Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 3.1s

Alternatives: 3

Speedup: N/A×

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, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - -1 \cdot x\\ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\frac{1}{t\_0} - \frac{\left(-1 \cdot x\right) \cdot x}{-1 \cdot t\_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (- 1.0 (* -1.0 x))))
   (*
    0.5
    (log1p (/ (* 2.0 x) (- (/ 1.0 t_0) (/ (* (* -1.0 x) x) (* -1.0 t_0))))))))

C

float code(float x) {
	float t_0 = 1.0f - (-1.0f * x);
	return 0.5f * log1pf(((2.0f * x) / ((1.0f / t_0) - (((-1.0f * x) * x) / (-1.0f * t_0)))));
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) - Float32(Float32(-1.0) * x))
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(2.0) * x) / Float32(Float32(Float32(1.0) / t_0) - Float32(Float32(Float32(Float32(-1.0) * x) * x) / Float32(Float32(-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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. unpow2 N/A

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

   5. div-sub N/A

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

   6. lower--.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. *-lft-identity N/A

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

   9. metadata-eval N/A

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

   10. fp-cancel-sub-sign-inv N/A

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

   11. lower--.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. *-lft-identity N/A

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

   17. metadata-eval N/A

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

   18. fp-cancel-sub-sign-inv N/A

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

   19. lower--.f32 N/A

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

   20. lower-*.f32 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, N/A× 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 3: 99.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.2857142857142857 - -0.4, x \cdot x, 0.6666666666666666\right) \cdot x\\ t_1 := t\_0 \cdot x\\ t_2 := -1 \cdot t\_1\\ 0.5 \cdot \left(\frac{\mathsf{fma}\left({t\_0}^{3}, \left(x \cdot x\right) \cdot x, 8\right)}{\mathsf{fma}\left(t\_2, t\_2, 4 - t\_1 \cdot 2\right)} \cdot x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0
         (*
          (fma
           (- (* (* x x) 0.2857142857142857) -0.4)
           (* x x)
           0.6666666666666666)
          x))
        (t_1 (* t_0 x))
        (t_2 (* -1.0 t_1)))
   (*
    0.5
    (*
     (/
      (fma (pow t_0 3.0) (* (* x x) x) 8.0)
      (fma t_2 t_2 (- 4.0 (* t_1 2.0))))
     x))))

C

float code(float x) {
	float t_0 = fmaf((((x * x) * 0.2857142857142857f) - -0.4f), (x * x), 0.6666666666666666f) * x;
	float t_1 = t_0 * x;
	float t_2 = -1.0f * t_1;
	return 0.5f * ((fmaf(powf(t_0, 3.0f), ((x * x) * x), 8.0f) / fmaf(t_2, t_2, (4.0f - (t_1 * 2.0f)))) * x);
}

Julia

function code(x)
	t_0 = Float32(fma(Float32(Float32(Float32(x * x) * Float32(0.2857142857142857)) - Float32(-0.4)), Float32(x * x), Float32(0.6666666666666666)) * x)
	t_1 = Float32(t_0 * x)
	t_2 = Float32(Float32(-1.0) * t_1)
	return Float32(Float32(0.5) * Float32(Float32(fma((t_0 ^ Float32(3.0)), Float32(Float32(x * x) * x), Float32(8.0)) / fma(t_2, t_2, Float32(Float32(4.0) - Float32(t_1 * Float32(2.0))))) * 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. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 99.1%

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

7. Applied rewrites 99.1%

   \[\leadsto 0.5 \cdot \left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.2857142857142857 - -0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right)}^{3}, \left(x \cdot x\right) \cdot x, 8\right)}{\mathsf{fma}\left(-\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.2857142857142857 - -0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, -\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.2857142857142857 - -0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, 4 - \left(\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.2857142857142857 - -0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right) \cdot 2\right)} \cdot x\right) \]

8. Final simplification 99.1%

   \[\leadsto 0.5 \cdot \left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.2857142857142857 - -0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right)}^{3}, \left(x \cdot x\right) \cdot x, 8\right)}{\mathsf{fma}\left(-1 \cdot \left(\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.2857142857142857 - -0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right), -1 \cdot \left(\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.2857142857142857 - -0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right), 4 - \left(\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.2857142857142857 - -0.4, x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x\right) \cdot 2\right)} \cdot x\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))