Rust f32::atanh

Percentage Accurate: 99.8% → 99.9%

Time: 9.5s

Alternatives: 8

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 99.7%

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

   1. lift-*.f32 N/A

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

   2. flip-- N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f32 N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. +-commutative N/A

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

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

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

   17. lower-fma.f32 N/A

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

   18. lower-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 99.7%

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

Alternative 3: 99.2% 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 99.7%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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. lower-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 99.1%

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

Alternative 4: 99.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(Float32(x * Float32(x * fma(Float32(x * x), Float32(0.2), Float32(0.3333333333333333)))), x, x)
end

Derivation

1. Initial program 99.7%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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. lower-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. lower-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. lower-*.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. lower-*.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. lower-*.f32 98.6

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

5. Simplified 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.6

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 5: 98.9% accurate, 4.5× speedup

Math

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

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(x, Float32(x * fma(Float32(x * x), Float32(0.2), Float32(0.3333333333333333))), Float32(1.0)))
end

Derivation

1. Initial program 99.7%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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. lower-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. lower-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. lower-*.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. lower-*.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. lower-*.f32 98.6

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

5. Simplified 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.6

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

7. Applied egg-rr 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. distribute-lft1-in N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f32 98.5

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

9. Applied egg-rr 98.5%

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

10. Final simplification 98.5%

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

Alternative 6: 98.4% 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 99.7%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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. lower-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. lower-*.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. lower-*.f32 97.7

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

5. Simplified 97.7%

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

Alternative 7: 98.4% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. flip-- N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f32 N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. +-commutative N/A

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

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

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

   17. lower-fma.f32 N/A

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

   18. lower-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. lower-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. lower-*.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. lower-*.f32 97.7

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

7. Simplified 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 97.6

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

9. Applied egg-rr 97.6%

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

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. lower-*.f32 97.6

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

11. Applied egg-rr 97.6%

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

12. Final simplification 97.6%

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

Alternative 8: 96.8% 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 99.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 95.8

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

5. Simplified 95.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

      \[\leadsto x \cdot \color{blue}{1} \]

   6. lower-*.f32 95.8

      \[\leadsto \color{blue}{x \cdot 1} \]

7. Applied egg-rr 95.8%

   \[\leadsto \color{blue}{x \cdot 1} \]
8. Step-by-step derivation

   1. *-rgt-identity 95.8

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

9. Applied egg-rr 95.8%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))