Rust f32::acosh

Percentage Accurate: 53.2% → 98.6%

Time: 5.1s

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: 53.2% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot \left(-x\right)\right)}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (-
   (* x 2.0)
   (+ (+ (/ 0.5 x) (/ -0.125 (* x (* x (- x))))) (/ 0.0625 (pow x 5.0))))))

C

float code(float x) {
	return logf(((x * 2.0f) - (((0.5f / x) + (-0.125f / (x * (x * -x)))) + (0.0625f / powf(x, 5.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(((x * 2.0e0) - (((0.5e0 / x) + ((-0.125e0) / (x * (x * -x)))) + (0.0625e0 / (x ** 5.0e0)))))
end function

Julia

function code(x)
	return log(Float32(Float32(x * Float32(2.0)) - Float32(Float32(Float32(Float32(0.5) / x) + Float32(Float32(-0.125) / Float32(x * Float32(x * Float32(-x))))) + Float32(Float32(0.0625) / (x ^ Float32(5.0))))))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * single(2.0)) - (((single(0.5) / x) + (single(-0.125) / (x * (x * -x)))) + (single(0.0625) / (x ^ single(5.0))))));
end

Derivation

1. Initial program 51.4%

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

2. Taylor expanded in x around inf 99.3%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.0625 \cdot \frac{1}{{x}^{5}} + \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \log \left(\color{blue}{x \cdot 2} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)\right) \]

   2. +-commutative 99.3%

      \[\leadsto \log \left(x \cdot 2 - \color{blue}{\left(\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right) \]

   3. +-commutative 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   4. associate-*r/ 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   5. metadata-eval 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{\color{blue}{0.5}}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   6. associate-*r/ 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   7. metadata-eval 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{\color{blue}{0.125}}{{x}^{3}}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   8. associate-*r/ 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right) + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{5}}}\right)\right) \]

   9. metadata-eval 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right) + \frac{\color{blue}{0.0625}}{{x}^{5}}\right)\right) \]

4. Simplified 99.3%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right) + \frac{0.0625}{{x}^{5}}\right)\right)} \]
5. Step-by-step derivation

   1. metadata-eval 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{\color{blue}{{0.5}^{3}}}{{x}^{3}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

   2. cube-div 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \color{blue}{{\left(\frac{0.5}{x}\right)}^{3}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

   3. cube-mult 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.5}{x} \cdot \left(\frac{0.5}{x} \cdot \frac{0.5}{x}\right)}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

   4. frac-times 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.5}{x} \cdot \color{blue}{\frac{0.5 \cdot 0.5}{x \cdot x}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

   5. metadata-eval 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.5}{x} \cdot \frac{\color{blue}{0.25}}{x \cdot x}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

6. Applied egg-rr 99.3%

   \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.5}{x} \cdot \frac{0.25}{x \cdot x}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.25}{x \cdot x} \cdot \frac{0.5}{x}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

   2. frac-2neg 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.25}{x \cdot x} \cdot \color{blue}{\frac{-0.5}{-x}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

   3. frac-times 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.25 \cdot \left(-0.5\right)}{\left(x \cdot x\right) \cdot \left(-x\right)}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

   4. metadata-eval 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.25 \cdot \color{blue}{-0.5}}{\left(x \cdot x\right) \cdot \left(-x\right)}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

   5. metadata-eval 99.3%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{\color{blue}{-0.125}}{\left(x \cdot x\right) \cdot \left(-x\right)}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

8. Applied egg-rr 99.3%

   \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \color{blue}{\frac{-0.125}{\left(x \cdot x\right) \cdot \left(-x\right)}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

9. Final simplification 99.3%

   \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot \left(-x\right)\right)}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

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 98.7%

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

   1. associate-*r/ 98.7%

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

   2. metadata-eval 98.7%

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

4. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 3: 97.7% 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-+ 7.7%

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

   2. div-inv 7.7%

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

   3. log-prod 7.7%

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

   4. add-sqr-sqrt 7.4%

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

   5. fma-neg 7.4%

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

   6. metadata-eval 7.4%

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

   7. fma-neg 7.4%

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

   8. metadata-eval 7.4%

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

3. Applied egg-rr 7.4%

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

   1. fma-def 7.4%

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

   2. associate--r+ 9.8%

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

   3. +-inverses 10.5%

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

   4. metadata-eval 10.5%

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

   5. metadata-eval 10.5%

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

   6. +-lft-identity 10.5%

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

   7. log-rec 9.8%

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

5. Simplified 9.8%

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

6. Taylor expanded in x around inf 97.6%

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

7. Final simplification 97.6%

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

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 51.4%

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

2. Taylor expanded in x around inf 97.2%

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

3. Final simplification 97.2%

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

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 97.2%

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

   1. count-2 97.2%

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

   2. sum-log 96.8%

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

   3. log1p-expm1-u 96.4%

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

   4. expm1-udef 96.4%

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

   5. exp-sum 96.4%

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

   6. add-exp-log 96.4%

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

   7. add-exp-log 97.2%

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

   8. count-2 97.2%

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

4. Applied egg-rr 97.2%

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

   1. count-2 97.2%

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

   2. *-commutative 97.2%

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

   3. add-exp-log 97.2%

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

   4. expm1-def 97.2%

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

   5. log1p-expm1-u 97.2%

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

   6. *-commutative 97.2%

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

   7. count-2 97.2%

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

   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. +-inverses -0.0%

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

   2. +-inverses -0.0%

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

   3. +-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 2023271 
(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)))))