Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

Alternatives: 12

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(x * Float32(Float32(2.0) / Float32(Float32(1.0) - x)))))
end

Derivation

1. Initial program 99.8%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\left(\frac{2}{1 - x}\right), x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(2, \left(1 - x\right)\right), x\right)\right)\right) \]

   6. --lowering--.f32 99.7%

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(2, \mathsf{\_.f32}\left(1, x\right)\right), x\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.1111111111111111 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\\ x \cdot \frac{1 - t\_0 \cdot t\_0}{\left(1 - x \cdot \left(x \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right) \cdot \left(1 + t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0
         (*
          (* x (* x (* x x)))
          (+ 0.1111111111111111 (* (* x x) 0.13333333333333333)))))
   (*
    x
    (/
     (- 1.0 (* t_0 t_0))
     (*
      (-
       1.0
       (*
        x
        (*
         x
         (+
          0.3333333333333333
          (* (* x x) (+ 0.2 (* (* x x) 0.14285714285714285)))))))
      (+ 1.0 t_0))))))

C

float code(float x) {
	float t_0 = (x * (x * (x * x))) * (0.1111111111111111f + ((x * x) * 0.13333333333333333f));
	return x * ((1.0f - (t_0 * t_0)) / ((1.0f - (x * (x * (0.3333333333333333f + ((x * x) * (0.2f + ((x * x) * 0.14285714285714285f))))))) * (1.0f + t_0)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    real(4) :: t_0
    t_0 = (x * (x * (x * x))) * (0.1111111111111111e0 + ((x * x) * 0.13333333333333333e0))
    code = x * ((1.0e0 - (t_0 * t_0)) / ((1.0e0 - (x * (x * (0.3333333333333333e0 + ((x * x) * (0.2e0 + ((x * x) * 0.14285714285714285e0))))))) * (1.0e0 + t_0)))
end function

Julia

function code(x)
	t_0 = Float32(Float32(x * Float32(x * Float32(x * x))) * Float32(Float32(0.1111111111111111) + Float32(Float32(x * x) * Float32(0.13333333333333333))))
	return Float32(x * Float32(Float32(Float32(1.0) - Float32(t_0 * t_0)) / Float32(Float32(Float32(1.0) - Float32(x * Float32(x * Float32(Float32(0.3333333333333333) + Float32(Float32(x * x) * Float32(Float32(0.2) + Float32(Float32(x * x) * Float32(0.14285714285714285)))))))) * Float32(Float32(1.0) + t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = (x * (x * (x * x))) * (single(0.1111111111111111) + ((x * x) * single(0.13333333333333333)));
	tmp = x * ((single(1.0) - (t_0 * t_0)) / ((single(1.0) - (x * (x * (single(0.3333333333333333) + ((x * x) * (single(0.2) + ((x * x) * single(0.14285714285714285)))))))) * (single(1.0) + t_0)));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5}} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \color{blue}{\left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \color{blue}{\left(\frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \left({x}^{2} \cdot \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f32 99.3%

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{*.f32}\left(x, \left(\frac{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right)}{\color{blue}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{*.f32}\left(x, \left(\left(1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\left(1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right)\right), \color{blue}{\left(\frac{1}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)}\right)}\right)\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto x \cdot \color{blue}{\left(\left(1 - \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right) \cdot \frac{1}{1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)}\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)}\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{4}\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. pow-sqr N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2}\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(x \cdot x\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \left(\frac{2}{15} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \left({x}^{2} \cdot \frac{2}{15}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{2}{15}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{2}{15}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 99.4%

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{2}{15}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.4%

    \[\leadsto x \cdot \left(\left(1 - \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.1111111111111111 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)}\right) \cdot \frac{1}{1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)}\right) \]
11. Step-by-step derivation

    1. un-div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(x, \left(\frac{1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)}{\color{blue}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)}}\right)\right) \]

    2. flip-- N/A

       \[\leadsto \mathsf{*.f32}\left(x, \left(\frac{\frac{1 \cdot 1 - \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right)}{1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)}}{\color{blue}{1} - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)}\right)\right) \]

    3. associate-/l/ N/A

       \[\leadsto \mathsf{*.f32}\left(x, \left(\frac{1 \cdot 1 - \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right)}{\color{blue}{\left(1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \left(1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right)}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(1 \cdot 1 - \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right)\right), \color{blue}{\left(\left(1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \left(1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right)\right)}\right)\right) \]

12. Applied egg-rr 99.4%

    \[\leadsto x \cdot \color{blue}{\frac{1 - \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.1111111111111111 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.1111111111111111 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)}{\left(1 - x \cdot \left(x \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.1111111111111111 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)}} \]
13. Add Preprocessing

Alternative 4: 99.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.1111111111111111 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)}{1 - x \cdot \left(x \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  x
  (/
   (-
    1.0
    (*
     (* x (* x (* x x)))
     (+ 0.1111111111111111 (* (* x x) 0.13333333333333333))))
   (-
    1.0
    (*
     x
     (*
      x
      (+
       0.3333333333333333
       (* (* x x) (+ 0.2 (* (* x x) 0.14285714285714285))))))))))

C

float code(float x) {
	return x * ((1.0f - ((x * (x * (x * x))) * (0.1111111111111111f + ((x * x) * 0.13333333333333333f)))) / (1.0f - (x * (x * (0.3333333333333333f + ((x * x) * (0.2f + ((x * x) * 0.14285714285714285f))))))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x * ((1.0e0 - ((x * (x * (x * x))) * (0.1111111111111111e0 + ((x * x) * 0.13333333333333333e0)))) / (1.0e0 - (x * (x * (0.3333333333333333e0 + ((x * x) * (0.2e0 + ((x * x) * 0.14285714285714285e0))))))))
end function

Julia

function code(x)
	return Float32(x * Float32(Float32(Float32(1.0) - Float32(Float32(x * Float32(x * Float32(x * x))) * Float32(Float32(0.1111111111111111) + Float32(Float32(x * x) * Float32(0.13333333333333333))))) / Float32(Float32(1.0) - Float32(x * Float32(x * Float32(Float32(0.3333333333333333) + Float32(Float32(x * x) * Float32(Float32(0.2) + Float32(Float32(x * x) * Float32(0.14285714285714285))))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * ((single(1.0) - ((x * (x * (x * x))) * (single(0.1111111111111111) + ((x * x) * single(0.13333333333333333))))) / (single(1.0) - (x * (x * (single(0.3333333333333333) + ((x * x) * (single(0.2) + ((x * x) * single(0.14285714285714285)))))))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5}} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \color{blue}{\left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \color{blue}{\left(\frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \left({x}^{2} \cdot \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f32 99.3%

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{*.f32}\left(x, \left(\frac{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right)}{\color{blue}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{*.f32}\left(x, \left(\left(1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\left(1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right)\right), \color{blue}{\left(\frac{1}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)}\right)}\right)\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto x \cdot \color{blue}{\left(\left(1 - \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right) \cdot \frac{1}{1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)}\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)}\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{4}\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. pow-sqr N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2}\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(x \cdot x\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \left(\frac{2}{15} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \left({x}^{2} \cdot \frac{2}{15}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{2}{15}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{2}{15}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 99.4%

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{2}{15}\right)\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.4%

    \[\leadsto x \cdot \left(\left(1 - \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.1111111111111111 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)}\right) \cdot \frac{1}{1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)}\right) \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\left(1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right) \cdot \frac{1}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)}\right) \cdot \color{blue}{x} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(\left(1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{9} + \left(x \cdot x\right) \cdot \frac{2}{15}\right)\right) \cdot \frac{1}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)}\right), \color{blue}{x}\right) \]

12. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\frac{1 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.1111111111111111 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)}{1 - x \cdot \left(x \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)} \cdot x} \]

13. Final simplification 99.4%

    \[\leadsto x \cdot \frac{1 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.1111111111111111 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)}{1 - x \cdot \left(x \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)} \]
14. Add Preprocessing

Alternative 5: 99.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x + ((x * (x * x)) * (0.3333333333333333e0 + ((x * x) * (0.2e0 + ((x * x) * 0.14285714285714285e0)))))
end function

Julia

function code(x)
	return Float32(x + Float32(Float32(x * Float32(x * x)) * Float32(Float32(0.3333333333333333) + Float32(Float32(x * x) * Float32(Float32(0.2) + Float32(Float32(x * x) * Float32(0.14285714285714285)))))))
end

MATLAB

function tmp = code(x)
	tmp = x + ((x * (x * x)) * (single(0.3333333333333333) + ((x * x) * (single(0.2) + ((x * x) * single(0.14285714285714285))))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5}} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \color{blue}{\left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \color{blue}{\left(\frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \left({x}^{2} \cdot \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f32 99.3%

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) + \color{blue}{1}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(x, \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \left(x \cdot x\right)\right) + 1\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{*.f32}\left(x, \left(\frac{1}{3} \cdot \left(x \cdot x\right) + \color{blue}{\left(\left(x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \left(x \cdot x\right) + 1\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\left(x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \left(x \cdot x\right) + 1\right)}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\left(\frac{1}{3} \cdot x\right) \cdot x\right), \left(\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \left(x \cdot x\right)} + 1\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(x \cdot \left(\frac{1}{3} \cdot x\right)\right), \left(\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \left(x \cdot x\right)} + 1\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{1}{3} \cdot x\right)\right), \left(\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \left(x \cdot x\right)} + 1\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot \frac{1}{3}\right)\right), \left(\left(x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \frac{1}{3}\right)\right), \left(\left(x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \frac{1}{3}\right)\right), \mathsf{+.f32}\left(\left(\left(x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{1}\right)\right)\right) \]

7. Applied egg-rr 99.3%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{+.f32}\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right) + 1\right)\right), \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{3}\right)\right)}\right) \]

9. Applied egg-rr 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

    \[\leadsto x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right) \]
13. Add Preprocessing

Alternative 6: 99.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x + (x * ((x * x) * (0.3333333333333333e0 + ((x * x) * (0.2e0 + ((x * x) * 0.14285714285714285e0))))))
end function

Julia

function code(x)
	return Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(0.3333333333333333) + Float32(Float32(x * x) * Float32(Float32(0.2) + Float32(Float32(x * x) * Float32(0.14285714285714285))))))))
end

MATLAB

function tmp = code(x)
	tmp = x + (x * ((x * x) * (single(0.3333333333333333) + ((x * x) * (single(0.2) + ((x * x) * single(0.14285714285714285)))))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5}} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \color{blue}{\left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \color{blue}{\left(\frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \left({x}^{2} \cdot \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f32 99.3%

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

      \[\leadsto \mathsf{+.f32}\left(\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{7}\right)\right)\right)\right) \cdot x\right), \color{blue}{x}\right) \]

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

   \[\leadsto x + x \cdot \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right) \]
9. Add Preprocessing

Alternative 7: 99.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x * (1.0e0 + ((x * x) * (0.3333333333333333e0 + (x * (x * (0.2e0 + ((x * x) * 0.14285714285714285e0)))))))
end function

Julia

function code(x)
	return Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(0.3333333333333333) + Float32(x * Float32(x * Float32(Float32(0.2) + Float32(Float32(x * x) * Float32(0.14285714285714285)))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (single(1.0) + ((x * x) * (single(0.3333333333333333) + (x * (x * (single(0.2) + ((x * x) * single(0.14285714285714285))))))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5}} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \color{blue}{\left(\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \color{blue}{\left(\frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \left({x}^{2} \cdot \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f32 99.3%

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{5}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

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

Alternative 8: 99.0% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x + (x * ((x * x) * (0.3333333333333333e0 + (x * (x * 0.2e0)))))
end function

Julia

function code(x)
	return Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(0.3333333333333333) + Float32(x * Float32(x * Float32(0.2)))))))
end

MATLAB

function tmp = code(x)
	tmp = x + (x * ((x * x) * (single(0.3333333333333333) + (x * (x * single(0.2))))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{5} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left({x}^{2} \cdot \color{blue}{\frac{1}{5}}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{5}\right)\right)\right)\right)\right) \]

5. Simplified 99.0%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

      \[\leadsto \mathsf{+.f32}\left(\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \frac{1}{5}\right)\right) \cdot x\right), \color{blue}{x}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \frac{1}{5}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \frac{1}{5}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \frac{1}{5}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \frac{1}{5}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(\left(x \cdot x\right) \cdot \frac{1}{5}\right)\right)\right)\right), x\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \left(x \cdot \frac{1}{5}\right)\right)\right)\right)\right), x\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \left(x \cdot \frac{1}{5}\right)\right)\right)\right)\right), x\right) \]

   12. *-lowering-*.f32 99.1%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \frac{1}{5}\right)\right)\right)\right)\right), x\right) \]

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

   \[\leadsto x + x \cdot \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot 0.2\right)\right)\right) \]
9. Add Preprocessing

Alternative 9: 98.9% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x * (1.0e0 + ((x * x) * (0.3333333333333333e0 + ((x * x) * 0.2e0))))
end function

Julia

function code(x)
	return Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(0.3333333333333333) + Float32(Float32(x * x) * Float32(0.2))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (single(1.0) + ((x * x) * (single(0.3333333333333333) + ((x * x) * single(0.2)))));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{5} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \left({x}^{2} \cdot \color{blue}{\frac{1}{5}}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{5}\right)\right)\right)\right)\right) \]

5. Simplified 99.0%

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

Alternative 10: 98.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = x + ((x * x) * (x * single(0.3333333333333333)));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(x, \color{blue}{\left(1 + \frac{1}{3} \cdot {x}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{3} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{3}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{3}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   5. *-lowering-*.f32 98.5%

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 98.5%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

      \[\leadsto \mathsf{+.f32}\left(\left(x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right), \color{blue}{x}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(\left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right), x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right), x\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{+.f32}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{3} \cdot x\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{3} \cdot x\right)\right), x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(x \cdot \frac{1}{3}\right)\right), x\right) \]

   11. *-lowering-*.f32 98.5%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, \frac{1}{3}\right)\right), x\right) \]

7. Applied egg-rr 98.5%

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

8. Final simplification 98.5%

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

Alternative 11: 98.4% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x * (1.0e0 + ((x * x) * 0.3333333333333333e0))
end function

Julia

function code(x)
	return Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(0.3333333333333333))))
end

MATLAB

function tmp = code(x)
	tmp = x * (single(1.0) + ((x * x) * single(0.3333333333333333)));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(x, \color{blue}{\left(1 + \frac{1}{3} \cdot {x}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{3} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{3}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{3}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   5. *-lowering-*.f32 98.5%

      \[\leadsto \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 98.5%

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

6. Final simplification 98.5%

   \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.3333333333333333\right) \]
7. Add Preprocessing

Alternative 12: 96.9% accurate, 109.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 x)

C

float code(float x) {
	return x;
}

Fortran

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

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 96.6%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))