Rust f32::asinh

Percentage Accurate: 37.0% → 99.5%

Time: 2.0s

Alternatives: 7

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: 37.0% 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 37.0%

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

3. Applied rewrites 99.5%

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

Alternative 2: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right|\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -4.0)
   (copysign (log (fabs x)) x)
   (if (<= x 0.05000000074505806)
     (copysign x x)
     (copysign (log (+ 1.0 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -4.0f) {
		tmp = copysignf(logf(fabsf(x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((1.0f + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-4.0))
		tmp = copysign(log(abs(x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(1.0) + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-4.0))
		tmp = sign(x) * abs(log(abs(x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((single(1.0) + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4

   1. Initial program 37.0%

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

   2. Taylor expanded in x around -inf

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 27.0%

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

   if -4 < x < 0.0500000007

   1. Initial program 37.0%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 55.0%

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

   if 0.0500000007 < x

   1. Initial program 37.0%

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

   3. Applied rewrites 27.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 20.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + x\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right|\right), x\right)\\ \mathbf{if}\;x \leq -4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (fabs x)) x)))
   (if (<= x -4.0) t_0 (if (<= x 2.0) (copysign x x) t_0))))

C

float code(float x) {
	float t_0 = copysignf(logf(fabsf(x)), x);
	float tmp;
	if (x <= -4.0f) {
		tmp = t_0;
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(abs(x)), x)
	tmp = Float32(0.0)
	if (x <= Float32(-4.0))
		tmp = t_0;
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log(abs(x)));
	tmp = single(0.0);
	if (x <= single(-4.0))
		tmp = t_0;
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4 or 2 < x

   1. Initial program 37.0%

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

   2. Taylor expanded in x around -inf

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

   4. Applied rewrites 29.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 27.0%

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

   if -4 < x < 2

   1. Initial program 37.0%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 55.0%

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

Alternative 4: 61.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(2 \cdot 2, x\right)\\ \mathbf{if}\;x \leq -4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (* 2.0 2.0) x)))
   (if (<= x -4.0) t_0 (if (<= x 2.0) (copysign x x) t_0))))

C

float code(float x) {
	float t_0 = copysignf((2.0f * 2.0f), x);
	float tmp;
	if (x <= -4.0f) {
		tmp = t_0;
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(Float32(Float32(2.0) * Float32(2.0)), x)
	tmp = Float32(0.0)
	if (x <= Float32(-4.0))
		tmp = t_0;
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs((single(2.0) * single(2.0)));
	tmp = single(0.0);
	if (x <= single(-4.0))
		tmp = t_0;
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4 or 2 < x

   1. Initial program 37.0%

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 16.3%

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

   7. Applied rewrites 16.9%

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

   if -4 < x < 2

   1. Initial program 37.0%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 55.0%

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

Alternative 5: 55.0% 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 37.0%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 55.0%

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

Alternative 6: 16.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(2, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 2.0 x))

C

float code(float x) {
	return copysignf(2.0f, x);
}

Julia

function code(x)
	return copysign(Float32(2.0), x)
end

MATLAB

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

Derivation

1. Initial program 37.0%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 16.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{2}, x\right) \]
6. Add Preprocessing

Alternative 7: 15.8% accurate, 5.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (copysign 1.0 x))

C

float code(float x) {
	return copysignf(1.0f, x);
}

Julia

function code(x)
	return copysign(Float32(1.0), x)
end

MATLAB

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

Derivation

1. Initial program 37.0%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 15.8%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{1}, x\right) \]
6. 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 2025159 
(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))