Rust f32::acosh

Percentage Accurate: 53.0% → 99.0%

Time: 4.4s

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: 53.0% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \left(\left(\left(x - \frac{0.125}{{x}^{3}}\right) + \frac{-0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (+ x (- (+ (- x (/ 0.125 (pow x 3.0))) (/ -0.5 x)) (/ 0.0625 (pow x 5.0))))))

C

float code(float x) {
	return logf((x + (((x - (0.125f / powf(x, 3.0f))) + (-0.5f / x)) - (0.0625f / powf(x, 5.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + (((x - (0.125e0 / (x ** 3.0e0))) + ((-0.5e0) / x)) - (0.0625e0 / (x ** 5.0e0)))))
end function

Julia

function code(x)
	return log(Float32(x + Float32(Float32(Float32(x - Float32(Float32(0.125) / (x ^ Float32(3.0)))) + Float32(Float32(-0.5) / x)) - Float32(Float32(0.0625) / (x ^ Float32(5.0))))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + (((x - (single(0.125) / (x ^ single(3.0)))) + (single(-0.5) / x)) - (single(0.0625) / (x ^ single(5.0))))));
end

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{\left(x - \left(0.0625 \cdot \frac{1}{{x}^{5}} + \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative 98.8%

      \[\leadsto \log \left(x + \left(x - \color{blue}{\left(\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right)\right) \]

   2. associate--r+ 98.8%

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right) - 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right) \]

   3. associate--r+ 98.8%

      \[\leadsto \log \left(x + \left(\color{blue}{\left(\left(x - 0.125 \cdot \frac{1}{{x}^{3}}\right) - 0.5 \cdot \frac{1}{x}\right)} - 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   4. sub-neg 98.8%

      \[\leadsto \log \left(x + \left(\color{blue}{\left(\left(x - 0.125 \cdot \frac{1}{{x}^{3}}\right) + \left(-0.5 \cdot \frac{1}{x}\right)\right)} - 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   5. associate-*r/ 98.8%

      \[\leadsto \log \left(x + \left(\left(\left(x - \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right) + \left(-0.5 \cdot \frac{1}{x}\right)\right) - 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   6. metadata-eval 98.8%

      \[\leadsto \log \left(x + \left(\left(\left(x - \frac{\color{blue}{0.125}}{{x}^{3}}\right) + \left(-0.5 \cdot \frac{1}{x}\right)\right) - 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   7. associate-*r/ 98.8%

      \[\leadsto \log \left(x + \left(\left(\left(x - \frac{0.125}{{x}^{3}}\right) + \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) - 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   8. metadata-eval 98.8%

      \[\leadsto \log \left(x + \left(\left(\left(x - \frac{0.125}{{x}^{3}}\right) + \left(-\frac{\color{blue}{0.5}}{x}\right)\right) - 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   9. distribute-neg-frac 98.8%

      \[\leadsto \log \left(x + \left(\left(\left(x - \frac{0.125}{{x}^{3}}\right) + \color{blue}{\frac{-0.5}{x}}\right) - 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   10. metadata-eval 98.8%

       \[\leadsto \log \left(x + \left(\left(\left(x - \frac{0.125}{{x}^{3}}\right) + \frac{\color{blue}{-0.5}}{x}\right) - 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   11. associate-*r/ 98.8%

       \[\leadsto \log \left(x + \left(\left(\left(x - \frac{0.125}{{x}^{3}}\right) + \frac{-0.5}{x}\right) - \color{blue}{\frac{0.0625 \cdot 1}{{x}^{5}}}\right)\right) \]

   12. metadata-eval 98.8%

       \[\leadsto \log \left(x + \left(\left(\left(x - \frac{0.125}{{x}^{3}}\right) + \frac{-0.5}{x}\right) - \frac{\color{blue}{0.0625}}{{x}^{5}}\right)\right) \]

5. Simplified 98.8%

   \[\leadsto \log \left(x + \color{blue}{\left(\left(\left(x - \frac{0.125}{{x}^{3}}\right) + \frac{-0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)}\right) \]

6. Final simplification 98.8%

   \[\leadsto \log \left(x + \left(\left(\left(x - \frac{0.125}{{x}^{3}}\right) + \frac{-0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\right) \]
7. Add Preprocessing

Alternative 2: 98.4% 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 56.9%

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

3. Taylor expanded in x around inf 98.1%

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

   1. associate-*r/ 98.1%

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

   2. metadata-eval 98.1%

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

5. Simplified 98.1%

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

6. Final simplification 98.1%

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

Alternative 3: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot 2 + -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log1p (+ (* x 2.0) -1.0)))

C

float code(float x) {
	return log1pf(((x * 2.0f) + -1.0f));
}

Julia

function code(x)
	return log1p(Float32(Float32(x * Float32(2.0)) + Float32(-1.0)))
end

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in x around inf 96.4%

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

   1. log1p-expm1-u 96.4%

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

   2. expm1-udef 96.4%

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

   3. count-2 96.4%

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

   4. sum-log 95.5%

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

   5. +-commutative 95.5%

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

   6. exp-sum 95.5%

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

   7. add-exp-log 96.4%

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

   8. rem-exp-log 96.4%

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

5. Applied egg-rr 96.4%

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

6. Final simplification 96.4%

   \[\leadsto \mathsf{log1p}\left(x \cdot 2 + -1\right) \]
7. Add Preprocessing

Alternative 4: 97.2% 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. Add Preprocessing

3. Taylor expanded in x around inf 96.4%

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

4. Final simplification 96.4%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Developer target: 99.4% 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 2024040 
(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)))))