Rust f32::atanh

Percentage Accurate: 92.5% → 98.4%

Time: 6.1s

Alternatives: 4

Speedup: 11.4×

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: 92.5% 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: 98.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.4%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 96.7

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

5. Applied rewrites 96.7%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, 2\right), x, 2\right) \cdot x}\right) \]
6. Step-by-step derivation

7. Applied rewrites 96.4%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 96.3

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

10. Applied rewrites 95.6%

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

12. Applied rewrites 98.2%

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

13. Final simplification 98.2%

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

Alternative 2: 98.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 88.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 96.7

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

5. Applied rewrites 96.7%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, 2\right), x, 2\right) \cdot x}\right) \]
6. Step-by-step derivation

7. Applied rewrites 96.4%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 96.3

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

10. Applied rewrites 95.6%

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

12. Applied rewrites 98.1%

    \[\leadsto 0.5 \cdot \left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right) \cdot x\right) \]

13. Final simplification 98.1%

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

Alternative 3: 96.7% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.6%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f32 96.7

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

5. Applied rewrites 96.7%

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

6. Final simplification 96.7%

   \[\leadsto \left(x \cdot 2\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 4: 7.7% accurate, 41.7× 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 Float32(-x)
end

MATLAB

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

Derivation

1. Initial program 89.3%

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

4. Applied rewrites 8.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(-8 \cdot {\left(\frac{x}{1 - x}\right)}^{3}\right) - \mathsf{log1p}\left(\frac{x}{1 - x} \cdot \mathsf{fma}\left(\frac{x}{1 - x}, 4, 2\right)\right)\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot x} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f32 8.0

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

7. Applied rewrites 8.0%

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

Reproduce

herbie shell --seed 2024295 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))