Rust f32::acosh

Percentage Accurate: 53.4% → 99.3%

Time: 7.8s

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: 53.4% 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: 99.3% 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

Derivation

1. Initial program 55.0%

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

   1. pow1/2 55.0%

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

   2. difference-of-sqr-1 55.0%

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

   3. unpow-prod-down 99.2%

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

   4. sub-neg 99.2%

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

   5. metadata-eval 99.2%

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

4. Applied egg-rr 99.2%

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

   1. unpow1/2 99.2%

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

   2. unpow1/2 99.2%

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

6. Simplified 99.2%

   \[\leadsto \log \left(x + \color{blue}{\sqrt{x + 1} \cdot \sqrt{x + -1}}\right) \]
7. Add Preprocessing

Alternative 2: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot \left(2 + \frac{-1 - \frac{0.5}{x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log1p (* x (+ 2.0 (/ (- -1.0 (/ 0.5 x)) x)))))

C

float code(float x) {
	return log1pf((x * (2.0f + ((-1.0f - (0.5f / x)) / x))));
}

Julia

function code(x)
	return log1p(Float32(x * Float32(Float32(2.0) + Float32(Float32(Float32(-1.0) - Float32(Float32(0.5) / x)) / x))))
end

Derivation

1. Initial program 55.0%

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

   1. log1p-expm1-u 55.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

   2. expm1-undefine 55.0%

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

   3. add-exp-log 55.0%

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

   4. fma-neg 55.0%

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

   5. metadata-eval 55.0%

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

4. Applied egg-rr 55.0%

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

5. Taylor expanded in x around inf 98.4%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + -1 \cdot \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 98.4%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + \color{blue}{\left(-\frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right)\right) \]

   2. unsub-neg 98.4%

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]

   3. un-div-inv 98.4%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \color{blue}{\frac{0.5}{x}}}{x}\right)\right) \]

7. Applied egg-rr 98.4%

   \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]

8. Final simplification 98.4%

   \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + \frac{-1 - \frac{0.5}{x}}{x}\right)\right) \]
9. Add Preprocessing

Alternative 3: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot \left(-1 + \left(3 + \frac{-1}{x}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log1p (* x (+ -1.0 (+ 3.0 (/ -1.0 x))))))

C

float code(float x) {
	return log1pf((x * (-1.0f + (3.0f + (-1.0f / x)))));
}

Julia

function code(x)
	return log1p(Float32(x * Float32(Float32(-1.0) + Float32(Float32(3.0) + Float32(Float32(-1.0) / x)))))
end

Derivation

1. Initial program 55.0%

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

   1. log1p-expm1-u 55.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

   2. expm1-undefine 55.0%

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

   3. add-exp-log 55.0%

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

   4. fma-neg 55.0%

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

   5. metadata-eval 55.0%

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

4. Applied egg-rr 55.0%

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

5. Taylor expanded in x around inf 97.0%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 - \frac{1}{x}\right)}\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 96.9%

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{1}{x}\right)\right)}\right) \]

   2. sub-neg 96.9%

      \[\leadsto \mathsf{log1p}\left(x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{2 + \left(-\frac{1}{x}\right)}\right)\right)\right) \]

   3. distribute-neg-frac 96.9%

      \[\leadsto \mathsf{log1p}\left(x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(2 + \color{blue}{\frac{-1}{x}}\right)\right)\right) \]

   4. metadata-eval 96.9%

      \[\leadsto \mathsf{log1p}\left(x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(2 + \frac{\color{blue}{-1}}{x}\right)\right)\right) \]

7. Applied egg-rr 96.9%

   \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 + \frac{-1}{x}\right)\right)}\right) \]
8. Step-by-step derivation

   1. expm1-undefine 96.9%

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

   2. sub-neg 96.9%

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

   3. log1p-undefine 96.9%

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

   4. rem-exp-log 96.9%

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

   5. associate-+r+ 97.0%

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

   6. metadata-eval 97.0%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(\left(\color{blue}{3} + \frac{-1}{x}\right) + \left(-1\right)\right)\right) \]

   7. metadata-eval 97.0%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(\left(3 + \frac{-1}{x}\right) + \color{blue}{-1}\right)\right) \]

9. Simplified 97.0%

   \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(\left(3 + \frac{-1}{x}\right) + -1\right)}\right) \]

10. Final simplification 97.0%

    \[\leadsto \mathsf{log1p}\left(x \cdot \left(-1 + \left(3 + \frac{-1}{x}\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 97.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.0%

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

   1. log1p-expm1-u 55.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

   2. expm1-undefine 55.0%

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

   3. add-exp-log 55.0%

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

   4. fma-neg 55.0%

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

   5. metadata-eval 55.0%

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

4. Applied egg-rr 55.0%

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

5. Taylor expanded in x around inf 97.0%

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

6. Taylor expanded in x around 0 97.0%

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

7. Final simplification 97.0%

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

Alternative 5: 97.1% 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 55.0%

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

3. Taylor expanded in x around inf 97.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
4. Add Preprocessing

Alternative 6: 44.2% 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 55.0%

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

3. Taylor expanded in x around inf 97.0%

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

4. Taylor expanded in x around 0 96.8%

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

6. Simplified 44.0%

   \[\leadsto \color{blue}{\log x} \]
7. Add Preprocessing

Alternative 7: 3.1% 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 55.0%

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

   1. log1p-expm1-u 55.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

   2. expm1-undefine 55.0%

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

   3. add-exp-log 55.0%

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

   4. fma-neg 55.0%

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

   5. metadata-eval 55.0%

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

4. Applied egg-rr 55.0%

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

5. Taylor expanded in x around 0 -0.0%

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

7. Simplified 2.1%

   \[\leadsto \color{blue}{-2 + \frac{x}{-2}} \]

8. Taylor expanded in x around 0 3.1%

   \[\leadsto \color{blue}{-2} \]
9. Add Preprocessing

Developer Target 1: 99.3% 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 2024132 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :alt
  (! :herbie-platform default (log (+ x (* (sqrt (- x 1)) (sqrt (+ x 1))))))

  (log (+ x (sqrt (- (* x x) 1.0)))))