Rust f32::acosh

Percentage Accurate: 53.0% → 97.6%

Time: 5.8s

Alternatives: 4

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: 53.0% 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: 97.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (* 2.0 (log (* (sqrt x) (sqrt 2.0)))))

C

float code(float x) {
	return 2.0f * logf((sqrtf(x) * sqrtf(2.0f)));
}

Fortran

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

Julia

function code(x)
	return Float32(Float32(2.0) * log(Float32(sqrt(x) * sqrt(Float32(2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = single(2.0) * log((sqrt(x) * sqrt(single(2.0))));
end

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around inf 97.5%

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

   1. count-2 97.5%

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

   2. sum-log 97.1%

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

   3. +-commutative 97.1%

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

   4. add-sqr-sqrt 96.2%

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

   5. fma-define 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)} \]

5. Applied egg-rr 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)} \]
6. Step-by-step derivation

   1. add-log-exp 95.9%

      \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}\right)} \]

   2. add-sqr-sqrt 95.9%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right)} \]

   3. log-prod 95.9%

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right)} \]

   4. fma-undefine 95.8%

      \[\leadsto \log \left(\sqrt{e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x} + \log 2}}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right) \]

   5. add-sqr-sqrt 96.4%

      \[\leadsto \log \left(\sqrt{e^{\color{blue}{\log x} + \log 2}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right) \]

   6. sum-log 96.7%

      \[\leadsto \log \left(\sqrt{e^{\color{blue}{\log \left(x \cdot 2\right)}}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right) \]

   7. add-exp-log 96.7%

      \[\leadsto \log \left(\sqrt{\color{blue}{x \cdot 2}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right) \]

   8. fma-undefine 96.6%

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

   9. add-sqr-sqrt 97.1%

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

7. Applied egg-rr 97.4%

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

   1. count-2 97.4%

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

   2. *-commutative 97.4%

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

9. Simplified 97.4%

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

10. Taylor expanded in x around 0 97.8%

    \[\leadsto 2 \cdot \log \color{blue}{\left(\sqrt{x} \cdot \sqrt{2}\right)} \]
11. Add Preprocessing

Alternative 2: 97.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log1p (fma x 2.0 -1.0)))

C

float code(float x) {
	return log1pf(fmaf(x, 2.0f, -1.0f));
}

Julia

function code(x)
	return log1p(fma(x, Float32(2.0), Float32(-1.0)))
end

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around inf 97.5%

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

   1. count-2 97.5%

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

   2. sum-log 97.1%

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

   3. +-commutative 97.1%

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

   4. add-sqr-sqrt 96.2%

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

   5. fma-define 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)} \]

5. Applied egg-rr 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)} \]
6. Step-by-step derivation

   1. log1p-expm1-u 95.9%

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

   2. expm1-undefine 95.9%

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

   3. fma-undefine 95.8%

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

   4. add-sqr-sqrt 96.8%

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

   5. sum-log 97.5%

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

   6. add-exp-log 97.5%

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

   7. fma-neg 97.5%

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

   8. metadata-eval 97.5%

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

7. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, 2, -1\right)\right)} \]
8. Add Preprocessing

Alternative 3: 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 56.5%

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

3. Taylor expanded in x around inf 97.5%

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

Alternative 4: 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 56.5%

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

3. Taylor expanded in x around inf 97.5%

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

   1. add-sqr-sqrt 97.4%

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

   2. log-prod 97.4%

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

   3. flip-+ -0.0%

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

   4. difference-of-squares -0.0%

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

   5. associate-*r/ -0.0%

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

   6. +-inverses -0.0%

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

   7. +-inverses -0.0%

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

   8. flip-+ 19.3%

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

   9. sqrt-unprod 30.4%

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

   10. add-sqr-sqrt 30.4%

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

   11. flip-+ -0.0%

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

   12. +-inverses -0.0%

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

   13. +-inverses -0.0%

       \[\leadsto \log \left(\frac{0}{\color{blue}{0}}\right) + \log \left(\sqrt{x + x}\right) \]

   14. flip-+ -0.0%

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

   15. difference-of-squares -0.0%

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

   16. associate-*r/ -0.0%

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

   17. +-inverses -0.0%

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

   18. +-inverses -0.0%

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

   19. flip-+ -0.0%

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

5. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\log \left(\frac{0}{0}\right) + \log \left(\frac{0}{0}\right)} \]

7. Simplified 6.1%

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

Developer target: 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 2024083 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

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

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