Rust f32::acosh

Percentage Accurate: 52.7% → 99.2%

Time: 4.7s

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: 52.7% 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: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.3%

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

   1. +-commutative 54.3%

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

   2. pow1/2 54.3%

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

   3. difference-of-sqr-1 54.3%

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

   4. unpow-prod-down 98.8%

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

   5. fma-def 98.9%

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

   6. sub-neg 98.9%

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

   7. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

   1. unpow1/2 98.9%

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

   2. unpow1/2 98.9%

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

   3. +-commutative 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

   \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x + -1}, x\right)\right) \]
8. Add Preprocessing

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 54.3%

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

3. Taylor expanded in x around inf 97.6%

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

   1. associate-*r/ 97.6%

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

   2. metadata-eval 97.6%

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

5. Simplified 97.6%

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

6. Final simplification 97.6%

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

Alternative 3: 96.8% 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 54.3%

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

3. Taylor expanded in x around inf 96.3%

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

4. Final simplification 96.3%

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

Alternative 4: 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 54.3%

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

   1. +-commutative 54.3%

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

   2. pow1/2 54.3%

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

   3. difference-of-sqr-1 54.3%

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

   4. unpow-prod-down 98.8%

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

   5. fma-def 98.9%

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

   6. sub-neg 98.9%

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

   7. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

   1. unpow1/2 98.9%

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

   2. unpow1/2 98.9%

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

   3. +-commutative 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in x around inf 44.0%

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

8. Final simplification 44.0%

   \[\leadsto \log x \]
9. Add Preprocessing

Alternative 5: 3.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -0.03125 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -0.03125)

C

float code(float x) {
	return -0.03125f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -0.03125e0
end function

Julia

function code(x)
	return Float32(-0.03125)
end

MATLAB

function tmp = code(x)
	tmp = single(-0.03125);
end

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around inf 97.6%

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

   1. associate-*r/ 97.6%

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

   2. metadata-eval 97.6%

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

5. Simplified 97.6%

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

6. Taylor expanded in x around inf 97.8%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - \left(0.03125 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

8. Simplified 3.2%

   \[\leadsto \color{blue}{-0.03125} \]

9. Final simplification 3.2%

   \[\leadsto -0.03125 \]
10. Add Preprocessing

Developer target: 99.2% 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 2024030 
(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)))))