Rust f32::asinh

Average Error: 20.4 → 0.2

Time: 5.9s

Precision: binary32

Cost: 9896

Math

\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

↓

(FPCore (x)
 :precision binary32
 (if (<= x -0.5)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.10000000149011612)
     (copysign (log1p (+ x (/ (* x x) (+ 2.0 (* (* x x) 0.5))))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

↓

float code(float x) {
	float tmp;
	if (x <= -0.5f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 0.10000000149011612f) {
		tmp = copysignf(log1pf((x + ((x * x) / (2.0f + ((x * x) * 0.5f))))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

↓

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.5))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(log1p(Float32(x + Float32(Float32(x * x) / Float32(Float32(2.0) + Float32(Float32(x * x) * Float32(0.5)))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

Target

Original	20.4
Target	0.1
Herbie	0.2

\[\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, \frac{1}{\left|x\right|}\right) + \frac{1}{\left|x\right|}}\right), x\right) \]

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 15.6

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Proof
      (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (hypot.f32 1 x))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (Rewrite<= hypot-1-def_binary32 (sqrt.f32 (+.f32 1 (*.f32 x x)))))) x): 69 points increase in error, 0 points decrease in error
      (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 x x) 1))))) x): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 29.7

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   4. Simplified 0.3

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]
      Proof
      (/.f32 1 (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 1 (Rewrite=> sub-neg_binary32 (+.f32 (hypot.f32 1 x) (neg.f32 x)))): 0 points increase in error, 0 points decrease in error
      (/.f32 1 (+.f32 (Rewrite<= remove-double-neg_binary32 (neg.f32 (neg.f32 (hypot.f32 1 x)))) (neg.f32 x))): 0 points increase in error, 0 points decrease in error
      (/.f32 1 (Rewrite<= distribute-neg-in_binary32 (neg.f32 (+.f32 (neg.f32 (hypot.f32 1 x)) x)))): 0 points increase in error, 0 points decrease in error
      (/.f32 1 (neg.f32 (Rewrite<= +-commutative_binary32 (+.f32 x (neg.f32 (hypot.f32 1 x)))))): 0 points increase in error, 0 points decrease in error
      (/.f32 1 (neg.f32 (Rewrite<= sub-neg_binary32 (-.f32 x (hypot.f32 1 x))))): 0 points increase in error, 0 points decrease in error
      (/.f32 1 (Rewrite=> neg-mul-1_binary32 (*.f32 -1 (-.f32 x (hypot.f32 1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r*_binary32 (/.f32 (/.f32 1 -1) (-.f32 x (hypot.f32 1 x)))): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite=> metadata-eval -1) (-.f32 x (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= metadata-eval (-.f32 0 1)) (-.f32 x (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f32 (-.f32 (Rewrite<= +-inverses_binary32 (-.f32 (*.f32 x x) (*.f32 x x))) 1) (-.f32 x (hypot.f32 1 x))): 30 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= associate--r+_binary32 (-.f32 (*.f32 x x) (+.f32 (*.f32 x x) 1))) (-.f32 x (hypot.f32 1 x))): 0 points increase in error, 30 points decrease in error
      (/.f32 (-.f32 (*.f32 x x) (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 x x)))) (-.f32 x (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (*.f32 x x) (-.f32 x (hypot.f32 1 x))) (/.f32 (+.f32 1 (*.f32 x x)) (-.f32 x (hypot.f32 1 x))))): 6 points increase in error, 4 points decrease in error

   if -0.5 < x < 0.100000001

   1. Initial program 25.3

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Simplified 25.3

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Proof
      (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (hypot.f32 1 x))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (Rewrite<= hypot-1-def_binary32 (sqrt.f32 (+.f32 1 (*.f32 x x)))))) x): 69 points increase in error, 0 points decrease in error
      (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 x x) 1))))) x): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 27.4

      \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\log \log \left(x + \mathsf{hypot}\left(1, x\right)\right)}}, x\right) \]

   4. Applied egg-rr 25.2

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)}, x\right) \]

   5. Simplified 0.9

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}, x\right) \]
      Proof
      (log1p.f32 (+.f32 x (-.f32 (hypot.f32 1 x) 1))): 0 points increase in error, 0 points decrease in error
      (log1p.f32 (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 x (hypot.f32 1 x)) 1))): 174 points increase in error, 3 points decrease in error

   6. Applied egg-rr 0.8

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

   7. Simplified 0.0

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{x \cdot x}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
      Proof
      (/.f32 (*.f32 x x) (+.f32 1 (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= unpow2_binary32 (pow.f32 x 2)) (+.f32 1 (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= +-rgt-identity_binary32 (+.f32 (pow.f32 x 2) 0)) (+.f32 1 (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f32 (+.f32 (pow.f32 x 2) (Rewrite<= metadata-eval (+.f32 1 -1))) (+.f32 1 (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= associate-+l+_binary32 (+.f32 (+.f32 (pow.f32 x 2) 1) -1)) (+.f32 1 (hypot.f32 1 x))): 40 points increase in error, 39 points decrease in error
      (/.f32 (+.f32 (Rewrite<= +-commutative_binary32 (+.f32 1 (pow.f32 x 2))) -1) (+.f32 1 (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f32 (+.f32 (+.f32 1 (Rewrite=> unpow2_binary32 (*.f32 x x))) -1) (+.f32 1 (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (+.f32 (+.f32 1 (*.f32 x x)) -1) 1)) (+.f32 1 (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary32 (*.f32 (+.f32 (+.f32 1 (*.f32 x x)) -1) (/.f32 1 (+.f32 1 (hypot.f32 1 x))))): 9 points increase in error, 10 points decrease in error

   8. Taylor expanded in x around 0 0.1

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

   9. Simplified 0.1

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{\color{blue}{2 + 0.5 \cdot \left(x \cdot x\right)}}\right), x\right) \]
      Proof
      (+.f32 2 (*.f32 1/2 (*.f32 x x))): 0 points increase in error, 0 points decrease in error
      (+.f32 2 (*.f32 1/2 (Rewrite<= unpow2_binary32 (pow.f32 x 2)))): 0 points increase in error, 0 points decrease in error

   if 0.100000001 < x

   1. Initial program 15.7

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Proof
      (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (hypot.f32 1 x))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (Rewrite<= hypot-1-def_binary32 (sqrt.f32 (+.f32 1 (*.f32 x x)))))) x): 69 points increase in error, 0 points decrease in error
      (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 x x) 1))))) x): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.2

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

   4. Simplified 0.2

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      Proof
      (log.f32 (+.f32 x (hypot.f32 1 x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary32 (+.f32 0 (log.f32 (+.f32 x (hypot.f32 1 x))))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	22916

\[\begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.0007999999797903001:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot 2 + -1\right), x\right)\\ \end{array} \]

Alternative 6
Error	4.9
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 7
Error	0.6
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 8
Error	9.9
Cost	6564

\[\begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 9
Error	12.0
Cost	6532

\[\begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 10
Error	14.6
Cost	3264

\[\mathsf{copysign}\left(x, x\right) \]

Reproduce

herbie shell --seed 2022332 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))