Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

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: 99.8% 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 2: 99.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(Float32(Float32(x * x) * fma(x, Float32(x * fma(Float32(x * x), Float32(0.14285714285714285), Float32(0.2))), Float32(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

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
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. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f32 N/A

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

7. Applied rewrites 98.8%

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

8. Final simplification 98.8%

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

Alternative 3: 99.2% accurate, 3.2× speedup

Math

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

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(x, Float32(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. *-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. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f32 N/A

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

7. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-*.f32 N/A

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

9. Applied rewrites 98.8%

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

10. Final simplification 98.8%

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

Alternative 4: 99.1% 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. 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. Applied rewrites 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. Add Preprocessing

Alternative 5: 99.0% 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

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
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. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f32 N/A

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

7. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-*.f32 N/A

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

9. Applied rewrites 98.8%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f32 N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 98.6

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

12. Applied rewrites 98.6%

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

13. Final simplification 98.6%

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

Alternative 6: 98.6% 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. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in 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 98.0

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

5. Applied rewrites 98.0%

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

Alternative 7: 98.5% 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

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
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. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f32 N/A

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

7. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-*.f32 N/A

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

9. Applied rewrites 98.8%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 98.0

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

12. Applied rewrites 98.0%

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

13. Final simplification 98.0%

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

Alternative 8: 97.0% 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

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

   1. *-commutative N/A

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

   2. lower-*.f32 96.3

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

5. Applied rewrites 96.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity 96.3

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

7. Applied rewrites 96.3%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))