Rust f32::atanh

Percentage Accurate: 99.8% → 99.7%

Time: 10.7s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \tanh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (atanh x))

C

float code(float x) {
	return atanhf(x);
}

Julia

function code(x)
	return atanh(x)
end

MATLAB

function tmp = code(x)
	tmp = atanh(x);
end

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. *-commutative 99.7%

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

   2. frac-2neg 99.7%

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

   3. metadata-eval 99.7%

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

   4. +-commutative 99.7%

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

   5. distribute-neg-in 99.7%

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

   6. metadata-eval 99.7%

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

   7. sub-neg 99.7%

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

   8. associate-*l/ 99.8%

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

   9. flip-- 99.7%

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

   10. associate-/r/ 99.8%

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

   11. *-commutative 99.8%

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

   12. metadata-eval 99.8%

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

   13. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

Alternative 4: 98.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\frac{x}{0.5} + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (* 0.5 (+ (/ x 0.5) (* x (* x (* x 0.6666666666666666))))))

C

float code(float x) {
	return 0.5f * ((x / 0.5f) + (x * (x * (x * 0.6666666666666666f))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * ((x / 0.5e0) + (x * (x * (x * 0.6666666666666666e0))))
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(Float32(x / Float32(0.5)) + Float32(x * Float32(x * Float32(x * Float32(0.6666666666666666))))))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * ((x / single(0.5)) + (x * (x * (x * single(0.6666666666666666)))));
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 98.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 98.3%

      \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)}\right) \]

   2. *-commutative 98.3%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot 0.6666666666666666} + 2\right)\right) \]

   3. fma-define 98.3%

      \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, 2\right)}\right) \]

   4. unpow2 98.3%

      \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.6666666666666666, 2\right)\right) \]

7. Simplified 98.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 98.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x\right)} \]

   2. fma-undefine 98.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right)} \cdot x\right) \]

9. Applied egg-rr 98.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right) \cdot x\right)} \]
10. Step-by-step derivation

    1. *-commutative 98.3%

       \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right)\right)} \]

    2. +-commutative 98.3%

       \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)}\right) \]

    3. distribute-lft-in 98.3%

       \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot 2 + x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right)\right)} \]

    4. metadata-eval 98.3%

       \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\frac{1}{0.5}} + x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right)\right) \]

    5. div-inv 98.3%

       \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{x}{0.5}} + x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right)\right) \]

    6. associate-*l* 98.3%

       \[\leadsto 0.5 \cdot \left(\frac{x}{0.5} + x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)}\right) \]

11. Applied egg-rr 98.3%

    \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{0.5} + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} \]
12. Add Preprocessing

Alternative 5: 98.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (* 0.5 (* x (+ 2.0 (* (* x x) 0.6666666666666666)))))

C

float code(float x) {
	return 0.5f * (x * (2.0f + ((x * x) * 0.6666666666666666f)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (x * (2.0e0 + ((x * x) * 0.6666666666666666e0)))
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(x * Float32(Float32(2.0) + Float32(Float32(x * x) * Float32(0.6666666666666666)))))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (x * (single(2.0) + ((x * x) * single(0.6666666666666666))));
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 98.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 98.3%

      \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)}\right) \]

   2. *-commutative 98.3%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot 0.6666666666666666} + 2\right)\right) \]

   3. fma-define 98.3%

      \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, 2\right)}\right) \]

   4. unpow2 98.3%

      \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.6666666666666666, 2\right)\right) \]

7. Simplified 98.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 98.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x\right)} \]

   2. fma-undefine 98.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right)} \cdot x\right) \]

9. Applied egg-rr 98.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right) \cdot x\right)} \]

10. Final simplification 98.3%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right) \]
11. Add Preprocessing

Alternative 6: 97.0% accurate, 21.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (* 0.5 (* x 2.0)))

C

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

Fortran

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

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(x * Float32(2.0)))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 96.4%

   \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot x\right)} \]

6. Final simplification 96.4%

   \[\leadsto 0.5 \cdot \left(x \cdot 2\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024097 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))