Rust f32::acosh

Percentage Accurate: 52.8% → 99.5%

Time: 7.5s

Alternatives: 6

Speedup: 1.2×

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: 52.8% 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: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\sqrt{\frac{1}{1 + x}}} \cdot \sqrt{x - 1} + x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log (+ (* (/ 1.0 (sqrt (/ 1.0 (+ 1.0 x)))) (sqrt (- x 1.0))) x)))

C

float code(float x) {
	return logf((((1.0f / sqrtf((1.0f / (1.0f + x)))) * sqrtf((x - 1.0f))) + x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((((1.0e0 / sqrt((1.0e0 / (1.0e0 + x)))) * sqrt((x - 1.0e0))) + x))
end function

Julia

function code(x)
	return log(Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) + x)))) * sqrt(Float32(x - Float32(1.0)))) + x))
end

MATLAB

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

Derivation

1. Initial program 54.2%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. lower-*.f32 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lower--.f32 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower--.f32 N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f32 N/A

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

   13. lower--.f32 54.3

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

6. Applied rewrites 54.3%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lift--.f32 N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. flip-+ N/A

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

   8. lift-+.f32 N/A

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

   9. lower-/.f32 99.2

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

8. Applied rewrites 99.2%

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

9. Final simplification 99.2%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float x) {
	return logf(((sqrtf((1.0f + x)) * sqrtf((x - 1.0f))) + x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(((sqrt((1.0e0 + x)) * sqrt((x - 1.0e0))) + x))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 54.2%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. lower-*.f32 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lower--.f32 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log (* (- 2.0 (/ 0.5 (* x x))) x)))

C

float code(float x) {
	return logf(((2.0f - (0.5f / (x * x))) * x));
}

Fortran

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

Julia

function code(x)
	return log(Float32(Float32(Float32(2.0) - Float32(Float32(0.5) / Float32(x * x))) * x))
end

MATLAB

function tmp = code(x)
	tmp = log(((single(2.0) - (single(0.5) / (x * x))) * x));
end

Derivation

1. Initial program 54.2%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. lower-*.f32 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lower--.f32 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. rem-square-sqrt N/A

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

   7. unpow2 N/A

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

   8. distribute-neg-in N/A

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

   9. associate--l+ N/A

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

   10. lower-*.f32 N/A

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

7. Applied rewrites 98.0%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. unsub-neg N/A

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

   6. remove-double-neg N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. distribute-lft-neg-out N/A

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. lower--.f32 N/A

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

   12. *-commutative N/A

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

   13. mul-1-neg N/A

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

   14. distribute-lft-neg-out N/A

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

   15. *-commutative N/A

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

   16. distribute-rgt-neg-in N/A

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

   17. metadata-eval N/A

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

   18. associate-*r* N/A

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

5. Applied rewrites 98.0%

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

6. Final simplification 98.0%

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

Alternative 5: 97.1% accurate, 1.2× 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 54.2%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. lower-*.f32 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lower--.f32 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. neg-mul-1 N/A

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

   5. mul-1-neg N/A

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

   6. remove-double-neg 96.5

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

7. Applied rewrites 96.5%

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

8. Final simplification 96.5%

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

Alternative 6: 1.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log 0 \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log 0.0))

C

float code(float x) {
	return logf(0.0f);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(0.0e0)
end function

Julia

function code(x)
	return log(Float32(0.0))
end

MATLAB

function tmp = code(x)
	tmp = log(single(0.0));
end

Derivation

1. Initial program 54.2%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower--.f32 N/A

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

   12. pow2 N/A

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

   13. lift-*.f32 N/A

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

   14. pow-prod-down N/A

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

   15. pow-prod-up N/A

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

   16. lower-pow.f32 N/A

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

   17. metadata-eval N/A

       \[\leadsto \log \left(x + \sqrt{\frac{1 - {x}^{\color{blue}{4}}}{\left(\mathsf{neg}\left(1\right)\right) - x \cdot x}}\right) \]

   18. lower--.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\frac{1 - {x}^{4}}{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) - x \cdot x}}}\right) \]

   19. metadata-eval 30.3

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

4. Applied rewrites 30.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. metadata-eval N/A

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

   5. mul0-rgt 1.3

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

7. Applied rewrites 1.3%

   \[\leadsto \log \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.9× 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 2024277 
(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)))))