Rust f32::acosh

Percentage Accurate: 52.9% → 98.4%

Time: 4.8s

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: 52.9% 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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.4%

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

2. Taylor expanded in x around inf 98.2%

   \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - \left(0.09375 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]
3. Step-by-step derivation

   1. associate--r+ 98.2%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - 0.09375 \cdot \frac{1}{{x}^{4}}\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]

   2. sub-neg 98.2%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - 0.09375 \cdot \frac{1}{{x}^{4}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   3. mul-1-neg 98.2%

      \[\leadsto \left(\left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \log 2\right) - 0.09375 \cdot \frac{1}{{x}^{4}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   4. log-rec 98.2%

      \[\leadsto \left(\left(\left(-\color{blue}{\left(-\log x\right)}\right) + \log 2\right) - 0.09375 \cdot \frac{1}{{x}^{4}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   5. remove-double-neg 98.2%

      \[\leadsto \left(\left(\color{blue}{\log x} + \log 2\right) - 0.09375 \cdot \frac{1}{{x}^{4}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   6. +-commutative 98.2%

      \[\leadsto \left(\color{blue}{\left(\log 2 + \log x\right)} - 0.09375 \cdot \frac{1}{{x}^{4}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   7. associate-*r/ 98.2%

      \[\leadsto \left(\left(\log 2 + \log x\right) - \color{blue}{\frac{0.09375 \cdot 1}{{x}^{4}}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   8. metadata-eval 98.2%

      \[\leadsto \left(\left(\log 2 + \log x\right) - \frac{\color{blue}{0.09375}}{{x}^{4}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   9. unpow2 98.2%

      \[\leadsto \left(\left(\log 2 + \log x\right) - \frac{0.09375}{{x}^{4}}\right) + \left(-0.25 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) \]

   10. associate-*r/ 98.2%

       \[\leadsto \left(\left(\log 2 + \log x\right) - \frac{0.09375}{{x}^{4}}\right) + \left(-\color{blue}{\frac{0.25 \cdot 1}{x \cdot x}}\right) \]

   11. metadata-eval 98.2%

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

   12. distribute-neg-frac 98.2%

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

   13. metadata-eval 98.2%

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

4. Simplified 98.2%

   \[\leadsto \color{blue}{\left(\left(\log 2 + \log x\right) - \frac{0.09375}{{x}^{4}}\right) + \frac{-0.25}{x \cdot x}} \]
5. Step-by-step derivation

   1. sum-log 98.7%

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

   2. count-2 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 2: 98.1% 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 51.4%

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

2. Taylor expanded in x around inf 98.2%

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

   1. associate-*r/ 98.2%

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

   2. metadata-eval 98.2%

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

4. Simplified 98.2%

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

5. Final simplification 98.2%

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

Alternative 3: 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 51.4%

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

   1. flip-+ 9.6%

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

   2. div-inv 9.6%

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

   3. log-prod 9.6%

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

   4. add-sqr-sqrt 9.3%

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

   5. fma-neg 9.3%

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

   6. metadata-eval 9.3%

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

   7. fma-neg 9.3%

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

   8. metadata-eval 9.3%

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

3. Applied egg-rr 9.3%

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

   1. fma-def 9.3%

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

   2. associate--r+ 11.6%

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

   3. +-inverses 12.2%

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

   4. metadata-eval 12.2%

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

   5. metadata-eval 12.2%

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

   6. +-lft-identity 12.2%

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

   7. log-rec 11.6%

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

5. Simplified 11.6%

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

6. Taylor expanded in x around inf 96.5%

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

7. Final simplification 96.5%

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

Alternative 4: 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 51.4%

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

2. Taylor expanded in x around inf 96.1%

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

3. Final simplification 96.1%

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

Alternative 5: 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 51.4%

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

2. Taylor expanded in x around inf 95.7%

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

   1. mul-1-neg 95.7%

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

   2. log-rec 95.7%

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

   3. remove-double-neg 95.7%

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

4. Simplified 95.7%

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

   1. log1p-expm1-u 95.3%

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

   2. expm1-udef 95.3%

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

   3. sum-log 96.1%

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

   4. count-2 96.1%

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

   5. add-exp-log 96.1%

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

6. Applied egg-rr 96.1%

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

   1. add-exp-log 96.1%

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

   2. expm1-udef 96.1%

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

   3. log1p-expm1-u 96.1%

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

   4. flip-+ -0.0%

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

   5. log-div -0.0%

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

   6. +-inverses -0.0%

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

   7. metadata-eval -0.0%

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

   8. metadata-eval -0.0%

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

   9. log1p-udef -0.0%

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

   10. metadata-eval -0.0%

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

   11. +-inverses -0.0%

       \[\leadsto \mathsf{log1p}\left(-1\right) - \log \color{blue}{0} \]

   12. metadata-eval -0.0%

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

   13. metadata-eval -0.0%

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

   14. log1p-udef -0.0%

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

   15. metadata-eval -0.0%

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

8. Applied egg-rr -0.0%

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

   1. +-inverses 6.1%

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

10. Simplified 6.1%

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

11. Final simplification 6.1%

    \[\leadsto 0 \]

Developer target: 99.0% 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 2023257 
(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)))))