Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 2.1s

Alternatives: 9

Speedup: 1.0×

Specification

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

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

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(x + 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. 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. count-2-rev N/A

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

   3. lower-+.f32 99.8

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(Float32(Float32(fma(fma(Float32(x * x), Float32(0.14285714285714285), Float32(0.2)), Float32(x * x), Float32(0.3333333333333333)) * 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. 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. count-2-rev N/A

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

   3. lower-+.f32 99.8

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

3. Applied rewrites 99.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. div-add N/A

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

   5. frac-add N/A

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

   6. lower-/.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lift--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 99.7

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. *-commutative N/A

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

    3. pow2 N/A

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

    4. lift-*.f32 N/A

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

    5. +-commutative N/A

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

    6. lift-*.f32 N/A

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

    7. *-commutative N/A

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

    8. pow2 N/A

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

    9. +-commutative N/A

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

    10. pow2 N/A

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

    11. associate-*l* N/A

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

11. Applied rewrites 99.2%

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

Alternative 3: 99.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(fma(fma(fma(Float32(0.14285714285714285), Float32(x * x), Float32(0.2)), Float32(x * x), Float32(0.3333333333333333)), Float32(x * x), 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. 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. count-2-rev N/A

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

   3. lower-+.f32 99.8

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. log-pow-rev 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. count-2-rev N/A

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

   3. log-pow-rev N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.1%

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

Alternative 4: 99.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. +-commutative N/A

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

   8. pow2 N/A

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

   9. pow2 N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. distribute-rgt-in N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. lower-*.f32 N/A

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

6. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. pow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-fma.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. pow2 N/A

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

   13. lift-*.f32 99.0

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

8. Applied rewrites 99.0%

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

Alternative 5: 99.0% accurate, 4.5× speedup

Math

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

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(Float32(Float32(fma(Float32(x * x), Float32(0.2), Float32(0.3333333333333333)) * 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. 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. count-2-rev N/A

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

   3. lower-+.f32 99.8

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. log-pow-rev 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. count-2-rev N/A

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

   3. log-pow-rev N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. pow2 N/A

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

   12. lift-*.f32 N/A

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

   13. pow2 N/A

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

   14. lift-*.f32 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-*.f32 N/A

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

   8. pow2 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. *-commutative N/A

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

8. Applied rewrites 99.0%

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

   1. lift-fma.f32 N/A

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

   2. *-rgt-identity N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

10. Applied rewrites 99.0%

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

Alternative 6: 98.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(fma(fma(Float32(0.2), Float32(x * x), Float32(0.3333333333333333)), Float32(x * x), 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. 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. count-2-rev N/A

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

   3. lower-+.f32 99.8

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. log-pow-rev 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. count-2-rev N/A

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

   3. log-pow-rev N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. pow2 N/A

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

   12. lift-*.f32 N/A

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

   13. pow2 N/A

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

   14. lift-*.f32 98.9

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

6. Applied rewrites 98.9%

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

Alternative 7: 98.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(x, Float32(1.0), Float32(x * Float32(Float32(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. 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. count-2-rev N/A

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

   3. lower-+.f32 99.8

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. log-pow-rev 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. count-2-rev N/A

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

   3. log-pow-rev N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.1%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. pow2 N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. pow2 N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, 1, x \cdot \left(\left(\frac{1}{3} \cdot x\right) \cdot x\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 98.4%

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

Alternative 8: 98.4% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. cube-mult N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-pow.f32 98.4

      \[\leadsto \mathsf{fma}\left({x}^{3}, 0.3333333333333333, x\right) \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.3333333333333333, x\right)} \]
5. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 98.4

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

6. Applied rewrites 98.4%

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

Alternative 9: 96.7% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(x)
use fmin_fmax_functions
    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. Taylor expanded in x around 0

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

4. Applied rewrites 96.7%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))