Rust f32::asinh

Average Error: 20.4 → 0.2

Time: 5.3s

Precision: binary32

Cost: 9896

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (if (<= x -0.10000000149011612)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.019999999552965164)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) 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 <= -0.10000000149011612f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 0.019999999552965164f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), 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(-0.10000000149011612))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

↓

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.10000000149011612))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (x <= single(0.019999999552965164))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = 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 < -0.100000001

   1. Initial program 15.7

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

   2. Applied egg-rr 28.9

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

   3. Simplified 0.2

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]
      Proof
      (/.f32 1 (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= metadata-eval (+.f32 1 0)) (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 (+.f32 1 (Rewrite<= +-inverses_binary32 (-.f32 (*.f32 x x) (*.f32 x x)))) (-.f32 (hypot.f32 1 x) x)): 30 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 1 (*.f32 x x)) (*.f32 x x))) (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 32 points decrease in error
      (/.f32 (-.f32 (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 x x) 1)) (*.f32 x x)) (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 (-.f32 (Rewrite<= fma-udef_binary32 (fma.f32 x x 1)) (*.f32 x x)) (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite=> sub-neg_binary32 (+.f32 (fma.f32 x x 1) (neg.f32 (*.f32 x x)))) (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite=> +-commutative_binary32 (+.f32 (neg.f32 (*.f32 x x)) (fma.f32 x x 1))) (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 (+.f32 (Rewrite=> neg-sub0_binary32 (-.f32 0 (*.f32 x x))) (fma.f32 x x 1)) (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= associate--r-_binary32 (-.f32 0 (-.f32 (*.f32 x x) (fma.f32 x x 1)))) (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite<= neg-sub0_binary32 (neg.f32 (-.f32 (*.f32 x x) (fma.f32 x x 1)))) (-.f32 (hypot.f32 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f32 (neg.f32 (-.f32 (*.f32 x x) (fma.f32 x x 1))) (Rewrite=> sub-neg_binary32 (+.f32 (hypot.f32 1 x) (neg.f32 x)))): 0 points increase in error, 0 points decrease in error
      (/.f32 (neg.f32 (-.f32 (*.f32 x x) (fma.f32 x x 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 (neg.f32 (-.f32 (*.f32 x x) (fma.f32 x x 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 (neg.f32 (-.f32 (*.f32 x x) (fma.f32 x x 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 (neg.f32 (-.f32 (*.f32 x x) (fma.f32 x x 1))) (neg.f32 (Rewrite<= sub-neg_binary32 (-.f32 x (hypot.f32 1 x))))): 0 points increase in error, 0 points decrease in error
      (/.f32 (Rewrite=> neg-mul-1_binary32 (*.f32 -1 (-.f32 (*.f32 x x) (fma.f32 x x 1)))) (neg.f32 (-.f32 x (hypot.f32 1 x)))): 0 points increase in error, 0 points decrease in error
      (/.f32 (*.f32 -1 (-.f32 (*.f32 x x) (fma.f32 x x 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=> times-frac_binary32 (*.f32 (/.f32 -1 -1) (/.f32 (-.f32 (*.f32 x x) (fma.f32 x x 1)) (-.f32 x (hypot.f32 1 x))))): 0 points increase in error, 0 points decrease in error
      (*.f32 (Rewrite=> metadata-eval 1) (/.f32 (-.f32 (*.f32 x x) (fma.f32 x x 1)) (-.f32 x (hypot.f32 1 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-lft-identity_binary32 (/.f32 (-.f32 (*.f32 x x) (fma.f32 x x 1)) (-.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 (fma.f32 x x 1) (-.f32 x (hypot.f32 1 x))))): 5 points increase in error, 8 points decrease in error

   if -0.100000001 < x < 0.0199999996

   1. Initial program 25.8

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

   2. Applied egg-rr 25.7

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

   3. Simplified 25.7

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

   4. Taylor expanded in x around 0 0.0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + x}, x\right) \]

   if 0.0199999996 < x

   1. Initial program 15.3

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

   2. Applied egg-rr 0.3

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

   3. Simplified 0.3

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	22916

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

Alternative 2
Error	0.4
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Error	0.8
Cost	6760

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 5
Error	5.4
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 6
Error	1.0
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 7
Error	1.0
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 8
Error	10.1
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	12.3
Cost	6532

\[\begin{array}{l} \mathbf{if}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 10
Error	14.9
Cost	3264

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

Reproduce

herbie shell --seed 2022330 
(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))