Rust f32::asinh

Percentage Accurate: 37.9% → 99.5%

Time: 2.0s

Alternatives: 5

Speedup: 5.8×

Specification

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 37.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\sinh^{-1} x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (asinh x) x))

C

float code(float x) {
	return copysignf(asinhf(x), x);
}

Julia

function code(x)
	return copysign(asinh(x), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(asinh(x));
end

Derivation

1. Initial program 37.9%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]
4. Add Preprocessing

Alternative 2: 83.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (copysign (log (- x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -100.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-100.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-100.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -100

   1. Initial program 50.0%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f32 44.8

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

   4. Applied rewrites 44.8%

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

   if -100 < x < 1

   1. Initial program 24.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 95.4%

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

   if 1 < x

   1. Initial program 53.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]

   4. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. log-pow-rev N/A

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

      3. sum-log N/A

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\frac{1}{x}\right)}^{-1} \cdot 2\right), x\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{copysign}\left(\log \left({\left({x}^{-1}\right)}^{-1} \cdot 2\right), x\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{\left(-1 \cdot -1\right)} \cdot 2\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{1} \cdot 2\right), x\right) \]

      7. unpow1 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right) \]

      8. lower-log.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right) \]

      9. lower-*.f32 96.7

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right) \]

   6. Applied rewrites 96.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x \cdot 2\right)}, x\right) \]
   7. Step-by-step derivation

      1. unpow1 96.7

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} \cdot 2\right), x\right) \]

      2. metadata-eval 96.7

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right) \]

      3. sqrt-pow1 96.7

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} \cdot 2\right), x\right) \]

      4. pow2 96.7

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right) \]

      5. rem-sqrt-square-rev 96.7

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} \cdot 2\right), x\right) \]

      6. asinh-def-rev 96.7

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)}, x\right) \]

      7. sqr-abs-rev 96.7

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(2 \cdot x\right), x\right) \]

      10. count-2-rev N/A

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

      11. lower-+.f32 96.7

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

   8. Applied rewrites 96.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + x\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (- x)) x)
     (if (<= t_0 2.0) (copysign x x) (copysign (log x) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (t_0 <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (t_0 <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-5.0))
		tmp = sign(x) * abs(log(-x));
	elseif (t_0 <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -5

   1. Initial program 50.2%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f32 44.8

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

   4. Applied rewrites 44.8%

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

   if -5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 2

   1. Initial program 24.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 95.0%

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

   if 2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 53.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{x}, x\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 44.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{x}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 2.0)
   (copysign x x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if ((sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))))) <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 2

   1. Initial program 32.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 67.9%

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

   if 2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 53.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{x}, x\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 44.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 53.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign x x))

C

float code(float x) {
	return copysignf(x, x);
}

Julia

function code(x)
	return copysign(x, x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(x);
end

Derivation

1. Initial program 37.9%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]

4. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
5. Step-by-step derivation

6. Applied rewrites 53.9%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
7. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2025117 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform c (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

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