Rust f32::atanh

Percentage Accurate: 99.8% → 99.9%

Time: 11.5s

Alternatives: 11

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: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(2.0) * fma(x, x, x)) / fma(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. Add Preprocessing
3. Step-by-step derivation

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f32 N/A

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

   5. associate-*l* N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. *-lowering-*.f32 N/A

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

   9. *-lft-identity N/A

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

   10. accelerator-lowering-fma.f32 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. +-commutative N/A

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

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

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

   15. accelerator-lowering-fma.f32 N/A

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

   16. neg-lowering-neg.f32 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 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

Derivation

1. Initial program 99.8%

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

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\\ 0.5 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(0.4 \cdot \left(x \cdot t\_0\right)\right)\right), -0.4444444444444444\right)}{\mathsf{fma}\left(x \cdot x, t\_0, -0.6666666666666666\right)}, x \cdot \left(x \cdot x\right), 2 \cdot x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (fma (* x x) 0.2857142857142857 0.4)))
   (*
    0.5
    (fma
     (/
      (fma x (* x (* x (* 0.4 (* x t_0)))) -0.4444444444444444)
      (fma (* x x) t_0 -0.6666666666666666))
     (* x (* x x))
     (* 2.0 x)))))

C

float code(float x) {
	float t_0 = fmaf((x * x), 0.2857142857142857f, 0.4f);
	return 0.5f * fmaf((fmaf(x, (x * (x * (0.4f * (x * t_0)))), -0.4444444444444444f) / fmaf((x * x), t_0, -0.6666666666666666f)), (x * (x * x)), (2.0f * x));
}

Julia

function code(x)
	t_0 = fma(Float32(x * x), Float32(0.2857142857142857), Float32(0.4))
	return Float32(Float32(0.5) * fma(Float32(fma(x, Float32(x * Float32(x * Float32(Float32(0.4) * Float32(x * t_0)))), Float32(-0.4444444444444444)) / fma(Float32(x * x), t_0, Float32(-0.6666666666666666))), Float32(x * Float32(x * x)), Float32(Float32(2.0) * 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. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) + 2\right)}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) + \frac{2}{3}}, 2\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right)}, 2\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right), 2\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right), 2\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{7} \cdot {x}^{2} + \frac{2}{5}}, \frac{2}{3}\right), 2\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{2}{7} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{2}{7} \cdot x\right) \cdot x} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{2}{7} \cdot x\right)} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   14. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{7} \cdot x, \frac{2}{5}\right)}, \frac{2}{3}\right), 2\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{7}}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

   16. *-lowering-*.f32 97.9

       \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.2857142857142857}, 0.4\right), 0.6666666666666666\right), 2\right)\right) \]

5. Simplified 97.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2857142857142857, 0.4\right), 0.6666666666666666\right), 2\right)\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) - \frac{2}{3}}}, 2\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) - \frac{2}{3}}}, 2\right)\right) \]

7. Applied egg-rr 97.9%

   \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right), -0.4444444444444444\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), -0.6666666666666666\right)}}, 2\right)\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right) \cdot \left(x \cdot \color{blue}{\frac{2}{5}}\right), \frac{-4}{9}\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right), \frac{-2}{3}\right)}, 2\right)\right) \]
9. Step-by-step derivation

10. Simplified 98.2%

    \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right) \cdot \left(x \cdot \color{blue}{0.4}\right), -0.4444444444444444\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), -0.6666666666666666\right)}, 2\right)\right) \]
11. Step-by-step derivation

    1. distribute-rgt-in N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) \cdot \left(x \cdot \frac{2}{5}\right)\right) + \frac{-4}{9}}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) + \frac{-2}{3}}\right) \cdot x + 2 \cdot x\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) \cdot \left(x \cdot \frac{2}{5}\right)\right) + \frac{-4}{9}}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) + \frac{-2}{3}} \cdot \left(x \cdot x\right)\right)} \cdot x + 2 \cdot x\right) \]

    3. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) \cdot \left(x \cdot \frac{2}{5}\right)\right) + \frac{-4}{9}}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) + \frac{-2}{3}} \cdot \left(\left(x \cdot x\right) \cdot x\right)} + 2 \cdot x\right) \]

    4. associate-*r* N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) \cdot \left(x \cdot \frac{2}{5}\right)\right) + \frac{-4}{9}}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) + \frac{-2}{3}} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + 2 \cdot x\right) \]

    5. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) \cdot \left(x \cdot \frac{2}{5}\right)\right) + \frac{-4}{9}}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{7} + \frac{2}{5}\right)\right) + \frac{-2}{3}}, x \cdot \left(x \cdot x\right), 2 \cdot x\right)} \]

12. Applied egg-rr 98.3%

    \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right) \cdot 0.4\right)\right), -0.4444444444444444\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), -0.6666666666666666\right)}, x \cdot \left(x \cdot x\right), x \cdot 2\right)} \]

13. Final simplification 98.3%

    \[\leadsto 0.5 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(0.4 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right)\right)\right), -0.4444444444444444\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), -0.6666666666666666\right)}, x \cdot \left(x \cdot x\right), 2 \cdot x\right) \]
14. Add Preprocessing

Alternative 4: 99.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.34285714285714286, 0.2857142857142857\right), 0.4\right), 0.6666666666666666\right), 2\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  0.5
  (*
   x
   (fma
    (* x x)
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) 0.34285714285714286 0.2857142857142857) 0.4)
     0.6666666666666666)
    2.0))))

C

float code(float x) {
	return 0.5f * (x * fmaf((x * x), fmaf((x * x), fmaf((x * x), fmaf((x * x), 0.34285714285714286f, 0.2857142857142857f), 0.4f), 0.6666666666666666f), 2.0f));
}

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(x * fma(Float32(x * x), fma(Float32(x * x), fma(Float32(x * x), fma(Float32(x * x), Float32(0.34285714285714286), Float32(0.2857142857142857)), Float32(0.4)), Float32(0.6666666666666666)), Float32(2.0))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) + 2\right)}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) + \frac{2}{3}}, 2\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right)}, 2\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right), 2\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right), 2\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{7} \cdot {x}^{2} + \frac{2}{5}}, \frac{2}{3}\right), 2\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{2}{7} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{2}{7} \cdot x\right) \cdot x} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{2}{7} \cdot x\right)} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   14. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{7} \cdot x, \frac{2}{5}\right)}, \frac{2}{3}\right), 2\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{7}}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

   16. *-lowering-*.f32 97.9

       \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.2857142857142857}, 0.4\right), 0.6666666666666666\right), 2\right)\right) \]

5. Simplified 97.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2857142857142857, 0.4\right), 0.6666666666666666\right), 2\right)\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) - \frac{2}{3}}}, 2\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) - \frac{2}{3}}}, 2\right)\right) \]

7. Applied egg-rr 97.9%

   \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right), -0.4444444444444444\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), -0.6666666666666666\right)}}, 2\right)\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{4}{25} \cdot {x}^{2}}, \frac{-4}{9}\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right), \frac{-2}{3}\right)}, 2\right)\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \frac{4}{25} \cdot \color{blue}{\left(x \cdot x\right)}, \frac{-4}{9}\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right), \frac{-2}{3}\right)}, 2\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{4}{25} \cdot x\right) \cdot x}, \frac{-4}{9}\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right), \frac{-2}{3}\right)}, 2\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{4}{25} \cdot x\right)}, \frac{-4}{9}\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right), \frac{-2}{3}\right)}, 2\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{4}{25} \cdot x\right)}, \frac{-4}{9}\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right), \frac{-2}{3}\right)}, 2\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(x \cdot \frac{4}{25}\right)}, \frac{-4}{9}\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right), \frac{-2}{3}\right)}, 2\right)\right) \]

   6. *-lowering-*.f32 97.9

      \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(x \cdot 0.16\right)}, -0.4444444444444444\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), -0.6666666666666666\right)}, 2\right)\right) \]

10. Simplified 97.9%

    \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot 0.16\right)}, -0.4444444444444444\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), -0.6666666666666666\right)}, 2\right)\right) \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{12}{35} \cdot {x}^{2}\right)\right)}, 2\right)\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{12}{35} \cdot {x}^{2}\right)\right) + \frac{2}{3}}, 2\right)\right) \]

    2. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{12}{35} \cdot {x}^{2}\right), \frac{2}{3}\right)}, 2\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{12}{35} \cdot {x}^{2}\right), \frac{2}{3}\right), 2\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{12}{35} \cdot {x}^{2}\right), \frac{2}{3}\right), 2\right)\right) \]

    5. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{7} + \frac{12}{35} \cdot {x}^{2}\right) + \frac{2}{5}}, \frac{2}{3}\right), 2\right)\right) \]

    6. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{7} + \frac{12}{35} \cdot {x}^{2}, \frac{2}{5}\right)}, \frac{2}{3}\right), 2\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{7} + \frac{12}{35} \cdot {x}^{2}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{7} + \frac{12}{35} \cdot {x}^{2}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{12}{35} \cdot {x}^{2} + \frac{2}{7}}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{12}{35}} + \frac{2}{7}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    11. accelerator-lowering-fma.f32 N/A

        \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{12}{35}, \frac{2}{7}\right)}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{12}{35}, \frac{2}{7}\right), \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    13. *-lowering-*.f32 98.2

        \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.34285714285714286, 0.2857142857142857\right), 0.4\right), 0.6666666666666666\right), 2\right)\right) \]

13. Simplified 98.2%

    \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.34285714285714286, 0.2857142857142857\right), 0.4\right), 0.6666666666666666\right)}, 2\right)\right) \]
14. Add Preprocessing

Alternative 5: 99.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.17142857142857143, 0.2857142857142857\right), 0.4\right), 0.6666666666666666\right), 2\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  0.5
  (*
   x
   (fma
    (* x x)
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) 0.17142857142857143 0.2857142857142857) 0.4)
     0.6666666666666666)
    2.0))))

C

float code(float x) {
	return 0.5f * (x * fmaf((x * x), fmaf((x * x), fmaf((x * x), fmaf((x * x), 0.17142857142857143f, 0.2857142857142857f), 0.4f), 0.6666666666666666f), 2.0f));
}

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(x * fma(Float32(x * x), fma(Float32(x * x), fma(Float32(x * x), fma(Float32(x * x), Float32(0.17142857142857143), Float32(0.2857142857142857)), Float32(0.4)), Float32(0.6666666666666666)), Float32(2.0))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) + 2\right)}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) + \frac{2}{3}}, 2\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right)}, 2\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right), 2\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right), 2\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{7} \cdot {x}^{2} + \frac{2}{5}}, \frac{2}{3}\right), 2\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{2}{7} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{2}{7} \cdot x\right) \cdot x} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{2}{7} \cdot x\right)} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   14. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{7} \cdot x, \frac{2}{5}\right)}, \frac{2}{3}\right), 2\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{7}}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

   16. *-lowering-*.f32 97.9

       \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.2857142857142857}, 0.4\right), 0.6666666666666666\right), 2\right)\right) \]

5. Simplified 97.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2857142857142857, 0.4\right), 0.6666666666666666\right), 2\right)\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) - \frac{2}{3}}}, 2\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) - \frac{2}{3}}}, 2\right)\right) \]

7. Applied egg-rr 97.9%

   \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right), -0.4444444444444444\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), -0.6666666666666666\right)}}, 2\right)\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right) \cdot \left(x \cdot \color{blue}{\frac{2}{5}}\right), \frac{-4}{9}\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right), \frac{-2}{3}\right)}, 2\right)\right) \]
9. Step-by-step derivation

10. Simplified 98.2%

    \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right)\right) \cdot \left(x \cdot \color{blue}{0.4}\right), -0.4444444444444444\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), -0.6666666666666666\right)}, 2\right)\right) \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{6}{35} \cdot {x}^{2}\right)\right)}, 2\right)\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{6}{35} \cdot {x}^{2}\right)\right) + \frac{2}{3}}, 2\right)\right) \]

    2. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{6}{35} \cdot {x}^{2}\right), \frac{2}{3}\right)}, 2\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{6}{35} \cdot {x}^{2}\right), \frac{2}{3}\right), 2\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + {x}^{2} \cdot \left(\frac{2}{7} + \frac{6}{35} \cdot {x}^{2}\right), \frac{2}{3}\right), 2\right)\right) \]

    5. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{7} + \frac{6}{35} \cdot {x}^{2}\right) + \frac{2}{5}}, \frac{2}{3}\right), 2\right)\right) \]

    6. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{7} + \frac{6}{35} \cdot {x}^{2}, \frac{2}{5}\right)}, \frac{2}{3}\right), 2\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{7} + \frac{6}{35} \cdot {x}^{2}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{7} + \frac{6}{35} \cdot {x}^{2}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{6}{35} \cdot {x}^{2} + \frac{2}{7}}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{6}{35}} + \frac{2}{7}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    11. accelerator-lowering-fma.f32 N/A

        \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{6}{35}, \frac{2}{7}\right)}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{6}{35}, \frac{2}{7}\right), \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

    13. *-lowering-*.f32 98.1

        \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.17142857142857143, 0.2857142857142857\right), 0.4\right), 0.6666666666666666\right), 2\right)\right) \]

13. Simplified 98.1%

    \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.17142857142857143, 0.2857142857142857\right), 0.4\right), 0.6666666666666666\right)}, 2\right)\right) \]
14. Add Preprocessing

Alternative 6: 99.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{fma}\left(x, 2, \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), 0.6666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  0.5
  (fma
   x
   2.0
   (*
    (* x (* x x))
    (fma x (* x (fma (* x x) 0.2857142857142857 0.4)) 0.6666666666666666)))))

C

float code(float x) {
	return 0.5f * fmaf(x, 2.0f, ((x * (x * x)) * fmaf(x, (x * fmaf((x * x), 0.2857142857142857f, 0.4f)), 0.6666666666666666f)));
}

Julia

function code(x)
	return Float32(Float32(0.5) * fma(x, Float32(2.0), Float32(Float32(x * Float32(x * x)) * fma(x, Float32(x * fma(Float32(x * x), Float32(0.2857142857142857), Float32(0.4))), Float32(0.6666666666666666)))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) + 2\right)}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), 2\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) + \frac{2}{3}}, 2\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right)}, 2\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right), 2\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, \frac{2}{3}\right), 2\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{7} \cdot {x}^{2} + \frac{2}{5}}, \frac{2}{3}\right), 2\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{2}{7} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{2}{7} \cdot x\right) \cdot x} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{2}{7} \cdot x\right)} + \frac{2}{5}, \frac{2}{3}\right), 2\right)\right) \]

   14. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{7} \cdot x, \frac{2}{5}\right)}, \frac{2}{3}\right), 2\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{7}}, \frac{2}{5}\right), \frac{2}{3}\right), 2\right)\right) \]

   16. *-lowering-*.f32 97.9

       \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.2857142857142857}, 0.4\right), 0.6666666666666666\right), 2\right)\right) \]

5. Simplified 97.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2857142857142857, 0.4\right), 0.6666666666666666\right), 2\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(2 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) + \frac{2}{3}\right)\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot 2 + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) + \frac{2}{3}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot 2 + \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) + \frac{2}{3}\right)\right) \cdot x}\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(x, 2, \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) + \frac{2}{3}\right)\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, 2, \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) + \frac{2}{3}\right) \cdot \left(x \cdot x\right)\right)} \cdot x\right) \]

   6. associate-*l* N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, 2, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) + \frac{2}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right) \]

   7. pow3 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, 2, \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) + \frac{2}{3}\right) \cdot \color{blue}{{x}^{3}}\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, 2, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{7}\right) + \frac{2}{5}\right) + \frac{2}{3}\right) \cdot {x}^{3}}\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), 0.6666666666666666\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]

8. Final simplification 98.0%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(x, 2, \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), 0.6666666666666666\right)\right) \]
9. Add Preprocessing

Alternative 7: 99.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.14285714285714285, 0.2\right), 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (fma
  (fma (* x x) (fma x (* x 0.14285714285714285) 0.2) 0.3333333333333333)
  (* x (* x x))
  x))

C

float code(float x) {
	return fmaf(fmaf((x * x), fmaf(x, (x * 0.14285714285714285f), 0.2f), 0.3333333333333333f), (x * (x * x)), x);
}

Julia

function code(x)
	return fma(fma(Float32(x * x), fma(x, Float32(x * Float32(0.14285714285714285)), Float32(0.2)), Float32(0.3333333333333333)), Float32(x * Float32(x * x)), 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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + 1\right)} \cdot x \]

   3. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) \cdot x + x} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)} \cdot x + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) \cdot \left({x}^{2} \cdot x\right)} + x \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x \]

   7. unpow3 N/A

      \[\leadsto \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{x}^{3}} + x \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), {x}^{3}, x\right)} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.14285714285714285, 0.2\right), 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
6. Add Preprocessing

Alternative 8: 99.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (fma (fma (* x x) 0.2 0.3333333333333333) (* x (* x x)) x))

C

float code(float x) {
	return fmaf(fmaf((x * x), 0.2f, 0.3333333333333333f), (x * (x * x)), x);
}

Julia

function code(x)
	return fma(fma(Float32(x * x), Float32(0.2), Float32(0.3333333333333333)), Float32(x * Float32(x * x)), 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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot x} \]

   2. *-lft-identity N/A

      \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot x \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot x + x} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot x + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot x\right)} + x \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x \]

   7. unpow3 N/A

      \[\leadsto \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{3}} + x \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}, {x}^{3}, x\right)} \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{5} \cdot {x}^{2} + \frac{1}{3}}, {x}^{3}, x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{1}{5}} + \frac{1}{3}, {x}^{3}, x\right) \]

   11. accelerator-lowering-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5}, \frac{1}{3}\right)}, {x}^{3}, x\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5}, \frac{1}{3}\right), {x}^{3}, x\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5}, \frac{1}{3}\right), {x}^{3}, x\right) \]

   14. cube-mult N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right), x \cdot \color{blue}{{x}^{2}}, x\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right), \color{blue}{x \cdot {x}^{2}}, x\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   18. *-lowering-*.f32 97.7

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

5. Simplified 97.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
6. Add Preprocessing

Alternative 9: 98.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333, x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (fma 0.3333333333333333 (* x (* x x)) x))

C

float code(float x) {
	return fmaf(0.3333333333333333f, (x * (x * x)), x);
}

Julia

function code(x)
	return fma(Float32(0.3333333333333333), Float32(x * Float32(x * x)), 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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{1 \cdot x + \left(\frac{1}{3} \cdot {x}^{2}\right) \cdot x} \]

   2. *-lft-identity N/A

      \[\leadsto \color{blue}{x} + \left(\frac{1}{3} \cdot {x}^{2}\right) \cdot x \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {x}^{2}\right) \cdot x + x} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left({x}^{2} \cdot x\right)} + x \]

   5. unpow2 N/A

      \[\leadsto \frac{1}{3} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x \]

   6. unpow3 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{{x}^{3}} + x \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, x\right)} \]

   8. cube-mult N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \color{blue}{x \cdot \left(x \cdot x\right)}, x\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, x \cdot \color{blue}{{x}^{2}}, x\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{3}, \color{blue}{x \cdot {x}^{2}}, x\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{3}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   12. *-lowering-*.f32 97.2

       \[\leadsto \mathsf{fma}\left(0.3333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

5. Simplified 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)} \]
6. Add Preprocessing

Alternative 10: 98.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* x (fma x (* x 0.3333333333333333) 1.0)))

C

float code(float x) {
	return x * fmaf(x, (x * 0.3333333333333333f), 1.0f);
}

Julia

function code(x)
	return Float32(x * fma(x, Float32(x * Float32(0.3333333333333333)), Float32(1.0)))
end

Derivation

1. Initial program 99.8%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f32 N/A

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

   5. associate-*l* N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. *-lowering-*.f32 N/A

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

   9. *-lft-identity N/A

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

   10. accelerator-lowering-fma.f32 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. +-commutative N/A

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

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

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

   15. accelerator-lowering-fma.f32 N/A

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

   16. neg-lowering-neg.f32 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot {x}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} \cdot {x}^{2}\right) + x \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto x \cdot \left(\frac{1}{3} \cdot {x}^{2}\right) + \color{blue}{x} \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{3} \cdot {x}^{2}, x\right)} \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{3}}, x\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{3}}, x\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{3}, x\right) \]

   8. *-lowering-*.f32 97.2

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot 0.3333333333333333, x\right) \]

7. Simplified 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.3333333333333333, x\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x} + x \]

   2. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{3} + 1\right) \cdot x} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{3} + 1\right) \cdot x} \]

   4. associate-*l* N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{3}\right)} + 1\right) \cdot x \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{3}, 1\right)} \cdot x \]

   6. *-lowering-*.f32 97.1

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot 0.3333333333333333}, 1\right) \cdot x \]

9. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.3333333333333333, 1\right) \cdot x} \]

10. Final simplification 97.1%

    \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 1\right) \]
11. Add Preprocessing

Alternative 11: 96.9% accurate, 125.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary32 x)

C

float code(float x) {
	return x;
}

Fortran

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

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = 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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 95.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))