.1%

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

2. Taylor expanded in x around inf 96.6%

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

   1. count-2 96.6%

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

   2. sum-log 96.4%

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

   3. log1p-expm1-u 95.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log 2 + \log x\right)\right)} \]

   4. expm1-udef 95.7%

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

   5. exp-sum 95.6%

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

   6. add-exp-log 95.6%

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

   7. add-exp-log 96.6%

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

   8. count-2 96.6%

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

4. Applied egg-rr 96.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + x\right) - 1\right)} \]
5. Step-by-step derivation

   1. count-2 96.6%

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

   2. *-commutative 96.6%

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

   3. add-exp-log 96.6%

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

   4. expm1-def 96.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x \cdot 2\right)\right)}\right) \]

   5. log1p-expm1-u 96.6%

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

   6. *-commutative 96.6%

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

   7. count-2 96.6%

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

   8. flip-+ -0.0%

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

   9. log-div -0.0%

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

6. Applied egg-rr -0.0%

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

   1. unpow2 -0.0%

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

   2. unpow2 -0.0%

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

   3. +-inverses -0.0%

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

   4. +-inverses -0.0%

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

   5. +-inverses 6.1%

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

8. Simplified 6.1%

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

9. Final simplification 6.1%

   \[\leadsto 0 \]

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 2023201 
(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)))))