Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 5.9s

Alternatives: 10

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: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(1.0) / fma(x, Float32(-x), Float32(1.0))) * Float32(Float32(2.0) * fma(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. Step-by-step derivation

   1. clear-num N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r/ N/A

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

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

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

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

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

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

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

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

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

   12. neg-lowering-neg.f32 N/A

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

   13. associate-*l* N/A

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

   14. +-commutative N/A

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

   15. distribute-rgt-out N/A

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

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

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

   17. *-lft-identity N/A

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

   18. accelerator-lowering-fma.f32 99.9

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

4. Applied egg-rr 99.9%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{fma}\left(x, -x, 1\right)} \cdot \left(2 \cdot \mathsf{fma}\left(x, x, x\right)\right)}\right) \]
5. 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: 99.4% accurate, 3.2× speedup

Math

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

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(x, Float32(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. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 99.5

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right) \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. pow3 N/A

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

   5. *-lft-identity N/A

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. cube-mult N/A

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

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

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

   15. *-lowering-*.f32 99.6

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

7. Applied egg-rr 99.6%

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

Alternative 4: 99.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  x
  (fma
   (* x x)
   (fma (* x x) (fma (* x x) 0.14285714285714285 0.2) 0.3333333333333333)
   1.0)))

C

float code(float x) {
	return x * fmaf((x * x), fmaf((x * x), fmaf((x * x), 0.14285714285714285f, 0.2f), 0.3333333333333333f), 1.0f);
}

Julia

function code(x)
	return Float32(x * fma(Float32(x * x), fma(Float32(x * x), fma(Float32(x * x), Float32(0.14285714285714285), Float32(0.2)), 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. 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. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 99.5

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right) \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right)} \]
6. Add Preprocessing

Alternative 5: 99.1% accurate, 4.5× speedup

Math

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

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(x, Float32(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. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 99.3

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.2, 0.3333333333333333\right), 1\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right), 1\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. pow3 N/A

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

   5. *-lft-identity N/A

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

   10. cube-mult N/A

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

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

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

   12. *-lowering-*.f32 99.4

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

7. Applied egg-rr 99.4%

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

Alternative 6: 99.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (* x (fma (* x x) (fma (* x x) 0.2 0.3333333333333333) 1.0)))

C

float code(float x) {
	return x * fmaf((x * x), fmaf((x * x), 0.2f, 0.3333333333333333f), 1.0f);
}

Julia

function code(x)
	return Float32(x * fma(Float32(x * x), fma(Float32(x * x), Float32(0.2), 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. 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. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 99.3

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.2, 0.3333333333333333\right), 1\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right), 1\right)} \]
6. Add Preprocessing

Alternative 7: 98.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x + (x * (0.3333333333333333e0 * (x * x)))
end function

Julia

function code(x)
	return Float32(x + Float32(x * Float32(Float32(0.3333333333333333) * Float32(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = x + (x * (single(0.3333333333333333) * (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. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f32 99.0

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

5. Simplified 99.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 1\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

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

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

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

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

   5. associate-*r* N/A

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

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

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

   7. *-lowering-*.f32 99.1

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.3333333333333333\right) + x \]

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right) + x} \]

8. Final simplification 99.1%

   \[\leadsto x + x \cdot \left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \]
9. Add Preprocessing

Alternative 8: 98.7% accurate, 7.4× speedup

Math

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

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(Float32(0.3333333333333333) * Float32(x * 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. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f32 99.0

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

5. Simplified 99.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 1\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

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

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

   4. associate-*r* N/A

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

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

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

   6. *-lowering-*.f32 99.1

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 9: 98.6% 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. 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. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f32 99.0

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

5. Simplified 99.0%

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

Alternative 10: 97.1% 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 98.2%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))