Rust f32::asinh

Percentage Accurate: 37.7% → 99.4%

Time: 11.1s

Alternatives: 12

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.7% 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: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (if (<= x 0.03999999910593033)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05000000074505806f) {
		tmp = copysignf(logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.03999999910593033f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(log(Float32(hypot(Float32(1.0), x) - x)), x);
	elseif (x <= Float32(0.03999999910593033))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05000000074505806))
		tmp = sign(x) * abs(log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.03999999910593033))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 57.8%

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

      1. log1p-expm1-u 57.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 57.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 57.7%

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

      4. +-commutative 57.7%

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

      5. metadata-eval 57.7%

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

      6. hypot-udef 98.4%

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

      7. associate--l+ 98.5%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 15.6%

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

   3. Applied egg-rr 15.6%

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

   5. Applied egg-rr 98.5%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{hypot}\left(1, x\right) + -1\right) - x}\right), x\right) \]
   6. Step-by-step derivation

      1. log1p-udef 98.5%

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

      2. +-commutative 98.5%

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

      3. associate--l+ 98.5%

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

      4. associate-+r+ 98.5%

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

      5. metadata-eval 98.5%

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

   7. Applied egg-rr 98.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0 + \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. +-lft-identity 98.5%

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

   9. Simplified 98.5%

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

   if -0.0500000007 < x < 0.0399999991

   1. Initial program 20.7%

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

      1. log1p-expm1-u 20.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 20.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 20.7%

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

      4. +-commutative 20.7%

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

      5. metadata-eval 20.7%

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

      6. hypot-udef 20.8%

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

      7. associate--l+ 97.3%

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

      8. add-sqr-sqrt 54.3%

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

      9. fabs-sqr 54.3%

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

      10. add-sqr-sqrt 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + -0.16666666666666666 \cdot {x}^{3}}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if 0.0399999991 < x

   1. Initial program 51.7%

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

      1. log1p-expm1-u 51.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 51.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 51.7%

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

      4. +-commutative 51.7%

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

      5. metadata-eval 51.7%

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

      6. hypot-udef 98.4%

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

      7. associate--l+ 98.5%

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

      8. add-sqr-sqrt 98.4%

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

      9. fabs-sqr 98.4%

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

      10. add-sqr-sqrt 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. associate-+r- 98.4%

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

      2. add-exp-log 98.4%

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

      3. expm1-def 98.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 98.4%

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

      5. +-commutative 98.4%

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

   5. Applied egg-rr 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 4:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left(\mathsf{hypot}\left(1, x\right) + -1\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 4.0)
   (copysign (log1p (- (+ (hypot 1.0 x) -1.0) x)) x)
   (copysign (log (/ 0.5 x)) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= 4.0f) {
		tmp = copysignf(log1pf(((hypotf(1.0f, x) + -1.0f) - x)), x);
	} else {
		tmp = copysignf(logf((0.5f / 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(4.0))
		tmp = copysign(log1p(Float32(Float32(hypot(Float32(1.0), x) + Float32(-1.0)) - x)), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return 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) 1)))) x) < 4

   1. Initial program 36.2%

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

      1. log1p-expm1-u 36.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 36.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 36.1%

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

      4. +-commutative 36.1%

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

      5. metadata-eval 36.1%

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

      6. hypot-udef 50.5%

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

      7. associate--l+ 97.7%

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

      8. add-sqr-sqrt 36.6%

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

      9. fabs-sqr 36.6%

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

      10. add-sqr-sqrt 68.6%

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

   3. Applied egg-rr 68.6%

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

   5. Applied egg-rr 97.7%

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

   if 4 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 47.0%

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

      1. log1p-expm1-u 47.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 47.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 47.0%

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

      4. +-commutative 47.0%

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

      5. metadata-eval 47.0%

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

      6. hypot-udef 98.4%

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

      7. associate--l+ 98.4%

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

      8. add-sqr-sqrt 98.4%

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

      9. fabs-sqr 98.4%

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

      10. add-sqr-sqrt 98.4%

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

   3. Applied egg-rr 98.4%

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

   5. Applied egg-rr 8.6%

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

   6. Taylor expanded in x around inf 13.5%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot \frac{1}{x} - 1}\right), x\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 13.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 1\right), x\right) \]

      2. metadata-eval 13.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{x} - 1\right), x\right) \]

      3. sub-neg 13.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\frac{0.5}{x} + \left(-1\right)}\right), x\right) \]

      4. metadata-eval 13.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\frac{0.5}{x} + \color{blue}{-1}\right), x\right) \]

      5. +-commutative 13.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 + \frac{0.5}{x}}\right), x\right) \]

   8. Simplified 13.5%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 + \frac{0.5}{x}}\right), x\right) \]

   9. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 0.5 + -1 \cdot \log x}, x\right) \]
   10. Step-by-step derivation

       1. mul-1-neg 99.0%

          \[\leadsto \mathsf{copysign}\left(\log 0.5 + \color{blue}{\left(-\log x\right)}, x\right) \]

       2. log-rec 99.0%

          \[\leadsto \mathsf{copysign}\left(\log 0.5 + \color{blue}{\log \left(\frac{1}{x}\right)}, x\right) \]

       3. log-prod 99.6%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0.5 \cdot \frac{1}{x}\right)}, x\right) \]

       4. associate-*r/ 99.6%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5 \cdot 1}{x}\right)}, x\right) \]

       5. metadata-eval 99.6%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0.5}}{x}\right), x\right) \]

   11. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\frac{0.5}{x}\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 4:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left(\mathsf{hypot}\left(1, x\right) + -1\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(-1 - x\right) - \frac{0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -40.0)
   (copysign (log1p (- (- (- -1.0 x) (/ 0.5 x)) x)) x)
   (if (<= x 0.03999999910593033)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -40.0f) {
		tmp = copysignf(log1pf((((-1.0f - x) - (0.5f / x)) - x)), x);
	} else if (x <= 0.03999999910593033f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-40.0))
		tmp = copysign(log1p(Float32(Float32(Float32(Float32(-1.0) - x) - Float32(Float32(0.5) / x)) - x)), x);
	elseif (x <= Float32(0.03999999910593033))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -40

   1. Initial program 56.7%

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

      1. log1p-expm1-u 56.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 56.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 56.7%

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

      4. +-commutative 56.7%

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

      5. metadata-eval 56.7%

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

      6. hypot-udef 98.6%

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

      7. associate--l+ 98.6%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 13.2%

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

   3. Applied egg-rr 13.2%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in x around -inf 98.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(-1 \cdot x - \left(1 + 0.5 \cdot \frac{1}{x}\right)\right)} - x\right), x\right) \]
   7. Step-by-step derivation

      1. associate--r+ 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left(-1 \cdot x - 1\right) - 0.5 \cdot \frac{1}{x}\right)} - x\right), x\right) \]

      2. mul-1-neg 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(\color{blue}{\left(-x\right)} - 1\right) - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      3. sub-neg 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\left(\left(-x\right) + \left(-1\right)\right)} - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      4. metadata-eval 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(\left(-x\right) + \color{blue}{-1}\right) - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      5. +-commutative 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\left(-1 + \left(-x\right)\right)} - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      6. sub-neg 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\left(-1 - x\right)} - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      7. associate-*r/ 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(-1 - x\right) - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) - x\right), x\right) \]

      8. metadata-eval 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(-1 - x\right) - \frac{\color{blue}{0.5}}{x}\right) - x\right), x\right) \]

   8. Simplified 98.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left(-1 - x\right) - \frac{0.5}{x}\right)} - x\right), x\right) \]

   if -40 < x < 0.0399999991

   1. Initial program 21.9%

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

      1. log1p-expm1-u 21.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 21.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 21.9%

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

      4. +-commutative 21.9%

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

      5. metadata-eval 21.9%

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

      6. hypot-udef 22.0%

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

      7. associate--l+ 97.2%

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

      8. add-sqr-sqrt 53.5%

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

      9. fabs-sqr 53.5%

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

      10. add-sqr-sqrt 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + -0.16666666666666666 \cdot {x}^{3}}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 99.5%

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

   6. Simplified 99.5%

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

   if 0.0399999991 < x

   1. Initial program 51.7%

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

      1. log1p-expm1-u 51.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 51.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 51.7%

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

      4. +-commutative 51.7%

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

      5. metadata-eval 51.7%

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

      6. hypot-udef 98.4%

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

      7. associate--l+ 98.5%

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

      8. add-sqr-sqrt 98.4%

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

      9. fabs-sqr 98.4%

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

      10. add-sqr-sqrt 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. associate-+r- 98.4%

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

      2. add-exp-log 98.4%

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

      3. expm1-def 98.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 98.4%

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

      5. +-commutative 98.4%

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

   5. Applied egg-rr 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(-1 - x\right) - \frac{0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.4000000059604645)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

float code(float x) {
	float tmp;
	if (x <= -0.4000000059604645f) {
		tmp = copysignf(logf((hypotf(1.0f, x) - x)), x);
	} else {
		tmp = copysignf(log1pf((x + (hypotf(1.0f, x) + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.4000000059604645))
		tmp = copysign(log(Float32(hypot(Float32(1.0), x) - x)), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(hypot(Float32(1.0), x) + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -0.400000006

   1. Initial program 57.3%

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

      1. log1p-expm1-u 57.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 57.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 57.3%

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

      4. +-commutative 57.3%

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

      5. metadata-eval 57.3%

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

      6. hypot-udef 98.5%

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

      7. associate--l+ 98.5%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 14.4%

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

   3. Applied egg-rr 14.4%

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

   5. Applied egg-rr 98.5%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{hypot}\left(1, x\right) + -1\right) - x}\right), x\right) \]
   6. Step-by-step derivation

      1. log1p-udef 98.6%

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

      2. +-commutative 98.6%

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

      3. associate--l+ 98.6%

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

      4. associate-+r+ 98.6%

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

      5. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0 + \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. +-lft-identity 98.6%

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

   9. Simplified 98.6%

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

   if -0.400000006 < x

   1. Initial program 32.0%

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

      1. log1p-expm1-u 32.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 32.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 32.0%

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

      4. +-commutative 32.0%

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

      5. metadata-eval 32.0%

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

      6. hypot-udef 48.4%

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

      7. associate--l+ 97.7%

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

      8. add-sqr-sqrt 69.5%

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

      9. fabs-sqr 69.5%

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

      10. add-sqr-sqrt 97.7%

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

   3. Applied egg-rr 97.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 5: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -40.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.5)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -40.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf(((x * 2.0f) + (0.5f * (1.0f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-40.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-40.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log(((x * single(2.0)) + (single(0.5) * (single(1.0) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -40

   1. Initial program 56.7%

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

      1. log1p-expm1-u 56.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 56.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 56.7%

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

      4. +-commutative 56.7%

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

      5. metadata-eval 56.7%

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

      6. hypot-udef 98.6%

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

      7. associate--l+ 98.6%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 13.2%

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

   3. Applied egg-rr 13.2%

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

      1. associate-+r- 13.5%

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

      2. add-exp-log 13.5%

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

      3. expm1-def 13.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 13.5%

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

      5. +-commutative 13.5%

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

   5. Applied egg-rr 13.5%

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

   6. Taylor expanded in x around -inf 98.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right) \]

   if -40 < x < 0.5

   1. Initial program 23.1%

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

      1. log1p-expm1-u 23.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 23.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 23.1%

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

      4. +-commutative 23.1%

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

      5. metadata-eval 23.1%

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

      6. hypot-udef 23.2%

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

      7. associate--l+ 97.3%

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

      8. add-sqr-sqrt 54.2%

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

      9. fabs-sqr 54.2%

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

      10. add-sqr-sqrt 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + -0.16666666666666666 \cdot {x}^{3}}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.8%

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

   6. Simplified 98.8%

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

   if 0.5 < x

   1. Initial program 50.3%

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

      1. log1p-expm1-u 50.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 50.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 50.3%

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

      4. +-commutative 50.3%

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

      5. metadata-eval 50.3%

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

      6. hypot-udef 98.5%

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

      7. associate--l+ 98.5%

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

      8. add-sqr-sqrt 98.4%

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

      9. fabs-sqr 98.4%

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

      10. add-sqr-sqrt 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. associate-+r- 98.5%

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

      2. add-exp-log 98.5%

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

      3. expm1-def 98.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 98.5%

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

      5. +-commutative 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in x around inf 96.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(-1 - x\right) - \frac{0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -40.0)
   (copysign (log1p (- (- (- -1.0 x) (/ 0.5 x)) x)) x)
   (if (<= x 0.5)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -40.0f) {
		tmp = copysignf(log1pf((((-1.0f - x) - (0.5f / x)) - x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf(((x * 2.0f) + (0.5f * (1.0f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-40.0))
		tmp = copysign(log1p(Float32(Float32(Float32(Float32(-1.0) - x) - Float32(Float32(0.5) / x)) - x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -40

   1. Initial program 56.7%

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

      1. log1p-expm1-u 56.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 56.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 56.7%

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

      4. +-commutative 56.7%

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

      5. metadata-eval 56.7%

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

      6. hypot-udef 98.6%

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

      7. associate--l+ 98.6%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 13.2%

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

   3. Applied egg-rr 13.2%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in x around -inf 98.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(-1 \cdot x - \left(1 + 0.5 \cdot \frac{1}{x}\right)\right)} - x\right), x\right) \]
   7. Step-by-step derivation

      1. associate--r+ 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left(-1 \cdot x - 1\right) - 0.5 \cdot \frac{1}{x}\right)} - x\right), x\right) \]

      2. mul-1-neg 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(\color{blue}{\left(-x\right)} - 1\right) - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      3. sub-neg 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\left(\left(-x\right) + \left(-1\right)\right)} - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      4. metadata-eval 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(\left(-x\right) + \color{blue}{-1}\right) - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      5. +-commutative 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\left(-1 + \left(-x\right)\right)} - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      6. sub-neg 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\left(-1 - x\right)} - 0.5 \cdot \frac{1}{x}\right) - x\right), x\right) \]

      7. associate-*r/ 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(-1 - x\right) - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) - x\right), x\right) \]

      8. metadata-eval 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(-1 - x\right) - \frac{\color{blue}{0.5}}{x}\right) - x\right), x\right) \]

   8. Simplified 98.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left(-1 - x\right) - \frac{0.5}{x}\right)} - x\right), x\right) \]

   if -40 < x < 0.5

   1. Initial program 23.1%

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

      1. log1p-expm1-u 23.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 23.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 23.1%

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

      4. +-commutative 23.1%

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

      5. metadata-eval 23.1%

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

      6. hypot-udef 23.2%

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

      7. associate--l+ 97.3%

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

      8. add-sqr-sqrt 54.2%

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

      9. fabs-sqr 54.2%

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

      10. add-sqr-sqrt 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + -0.16666666666666666 \cdot {x}^{3}}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.8%

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

   6. Simplified 98.8%

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

   if 0.5 < x

   1. Initial program 50.3%

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

      1. log1p-expm1-u 50.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 50.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 50.3%

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

      4. +-commutative 50.3%

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

      5. metadata-eval 50.3%

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

      6. hypot-udef 98.5%

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

      7. associate--l+ 98.5%

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

      8. add-sqr-sqrt 98.4%

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

      9. fabs-sqr 98.4%

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

      10. add-sqr-sqrt 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. associate-+r- 98.5%

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

      2. add-exp-log 98.5%

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

      3. expm1-def 98.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 98.5%

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

      5. +-commutative 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in x around inf 96.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left(-1 - x\right) - \frac{0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \]

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -40.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.5)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -40.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-40.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-40.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -40

   1. Initial program 56.7%

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

      1. log1p-expm1-u 56.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 56.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 56.7%

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

      4. +-commutative 56.7%

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

      5. metadata-eval 56.7%

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

      6. hypot-udef 98.6%

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

      7. associate--l+ 98.6%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 13.2%

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

   3. Applied egg-rr 13.2%

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

      1. associate-+r- 13.5%

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

      2. add-exp-log 13.5%

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

      3. expm1-def 13.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 13.5%

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

      5. +-commutative 13.5%

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

   5. Applied egg-rr 13.5%

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

   6. Taylor expanded in x around -inf 98.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right) \]

   if -40 < x < 0.5

   1. Initial program 23.1%

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

      1. log1p-expm1-u 23.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 23.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 23.1%

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

      4. +-commutative 23.1%

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

      5. metadata-eval 23.1%

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

      6. hypot-udef 23.2%

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

      7. associate--l+ 97.3%

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

      8. add-sqr-sqrt 54.2%

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

      9. fabs-sqr 54.2%

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

      10. add-sqr-sqrt 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + -0.16666666666666666 \cdot {x}^{3}}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.8%

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

   6. Simplified 98.8%

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

   if 0.5 < x

   1. Initial program 50.3%

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

      1. log1p-expm1-u 50.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 50.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 50.3%

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

      4. +-commutative 50.3%

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

      5. metadata-eval 50.3%

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

      6. hypot-udef 98.5%

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

      7. associate--l+ 98.5%

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

      8. add-sqr-sqrt 98.4%

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

      9. fabs-sqr 98.4%

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

      10. add-sqr-sqrt 98.5%

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

   3. Applied egg-rr 98.5%

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

   5. Applied egg-rr 14.0%

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

   6. Taylor expanded in x around inf 15.7%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot \frac{1}{x} - 1}\right), x\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 1\right), x\right) \]

      2. metadata-eval 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{x} - 1\right), x\right) \]

      3. sub-neg 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\frac{0.5}{x} + \left(-1\right)}\right), x\right) \]

      4. metadata-eval 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\frac{0.5}{x} + \color{blue}{-1}\right), x\right) \]

      5. +-commutative 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 + \frac{0.5}{x}}\right), x\right) \]

   8. Simplified 15.7%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 + \frac{0.5}{x}}\right), x\right) \]

   9. Taylor expanded in x around 0 95.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 0.5 + -1 \cdot \log x}, x\right) \]
   10. Step-by-step derivation

       1. mul-1-neg 95.8%

          \[\leadsto \mathsf{copysign}\left(\log 0.5 + \color{blue}{\left(-\log x\right)}, x\right) \]

       2. log-rec 95.8%

          \[\leadsto \mathsf{copysign}\left(\log 0.5 + \color{blue}{\log \left(\frac{1}{x}\right)}, x\right) \]

       3. log-prod 96.4%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0.5 \cdot \frac{1}{x}\right)}, x\right) \]

       4. associate-*r/ 96.4%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5 \cdot 1}{x}\right)}, x\right) \]

       5. metadata-eval 96.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0.5}}{x}\right), x\right) \]

   11. Simplified 96.4%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\frac{0.5}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 8: 81.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -40.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.9999999949504854e-6)
     (copysign (log1p (- x)) x)
     (copysign (log (+ x 1.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -40.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.9999999949504854e-6f) {
		tmp = copysignf(log1pf(-x), x);
	} else {
		tmp = copysignf(logf((x + 1.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-40.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.9999999949504854e-6))
		tmp = copysign(log1p(Float32(-x)), x);
	else
		tmp = copysign(log(Float32(x + Float32(1.0))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -40

   1. Initial program 56.7%

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

      1. log1p-expm1-u 56.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 56.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 56.7%

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

      4. +-commutative 56.7%

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

      5. metadata-eval 56.7%

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

      6. hypot-udef 98.6%

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

      7. associate--l+ 98.6%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 13.2%

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

   3. Applied egg-rr 13.2%

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

      1. associate-+r- 13.5%

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

      2. add-exp-log 13.5%

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

      3. expm1-def 13.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 13.5%

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

      5. +-commutative 13.5%

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

   5. Applied egg-rr 13.5%

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

   6. Taylor expanded in x around -inf 98.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right) \]

   if -40 < x < 1.99999999e-6

   1. Initial program 18.4%

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

      1. log1p-expm1-u 18.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 18.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 18.4%

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

      4. +-commutative 18.4%

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

      5. metadata-eval 18.4%

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

      6. hypot-udef 18.4%

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

      7. associate--l+ 98.0%

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

      8. add-sqr-sqrt 51.6%

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

      9. fabs-sqr 51.6%

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

      10. add-sqr-sqrt 98.0%

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

   3. Applied egg-rr 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in x around 0 96.3%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 \cdot x}\right), x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 96.3%

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

   8. Simplified 96.3%

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

   if 1.99999999e-6 < x

   1. Initial program 54.4%

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

      1. log1p-expm1-u 54.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 54.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 54.4%

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

      4. +-commutative 54.4%

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

      5. metadata-eval 54.4%

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

      6. hypot-udef 96.8%

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

      7. associate--l+ 97.1%

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

      8. add-sqr-sqrt 97.1%

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

      9. fabs-sqr 97.1%

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

      10. add-sqr-sqrt 97.1%

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

   3. Applied egg-rr 97.1%

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

      1. associate-+r- 96.8%

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

      2. add-exp-log 96.8%

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

      3. expm1-def 96.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 96.8%

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

      5. +-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in x around 0 45.4%

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

      1. +-commutative 45.4%

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

   8. Simplified 45.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\ \end{array} \]

Alternative 9: 95.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -40.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.5) (copysign (log1p (- x)) x) (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -40.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf(log1pf(-x), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-40.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(log1p(Float32(-x)), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -40

   1. Initial program 56.7%

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

      1. log1p-expm1-u 56.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 56.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 56.7%

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

      4. +-commutative 56.7%

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

      5. metadata-eval 56.7%

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

      6. hypot-udef 98.6%

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

      7. associate--l+ 98.6%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 13.2%

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

   3. Applied egg-rr 13.2%

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

      1. associate-+r- 13.5%

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

      2. add-exp-log 13.5%

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

      3. expm1-def 13.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 13.5%

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

      5. +-commutative 13.5%

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

   5. Applied egg-rr 13.5%

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

   6. Taylor expanded in x around -inf 98.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right) \]

   if -40 < x < 0.5

   1. Initial program 23.1%

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

      1. log1p-expm1-u 23.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 23.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 23.1%

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

      4. +-commutative 23.1%

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

      5. metadata-eval 23.1%

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

      6. hypot-udef 23.2%

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

      7. associate--l+ 97.3%

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

      8. add-sqr-sqrt 54.2%

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

      9. fabs-sqr 54.2%

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

      10. add-sqr-sqrt 97.2%

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

   3. Applied egg-rr 97.2%

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

   5. Applied egg-rr 97.2%

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

   6. Taylor expanded in x around 0 93.1%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 \cdot x}\right), x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 93.1%

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

   8. Simplified 93.1%

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

   if 0.5 < x

   1. Initial program 50.3%

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

      1. log1p-expm1-u 50.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 50.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 50.3%

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

      4. +-commutative 50.3%

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

      5. metadata-eval 50.3%

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

      6. hypot-udef 98.5%

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

      7. associate--l+ 98.5%

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

      8. add-sqr-sqrt 98.4%

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

      9. fabs-sqr 98.4%

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

      10. add-sqr-sqrt 98.5%

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

   3. Applied egg-rr 98.5%

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

   5. Applied egg-rr 14.0%

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

   6. Taylor expanded in x around inf 15.7%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot \frac{1}{x} - 1}\right), x\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 1\right), x\right) \]

      2. metadata-eval 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{x} - 1\right), x\right) \]

      3. sub-neg 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\frac{0.5}{x} + \left(-1\right)}\right), x\right) \]

      4. metadata-eval 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\frac{0.5}{x} + \color{blue}{-1}\right), x\right) \]

      5. +-commutative 15.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 + \frac{0.5}{x}}\right), x\right) \]

   8. Simplified 15.7%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 + \frac{0.5}{x}}\right), x\right) \]

   9. Taylor expanded in x around 0 95.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 0.5 + -1 \cdot \log x}, x\right) \]
   10. Step-by-step derivation

       1. mul-1-neg 95.8%

          \[\leadsto \mathsf{copysign}\left(\log 0.5 + \color{blue}{\left(-\log x\right)}, x\right) \]

       2. log-rec 95.8%

          \[\leadsto \mathsf{copysign}\left(\log 0.5 + \color{blue}{\log \left(\frac{1}{x}\right)}, x\right) \]

       3. log-prod 96.4%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0.5 \cdot \frac{1}{x}\right)}, x\right) \]

       4. associate-*r/ 96.4%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5 \cdot 1}{x}\right)}, x\right) \]

       5. metadata-eval 96.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0.5}}{x}\right), x\right) \]

   11. Simplified 96.4%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\frac{0.5}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 10: 68.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 1.9999999949504854e-6)
   (copysign (log1p (- x)) x)
   (copysign (log (+ x 1.0)) x)))

C

float code(float x) {
	float tmp;
	if (x <= 1.9999999949504854e-6f) {
		tmp = copysignf(log1pf(-x), x);
	} else {
		tmp = copysignf(logf((x + 1.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999949504854e-6))
		tmp = copysign(log1p(Float32(-x)), x);
	else
		tmp = copysign(log(Float32(x + Float32(1.0))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-6

   1. Initial program 32.4%

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

      1. log1p-expm1-u 32.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 32.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 32.4%

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

      4. +-commutative 32.4%

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

      5. metadata-eval 32.4%

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

      6. hypot-udef 47.7%

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

      7. associate--l+ 98.2%

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

      8. add-sqr-sqrt 32.7%

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

      9. fabs-sqr 32.7%

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

      10. add-sqr-sqrt 67.0%

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

   3. Applied egg-rr 67.0%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in x around 0 77.1%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 \cdot x}\right), x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 77.1%

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

   8. Simplified 77.1%

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

   if 1.99999999e-6 < x

   1. Initial program 54.4%

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

      1. log1p-expm1-u 54.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 54.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 54.4%

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

      4. +-commutative 54.4%

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

      5. metadata-eval 54.4%

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

      6. hypot-udef 96.8%

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

      7. associate--l+ 97.1%

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

      8. add-sqr-sqrt 97.1%

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

      9. fabs-sqr 97.1%

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

      10. add-sqr-sqrt 97.1%

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

   3. Applied egg-rr 97.1%

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

      1. associate-+r- 96.8%

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

      2. add-exp-log 96.8%

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

      3. expm1-def 96.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

      4. log1p-expm1-u 96.8%

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

      5. +-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in x around 0 45.4%

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

      1. +-commutative 45.4%

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

   8. Simplified 45.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + 1\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\ \end{array} \]

Alternative 11: 57.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (log1p (- x)) x))

C

float code(float x) {
	return copysignf(log1pf(-x), x);
}

Julia

function code(x)
	return copysign(log1p(Float32(-x)), x)
end

Derivation

1. Initial program 38.7%

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

   1. log1p-expm1-u 38.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

   2. expm1-udef 38.7%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

   3. add-exp-log 38.7%

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

   4. +-commutative 38.7%

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

   5. metadata-eval 38.7%

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

   6. hypot-udef 61.7%

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

   7. associate--l+ 97.9%

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

   8. add-sqr-sqrt 51.1%

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

   9. fabs-sqr 51.1%

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

   10. add-sqr-sqrt 75.6%

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

3. Applied egg-rr 75.6%

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

5. Applied egg-rr 76.8%

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

6. Taylor expanded in x around 0 56.9%

   \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{-1 \cdot x}\right), x\right) \]
7. Step-by-step derivation

   1. mul-1-neg 56.9%

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

8. Simplified 56.9%

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

9. Final simplification 56.9%

   \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right) \]

Alternative 12: 8.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(0, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 0.0 x))

C

float code(float x) {
	return copysignf(0.0f, x);
}

Julia

function code(x)
	return copysign(Float32(0.0), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(0.0));
end

Derivation

1. Initial program 38.7%

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

   1. log1p-expm1-u 38.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

   2. expm1-udef 38.7%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

   3. add-exp-log 38.7%

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

   4. +-commutative 38.7%

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

   5. metadata-eval 38.7%

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

   6. hypot-udef 61.7%

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

   7. associate--l+ 97.9%

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

   8. add-sqr-sqrt 51.1%

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

   9. fabs-sqr 51.1%

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

   10. add-sqr-sqrt 75.6%

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

3. Applied egg-rr 75.6%

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

   1. associate-+r- 39.5%

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

   2. add-exp-log 39.5%

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

   3. expm1-def 39.5%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]

   4. log1p-expm1-u 39.5%

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

   5. +-commutative 39.5%

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

5. Applied egg-rr 39.5%

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

6. Taylor expanded in x around 0 8.3%

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

7. Final simplification 8.3%

   \[\leadsto \mathsf{copysign}\left(0, x\right) \]

Developer target: 99.5% accurate, 0.6× 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 2023305 
(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))