Rust f32::acosh

Percentage Accurate: 54.8% → 100.0%

Time: 9.4s

Alternatives: 13

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: 54.8% 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: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( \log \left(x + \sqrt{x \cdot x + -1}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (cast (! :precision binary64 (log (+ x (sqrt (+ (* x x) -1.0)))))))

C

float code(float x) {
	double tmp = log((((double) x) + sqrt(((((double) x) * ((double) x)) + -1.0))));
	return (float) tmp;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    real(8) :: tmp
    tmp = log((real(x, 8) + sqrt(((real(x, 8) * real(x, 8)) + (-1.0d0)))))
    code = real(tmp, 4)
end function

Julia

function code(x)
	tmp = log(Float64(Float64(x) + sqrt(Float64(Float64(Float64(x) * Float64(x)) + -1.0))))
	return Float32(tmp)
end

MATLAB

function tmp_2 = code(x)
	tmp = log((double(x) + sqrt(((double(x) * double(x)) + -1.0))));
	tmp_2 = single(tmp);
end

Derivation

1. Initial program 56.8%

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

   1. rewrite-binary32/binary64 100.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\log \left(x + \sqrt{x \cdot x - 1}\right)} \rangle_{\text{binary64}}} \]

3. Applied rewrite-once 100.0%

   \[\leadsto \color{blue}{\langle \color{blue}{\log \left(x + \sqrt{x \cdot x - 1}\right)} \rangle_{\text{binary64}}} \]

4. Final simplification 100.0%

   \[\leadsto \langle \log \left(x + \sqrt{x \cdot x + -1}\right) \rangle_{\text{binary64}} \]

Alternative 2: 98.8% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (- (log (+ (/ 0.5 x) (/ (/ (/ 0.125 x) x) x)))))

C

float code(float x) {
	return -logf(((0.5f / x) + (((0.125f / x) / x) / x)));
}

Fortran

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

Julia

function code(x)
	return Float32(-log(Float32(Float32(Float32(0.5) / x) + Float32(Float32(Float32(Float32(0.125) / x) / x) / x))))
end

MATLAB

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

Derivation

1. Initial program 56.8%

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

   1. flip-+ 7.8%

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

   2. clear-num 7.8%

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

   3. log-rec 7.8%

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

   4. clear-num 7.8%

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

   5. flip-+ 56.8%

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

   6. fma-neg 56.8%

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

   7. metadata-eval 56.8%

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

3. Applied egg-rr 56.8%

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

4. Taylor expanded in x around inf 99.1%

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

   1. associate-*r/ 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

6. Simplified 99.1%

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

   1. metadata-eval 99.1%

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

   2. cube-div 99.1%

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

   3. unpow3 99.1%

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

   4. associate-*l/ 99.1%

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

   5. associate-*r/ 99.1%

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

   6. metadata-eval 99.1%

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

8. Applied egg-rr 99.1%

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

   1. associate-*l/ 99.1%

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

   2. associate-*r/ 99.1%

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

   3. associate-*l/ 99.1%

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

   4. metadata-eval 99.1%

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

10. Applied egg-rr 99.1%

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

11. Final simplification 99.1%

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

Alternative 3: 98.3% 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 56.8%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 97.5% 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 56.8%

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

   1. flip-+ 7.8%

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

   2. clear-num 7.8%

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

   3. log-rec 7.8%

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

   4. clear-num 7.8%

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

   5. flip-+ 56.8%

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

   6. fma-neg 56.8%

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

   7. metadata-eval 56.8%

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

3. Applied egg-rr 56.8%

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

4. Taylor expanded in x around inf 97.4%

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

5. Final simplification 97.4%

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

Alternative 5: 97.0% 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 56.8%

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

2. Taylor expanded in x around inf 97.0%

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

3. Final simplification 97.0%

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

Alternative 6: 20.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (x) :precision binary32 0.75)

C

float code(float x) {
	return 0.75f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.75e0
end function

Julia

function code(x)
	return Float32(0.75)
end

MATLAB

function tmp = code(x)
	tmp = single(0.75);
end

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in x around inf 97.0%

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

4. Applied egg-rr 20.7%

   \[\leadsto \color{blue}{0.75} \]

5. Final simplification 20.7%

   \[\leadsto 0.75 \]

Alternative 7: 21.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.5 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.5)

C

float code(float x) {
	return 1.5f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.5e0
end function

Julia

function code(x)
	return Float32(1.5)
end

MATLAB

function tmp = code(x)
	tmp = single(1.5);
end

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in x around inf 97.0%

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

4. Applied egg-rr 21.7%

   \[\leadsto \color{blue}{1.5} \]

5. Final simplification 21.7%

   \[\leadsto 1.5 \]

Alternative 8: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.0)

C

float code(float x) {
	return 2.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.0e0
end function

Julia

function code(x)
	return Float32(2.0)
end

MATLAB

function tmp = code(x)
	tmp = single(2.0);
end

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in x around inf 97.0%

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

4. Applied egg-rr 22.3%

   \[\leadsto \color{blue}{2} \]

5. Final simplification 22.3%

   \[\leadsto 2 \]

Alternative 9: 22.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 3 \end{array} \]

FPCore

(FPCore (x) :precision binary32 3.0)

C

float code(float x) {
	return 3.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 3.0e0
end function

Julia

function code(x)
	return Float32(3.0)
end

MATLAB

function tmp = code(x)
	tmp = single(3.0);
end

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in x around inf 97.0%

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

4. Applied egg-rr 23.1%

   \[\leadsto \color{blue}{3} \]

5. Final simplification 23.1%

   \[\leadsto 3 \]

Alternative 10: 23.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x) :precision binary32 4.0)

C

float code(float x) {
	return 4.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 4.0e0
end function

Julia

function code(x)
	return Float32(4.0)
end

MATLAB

function tmp = code(x)
	tmp = single(4.0);
end

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in x around inf 97.0%

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

4. Applied egg-rr 24.0%

   \[\leadsto \color{blue}{4} \]

5. Final simplification 24.0%

   \[\leadsto 4 \]

Alternative 11: 24.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 6 \end{array} \]

FPCore

(FPCore (x) :precision binary32 6.0)

C

float code(float x) {
	return 6.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 6.0e0
end function

Julia

function code(x)
	return Float32(6.0)
end

MATLAB

function tmp = code(x)
	tmp = single(6.0);
end

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in x around inf 97.0%

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

4. Applied egg-rr 24.8%

   \[\leadsto \color{blue}{6} \]

5. Final simplification 24.8%

   \[\leadsto 6 \]

Alternative 12: 25.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 8 \end{array} \]

FPCore

(FPCore (x) :precision binary32 8.0)

C

float code(float x) {
	return 8.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 8.0e0
end function

Julia

function code(x)
	return Float32(8.0)
end

MATLAB

function tmp = code(x)
	tmp = single(8.0);
end

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in x around inf 97.0%

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

4. Applied egg-rr 25.6%

   \[\leadsto \color{blue}{8} \]

5. Final simplification 25.6%

   \[\leadsto 8 \]

Alternative 13: 27.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 16 \end{array} \]

FPCore

(FPCore (x) :precision binary32 16.0)

C

float code(float x) {
	return 16.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 16.0e0
end function

Julia

function code(x)
	return Float32(16.0)
end

MATLAB

function tmp = code(x)
	tmp = single(16.0);
end

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in x around inf 97.0%

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

4. Applied egg-rr 27.6%

   \[\leadsto \color{blue}{16} \]

5. Final simplification 27.6%

   \[\leadsto 16 \]

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 2023297 
(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)))))