Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \tanh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (atanh x))

C

float code(float x) {
	return atanhf(x);
}

Julia

function code(x)
	return atanh(x)
end

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Alternative 1: 99.8% accurate, 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{-2}{x + -1}\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 3: 99.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \left(2 + \left(-1 + \left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + x \cdot \left(x \cdot 0.2857142857142857\right)\right)\right)\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  0.5
  (*
   x
   (+
    2.0
    (+
     -1.0
     (+
      1.0
      (*
       x
       (*
        x
        (+
         0.6666666666666666
         (* x (* x (+ 0.4 (* x (* x 0.2857142857142857))))))))))))))

C

float code(float x) {
	return 0.5f * (x * (2.0f + (-1.0f + (1.0f + (x * (x * (0.6666666666666666f + (x * (x * (0.4f + (x * (x * 0.2857142857142857f))))))))))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (x * (2.0e0 + ((-1.0e0) + (1.0e0 + (x * (x * (0.6666666666666666e0 + (x * (x * (0.4e0 + (x * (x * 0.2857142857142857e0))))))))))))
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(x * Float32(Float32(2.0) + Float32(Float32(-1.0) + Float32(Float32(1.0) + Float32(x * Float32(x * Float32(Float32(0.6666666666666666) + Float32(x * Float32(x * Float32(Float32(0.4) + Float32(x * Float32(x * Float32(0.2857142857142857))))))))))))))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (x * (single(2.0) + (single(-1.0) + (single(1.0) + (x * (x * (single(0.6666666666666666) + (x * (x * (single(0.4) + (x * (x * single(0.2857142857142857)))))))))))));
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right) \]

   2. associate-*l* 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{x \cdot \left(x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   3. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   4. *-commutative 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \color{blue}{{x}^{2} \cdot 0.2857142857142857}\right)\right)\right)\right)\right) \]

   5. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2857142857142857\right)\right)\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)\right)}\right)\right) \]

   2. expm1-undefine 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)} - 1\right)}\right)\right) \]

   3. associate-*l* 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(e^{\mathsf{log1p}\left(x \cdot \left(x \cdot \left(0.6666666666666666 + \color{blue}{x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)}\right)\right)\right)} - 1\right)\right)\right) \]

9. Applied egg-rr 99.5%

   \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)\right)} - 1\right)}\right)\right) \]
10. Step-by-step derivation

    1. sub-neg 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)\right)} + \left(-1\right)\right)}\right)\right) \]

    2. log1p-undefine 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(e^{\color{blue}{\log \left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)\right)}} + \left(-1\right)\right)\right)\right) \]

    3. rem-exp-log 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)\right)} + \left(-1\right)\right)\right)\right) \]

    4. associate-*r* 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(\left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \color{blue}{x \cdot \left(x \cdot 0.2857142857142857\right)}\right)\right)\right)\right)\right) + \left(-1\right)\right)\right)\right) \]

    5. metadata-eval 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(\left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + x \cdot \left(x \cdot 0.2857142857142857\right)\right)\right)\right)\right)\right) + \color{blue}{-1}\right)\right)\right) \]

11. Simplified 99.5%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(\left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + x \cdot \left(x \cdot 0.2857142857142857\right)\right)\right)\right)\right)\right) + -1\right)}\right)\right) \]

12. Final simplification 99.5%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(-1 + \left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + x \cdot \left(x \cdot 0.2857142857142857\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 4: 99.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(-1 + \left(1 + x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + x \cdot \left(x \cdot 0.2857142857142857\right)\right)\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  0.5
  (*
   x
   (+
    2.0
    (*
     x
     (+
      -1.0
      (+
       1.0
       (*
        x
        (+
         0.6666666666666666
         (* x (* x (+ 0.4 (* x (* x 0.2857142857142857))))))))))))))

C

float code(float x) {
	return 0.5f * (x * (2.0f + (x * (-1.0f + (1.0f + (x * (0.6666666666666666f + (x * (x * (0.4f + (x * (x * 0.2857142857142857f))))))))))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (x * (2.0e0 + (x * ((-1.0e0) + (1.0e0 + (x * (0.6666666666666666e0 + (x * (x * (0.4e0 + (x * (x * 0.2857142857142857e0))))))))))))
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(x * Float32(Float32(2.0) + Float32(x * Float32(Float32(-1.0) + Float32(Float32(1.0) + Float32(x * Float32(Float32(0.6666666666666666) + Float32(x * Float32(x * Float32(Float32(0.4) + Float32(x * Float32(x * Float32(0.2857142857142857))))))))))))))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (x * (single(2.0) + (x * (single(-1.0) + (single(1.0) + (x * (single(0.6666666666666666) + (x * (x * (single(0.4) + (x * (x * single(0.2857142857142857)))))))))))));
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right) \]

   2. associate-*l* 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{x \cdot \left(x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   3. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   4. *-commutative 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \color{blue}{{x}^{2} \cdot 0.2857142857142857}\right)\right)\right)\right)\right) \]

   5. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2857142857142857\right)\right)\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)}\right)\right) \]

   2. expm1-undefine 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)} - 1\right)}\right)\right) \]

   3. associate-*l* 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(e^{\mathsf{log1p}\left(x \cdot \left(0.6666666666666666 + \color{blue}{x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)}\right)\right)} - 1\right)\right)\right) \]

9. Applied egg-rr 99.5%

   \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)} - 1\right)}\right)\right) \]
10. Step-by-step derivation

    1. sub-neg 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)} + \left(-1\right)\right)}\right)\right) \]

    2. log1p-undefine 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(e^{\color{blue}{\log \left(1 + x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)}} + \left(-1\right)\right)\right)\right) \]

    3. rem-exp-log 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(\color{blue}{\left(1 + x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)} + \left(-1\right)\right)\right)\right) \]

    4. associate-*r* 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(\left(1 + x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + \color{blue}{x \cdot \left(x \cdot 0.2857142857142857\right)}\right)\right)\right)\right) + \left(-1\right)\right)\right)\right) \]

    5. metadata-eval 99.5%

       \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(\left(1 + x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + x \cdot \left(x \cdot 0.2857142857142857\right)\right)\right)\right)\right) + \color{blue}{-1}\right)\right)\right) \]

11. Simplified 99.5%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \color{blue}{\left(\left(1 + x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + x \cdot \left(x \cdot 0.2857142857142857\right)\right)\right)\right)\right) + -1\right)}\right)\right) \]

12. Final simplification 99.5%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(-1 + \left(1 + x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.4 + x \cdot \left(x \cdot 0.2857142857142857\right)\right)\right)\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 5: 99.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + 0.2857142857142857 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  0.5
  (*
   x
   (+
    2.0
    (*
     x
     (*
      x
      (+
       0.6666666666666666
       (* (* x x) (+ 0.4 (* 0.2857142857142857 (* x x)))))))))))

C

float code(float x) {
	return 0.5f * (x * (2.0f + (x * (x * (0.6666666666666666f + ((x * x) * (0.4f + (0.2857142857142857f * (x * x)))))))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (x * (2.0e0 + (x * (x * (0.6666666666666666e0 + ((x * x) * (0.4e0 + (0.2857142857142857e0 * (x * x)))))))))
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(x * Float32(Float32(2.0) + Float32(x * Float32(x * Float32(Float32(0.6666666666666666) + Float32(Float32(x * x) * Float32(Float32(0.4) + Float32(Float32(0.2857142857142857) * Float32(x * x))))))))))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (x * (single(2.0) + (x * (x * (single(0.6666666666666666) + ((x * x) * (single(0.4) + (single(0.2857142857142857) * (x * x)))))))));
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right) \]

   2. associate-*l* 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{x \cdot \left(x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   3. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   4. *-commutative 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \color{blue}{{x}^{2} \cdot 0.2857142857142857}\right)\right)\right)\right)\right) \]

   5. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2857142857142857\right)\right)\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)\right)} \]

8. Final simplification 99.5%

   \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + 0.2857142857142857 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 6: 99.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * ((x * ((x * x) * (0.6666666666666666e0 + (x * (x * 0.4e0))))) + (2.0e0 * x))
end function

Julia

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

MATLAB

function tmp = code(x)
	tmp = single(0.5) * ((x * ((x * x) * (single(0.6666666666666666) + (x * (x * single(0.4)))))) + (single(2.0) * x));
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.4 \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.6666666666666666 + 0.4 \cdot {x}^{2}\right)\right)\right) \]

   2. *-commutative 99.1%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \color{blue}{{x}^{2} \cdot 0.4}\right)\right)\right) \]

   3. unpow2 99.1%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.4\right)\right)\right) \]

7. Simplified 99.1%

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

   1. +-commutative 99.1%

      \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.4\right) + 2\right)}\right) \]

   2. distribute-rgt-in 99.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.4\right)\right) \cdot x + 2 \cdot x\right)} \]

   3. associate-*l* 99.2%

      \[\leadsto 0.5 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(0.6666666666666666 + \color{blue}{x \cdot \left(x \cdot 0.4\right)}\right)\right) \cdot x + 2 \cdot x\right) \]

   4. *-commutative 99.2%

      \[\leadsto 0.5 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot 0.4\right)\right)\right) \cdot x + \color{blue}{x \cdot 2}\right) \]

9. Applied egg-rr 99.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot 0.4\right)\right)\right) \cdot x + x \cdot 2\right)} \]

10. Final simplification 99.2%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot 0.4\right)\right)\right) + 2 \cdot x\right) \]
11. Add Preprocessing

Alternative 7: 99.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (x * (2.0e0 + ((x * x) * (0.6666666666666666e0 + (0.4e0 * (x * x))))))
end function

Julia

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

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (x * (single(2.0) + ((x * x) * (single(0.6666666666666666) + (single(0.4) * (x * x))))));
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.4 \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.6666666666666666 + 0.4 \cdot {x}^{2}\right)\right)\right) \]

   2. *-commutative 99.1%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \color{blue}{{x}^{2} \cdot 0.4}\right)\right)\right) \]

   3. unpow2 99.1%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.4\right)\right)\right) \]

7. Simplified 99.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.4\right)\right)\right)} \]

8. Final simplification 99.1%

   \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + 0.4 \cdot \left(x \cdot x\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 8: 98.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(2 \cdot x + x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (* 0.5 (+ (* 2.0 x) (* x (* 0.6666666666666666 (* x x))))))

C

float code(float x) {
	return 0.5f * ((2.0f * x) + (x * (0.6666666666666666f * (x * x))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * ((2.0e0 * x) + (x * (0.6666666666666666e0 * (x * x))))
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(Float32(Float32(2.0) * x) + Float32(x * Float32(Float32(0.6666666666666666) * Float32(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * ((single(2.0) * x) + (x * (single(0.6666666666666666) * (x * x))));
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 98.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 98.6%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{{x}^{2} \cdot 0.6666666666666666}\right)\right) \]

   2. unpow2 98.6%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(x \cdot x\right)} \cdot 0.6666666666666666\right)\right) \]

7. Simplified 98.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 98.6%

      \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right)}\right) \]

   2. distribute-rgt-in 98.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot x + 2 \cdot x\right)} \]

   3. *-commutative 98.6%

      \[\leadsto 0.5 \cdot \left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot x + \color{blue}{x \cdot 2}\right) \]

9. Applied egg-rr 98.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot x + x \cdot 2\right)} \]

10. Final simplification 98.6%

    \[\leadsto 0.5 \cdot \left(2 \cdot x + x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \]
11. Add Preprocessing

Alternative 9: 98.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (* 0.5 (* x (+ 2.0 (* 0.6666666666666666 (* x x))))))

C

float code(float x) {
	return 0.5f * (x * (2.0f + (0.6666666666666666f * (x * x))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (x * (2.0e0 + (0.6666666666666666e0 * (x * x))))
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(x * Float32(Float32(2.0) + Float32(Float32(0.6666666666666666) * Float32(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (x * (single(2.0) + (single(0.6666666666666666) * (x * x))));
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 98.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 98.6%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{{x}^{2} \cdot 0.6666666666666666}\right)\right) \]

   2. unpow2 98.6%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(x \cdot x\right)} \cdot 0.6666666666666666\right)\right) \]

7. Simplified 98.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right)} \]

8. Final simplification 98.6%

   \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \]
9. Add Preprocessing

Alternative 10: 96.9% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(2 \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* 0.5 (* 2.0 x)))

C

float code(float x) {
	return 0.5f * (2.0f * x);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0 * (2.0e0 * x)
end function

Julia

function code(x)
	return Float32(Float32(0.5) * Float32(Float32(2.0) * x))
end

MATLAB

function tmp = code(x)
	tmp = single(0.5) * (single(2.0) * x);
end

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. neg-sub0 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 99.7%

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

   8. distribute-frac-neg2 99.7%

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

   9. distribute-neg-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. sub-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 97.4%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))