Rust f32::acosh

Percentage Accurate: 52.7% → 99.1%

Time: 9.8s

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: 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: 99.1% 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 51.7%

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

   1. +-commutative 51.7%

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

   2. difference-of-sqr-1 51.7%

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

   3. sqrt-prod 99.5%

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

   4. fma-define 99.6%

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

   5. sub-neg 99.6%

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

   6. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{0.5 + \frac{0.125}{{x}^{2}}}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (- (log (/ (+ 0.5 (/ 0.125 (pow x 2.0))) x))))

C

float code(float x) {
	return -logf(((0.5f + (0.125f / powf(x, 2.0f))) / x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -log(((0.5e0 + (0.125e0 / (x ** 2.0e0))) / x))
end function

Julia

function code(x)
	return Float32(-log(Float32(Float32(Float32(0.5) + Float32(Float32(0.125) / (x ^ Float32(2.0)))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = -log(((single(0.5) + (single(0.125) / (x ^ single(2.0)))) / x));
end

Derivation

1. Initial program 51.7%

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

   1. fmm-def 51.7%

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

   2. metadata-eval 51.7%

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

   3. flip-+ 6.1%

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

   4. metadata-eval 6.1%

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

   5. fmm-def 6.1%

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

   6. div-inv 6.0%

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

   7. log-prod 6.1%

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

   8. pow2 6.1%

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

   9. add-sqr-sqrt 6.0%

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

   10. fmm-def 6.0%

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

   11. metadata-eval 6.0%

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

4. Applied egg-rr 6.0%

   \[\leadsto \color{blue}{\log \left({x}^{2} - \mathsf{fma}\left(x, x, -1\right)\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right)} \]
5. Step-by-step derivation

   1. log-rec 6.0%

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

   2. sub-neg 6.0%

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

   3. fma-undefine 6.0%

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

   4. unpow2 6.0%

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

   5. associate--r+ 8.6%

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

   6. +-inverses 8.6%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   7. metadata-eval 8.6%

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

   8. metadata-eval 8.6%

      \[\leadsto \color{blue}{0} - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   9. neg-sub0 8.6%

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

6. Simplified 8.6%

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

7. Taylor expanded in x around inf 98.9%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 98.9%

      \[\leadsto -\log \left(\frac{0.5 + \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}}{x}\right) \]

   2. metadata-eval 98.9%

      \[\leadsto -\log \left(\frac{0.5 + \frac{\color{blue}{0.125}}{{x}^{2}}}{x}\right) \]

9. Simplified 98.9%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5 + \frac{0.125}{{x}^{2}}}{x}\right)} \]

10. Final simplification 98.9%

    \[\leadsto -\log \left(\frac{0.5 + \frac{0.125}{{x}^{2}}}{x}\right) \]
11. Add Preprocessing

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 51.7%

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

   1. log1p-expm1-u 51.7%

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

   2. expm1-undefine 51.7%

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

   3. add-exp-log 51.7%

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

   4. fmm-def 51.7%

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

   5. metadata-eval 51.7%

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

4. Applied egg-rr 51.7%

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

5. Taylor expanded in x around inf 98.6%

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

6. Taylor expanded in x around 0 49.6%

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

   1. div-sub 49.6%

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

   2. sub-neg 49.6%

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

   3. *-commutative 49.6%

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

   4. associate-/l* 98.6%

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

   5. *-commutative 98.6%

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

   6. fmm-def 98.6%

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

   7. metadata-eval 98.6%

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

8. Applied egg-rr 98.6%

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

   1. *-inverses 98.6%

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

   2. *-rgt-identity 98.6%

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

   3. distribute-neg-frac 98.6%

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

   4. metadata-eval 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 4: 98.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 51.7%

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

   1. log1p-expm1-u 51.7%

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

   2. expm1-undefine 51.7%

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

   3. add-exp-log 51.7%

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

   4. fmm-def 51.7%

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

   5. metadata-eval 51.7%

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

4. Applied egg-rr 51.7%

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

5. Taylor expanded in x around inf 98.6%

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

6. Final simplification 98.6%

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

Alternative 5: 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 51.7%

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

   1. fmm-def 51.7%

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

   2. metadata-eval 51.7%

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

   3. flip-+ 6.1%

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

   4. metadata-eval 6.1%

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

   5. fmm-def 6.1%

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

   6. div-inv 6.0%

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

   7. log-prod 6.1%

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

   8. pow2 6.1%

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

   9. add-sqr-sqrt 6.0%

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

   10. fmm-def 6.0%

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

   11. metadata-eval 6.0%

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

4. Applied egg-rr 6.0%

   \[\leadsto \color{blue}{\log \left({x}^{2} - \mathsf{fma}\left(x, x, -1\right)\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right)} \]
5. Step-by-step derivation

   1. log-rec 6.0%

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

   2. sub-neg 6.0%

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

   3. fma-undefine 6.0%

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

   4. unpow2 6.0%

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

   5. associate--r+ 8.6%

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

   6. +-inverses 8.6%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   7. metadata-eval 8.6%

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

   8. metadata-eval 8.6%

      \[\leadsto \color{blue}{0} - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   9. neg-sub0 8.6%

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

6. Simplified 8.6%

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

7. Taylor expanded in x around inf 98.9%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 98.9%

      \[\leadsto -\log \left(\frac{0.5 + \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}}{x}\right) \]

   2. metadata-eval 98.9%

      \[\leadsto -\log \left(\frac{0.5 + \frac{\color{blue}{0.125}}{{x}^{2}}}{x}\right) \]

9. Simplified 98.9%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5 + \frac{0.125}{{x}^{2}}}{x}\right)} \]

10. Taylor expanded in x around inf 97.9%

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

11. Final simplification 97.9%

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

Alternative 6: 96.7% 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 51.7%

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

3. Taylor expanded in x around inf 97.5%

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

4. Final simplification 97.5%

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

Alternative 7: 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 51.7%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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-cube-cbrt 97.4%

      \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{x + x} \cdot \sqrt[3]{x + x}\right) \cdot \sqrt[3]{x + x}\right)} \]

   2. log-prod 96.7%

      \[\leadsto \color{blue}{\log \left(\sqrt[3]{x + x} \cdot \sqrt[3]{x + x}\right) + \log \left(\sqrt[3]{x + x}\right)} \]

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. Final simplification 6.1%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer target: 99.1% 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 2024109 
(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)))))