Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 11.1s

Alternatives: 14

Speedup: 1.5×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ \frac{\frac{t\_0}{t\_0 + 1}}{\mathsf{fma}\left(s, t\_0, s\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s))))) (/ (/ t_0 (+ t_0 1.0)) (fma s t_0 s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	return (t_0 / (t_0 + 1.0f)) / fmaf(s, t_0, s);
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	return Float32(Float32(t_0 / Float32(t_0 + Float32(1.0))) / fma(s, t_0, s))
end

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-fabs.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   2. remove-double-neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   3. lift-neg.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left|x\right|\right)\right)}\right)\right)\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   4. remove-double-neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(s\right)\right)\right)}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   5. frac-2neg N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)}{\mathsf{neg}\left(s\right)}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   6. frac-2neg N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   8. lift-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{\left|x\right|}{s}}}{1 + e^{-\frac{\left|x\right|}{s}}}}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right)}} \]

5. Final simplification 99.8%

   \[\leadsto \frac{\frac{e^{-\frac{\left|x\right|}{s}}}{e^{-\frac{\left|x\right|}{s}} + 1}}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right)} \]
6. Add Preprocessing

Alternative 2: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 2, \left|x\right|\right), 0.5 \cdot \left(x \cdot x\right)\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0)
     (/ 0.5 (/ (fma s (fma s 2.0 (fabs x)) (* 0.5 (* x x))) s))
     (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = 0.5f / (fmaf(s, fmaf(s, 2.0f, fabsf(x)), (0.5f * (x * x))) / s);
	} else {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(Float32(0.5) / Float32(fma(s, fma(s, Float32(2.0), abs(x)), Float32(Float32(0.5) * Float32(x * x))) / s));
	else
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

      1. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      4. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      5. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      8. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      9. lift-+.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left|x\right| + \left(-1 \cdot \frac{-1 \cdot {\left(\left|x\right|\right)}^{2} + \left(\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + 2 \cdot \left|x\right|\right)}{s} - 2\right)\right)}} \]

   10. Simplified 73.7%

       \[\leadsto \frac{0.5}{\color{blue}{\left(\frac{\left|x\right| + \left(-\frac{-0.5 \cdot \left(x \cdot x\right)}{s}\right)}{-s} + -2\right) \cdot \left(-s\right)}} \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{2} + s \cdot \left(\left|x\right| + 2 \cdot s\right)}{s}}} \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{2} + s \cdot \left(\left|x\right| + 2 \cdot s\right)}{s}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{s \cdot \left(\left|x\right| + 2 \cdot s\right) + \frac{1}{2} \cdot {x}^{2}}}{s}} \]

       3. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\mathsf{fma}\left(s, \left|x\right| + 2 \cdot s, \frac{1}{2} \cdot {x}^{2}\right)}}{s}} \]

       4. +-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(s, \color{blue}{2 \cdot s + \left|x\right|}, \frac{1}{2} \cdot {x}^{2}\right)}{s}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(s, \color{blue}{s \cdot 2} + \left|x\right|, \frac{1}{2} \cdot {x}^{2}\right)}{s}} \]

       6. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(s, 2, \left|x\right|\right)}, \frac{1}{2} \cdot {x}^{2}\right)}{s}} \]

       7. lower-fabs.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 2, \color{blue}{\left|x\right|}\right), \frac{1}{2} \cdot {x}^{2}\right)}{s}} \]

       8. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 2, \left|x\right|\right), \color{blue}{\frac{1}{2} \cdot {x}^{2}}\right)}{s}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 2, \left|x\right|\right), \frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)}{s}} \]

       10. lower-*.f32 63.1

           \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 2, \left|x\right|\right), 0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right)}{s}} \]

   13. Simplified 63.1%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 2, \left|x\right|\right), 0.5 \cdot \left(x \cdot x\right)\right)}{s}}} \]

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   5. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s}}{s}} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s}}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s}}{s} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{\color{blue}{s \cdot s}}}{s} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}}}{s} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}}{s} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}} + \frac{1}{4}}{s} \]

      7. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s} + \frac{1}{4}}{s} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}{s} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \frac{-1}{16}\right)}}{s \cdot s} + \frac{1}{4}}{s} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{x \cdot \left(x \cdot \frac{-1}{16}\right)}{\color{blue}{s \cdot s}} + \frac{1}{4}}{s} \]

      11. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{s} \cdot \frac{x \cdot \frac{-1}{16}}{s}} + \frac{1}{4}}{s} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}}{s} \]

      13. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}{s} \]

      14. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x \cdot \frac{-1}{16}}{s}}, \frac{1}{4}\right)}{s} \]

      15. lower-*.f32 91.9

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{\color{blue}{x \cdot -0.0625}}{s}, 0.25\right)}{s} \]

   7. Applied egg-rr 91.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 2, \left|x\right|\right), 0.5 \cdot \left(x \cdot x\right)\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right| \cdot s\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0)
     (/ 0.5 (/ (fma x (* x 0.5) (* (fabs x) s)) s))
     (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = 0.5f / (fmaf(x, (x * 0.5f), (fabsf(x) * s)) / s);
	} else {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(Float32(0.5) / Float32(fma(x, Float32(x * Float32(0.5)), Float32(abs(x) * s)) / s));
	else
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

      1. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      4. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      5. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      8. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      9. lift-+.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left|x\right| + \left(-1 \cdot \frac{-1 \cdot {\left(\left|x\right|\right)}^{2} + \left(\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + 2 \cdot \left|x\right|\right)}{s} - 2\right)\right)}} \]

   10. Simplified 73.7%

       \[\leadsto \frac{0.5}{\color{blue}{\left(\frac{\left|x\right| + \left(-\frac{-0.5 \cdot \left(x \cdot x\right)}{s}\right)}{-s} + -2\right) \cdot \left(-s\right)}} \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{2} + s \cdot \left|x\right|}{s}}} \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{2} + s \cdot \left|x\right|}{s}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{{x}^{2} \cdot \frac{1}{2}} + s \cdot \left|x\right|}{s}} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{2} + s \cdot \left|x\right|}{s}} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{x \cdot \left(x \cdot \frac{1}{2}\right)} + s \cdot \left|x\right|}{s}} \]

       5. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, s \cdot \left|x\right|\right)}}{s}} \]

       6. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, s \cdot \left|x\right|\right)}{s}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, \color{blue}{\left|x\right| \cdot s}\right)}{s}} \]

       8. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, \color{blue}{\left|x\right| \cdot s}\right)}{s}} \]

       9. lower-fabs.f32 63.1

          \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, \color{blue}{\left|x\right|} \cdot s\right)}{s}} \]

   13. Simplified 63.1%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right| \cdot s\right)}{s}}} \]

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   5. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s}}{s}} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s}}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s}}{s} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{\color{blue}{s \cdot s}}}{s} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}}}{s} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}}{s} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}} + \frac{1}{4}}{s} \]

      7. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s} + \frac{1}{4}}{s} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}{s} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \frac{-1}{16}\right)}}{s \cdot s} + \frac{1}{4}}{s} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{x \cdot \left(x \cdot \frac{-1}{16}\right)}{\color{blue}{s \cdot s}} + \frac{1}{4}}{s} \]

      11. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{s} \cdot \frac{x \cdot \frac{-1}{16}}{s}} + \frac{1}{4}}{s} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}}{s} \]

      13. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}{s} \]

      14. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x \cdot \frac{-1}{16}}{s}}, \frac{1}{4}\right)}{s} \]

      15. lower-*.f32 91.9

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{\color{blue}{x \cdot -0.0625}}{s}, 0.25\right)}{s} \]

   7. Applied egg-rr 91.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right| \cdot s\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right| \cdot s\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0)
     (/ 0.5 (/ (fma x (* x 0.5) (* (fabs x) s)) s))
     (/ 0.25 s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = 0.5f / (fmaf(x, (x * 0.5f), (fabsf(x) * s)) / s);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(Float32(0.5) / Float32(fma(x, Float32(x * Float32(0.5)), Float32(abs(x) * s)) / s));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

      1. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      4. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      5. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      8. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      9. lift-+.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left|x\right| + \left(-1 \cdot \frac{-1 \cdot {\left(\left|x\right|\right)}^{2} + \left(\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + 2 \cdot \left|x\right|\right)}{s} - 2\right)\right)}} \]

   10. Simplified 73.7%

       \[\leadsto \frac{0.5}{\color{blue}{\left(\frac{\left|x\right| + \left(-\frac{-0.5 \cdot \left(x \cdot x\right)}{s}\right)}{-s} + -2\right) \cdot \left(-s\right)}} \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{2} + s \cdot \left|x\right|}{s}}} \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{2} + s \cdot \left|x\right|}{s}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{{x}^{2} \cdot \frac{1}{2}} + s \cdot \left|x\right|}{s}} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{2} + s \cdot \left|x\right|}{s}} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{x \cdot \left(x \cdot \frac{1}{2}\right)} + s \cdot \left|x\right|}{s}} \]

       5. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, s \cdot \left|x\right|\right)}}{s}} \]

       6. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, s \cdot \left|x\right|\right)}{s}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, \color{blue}{\left|x\right| \cdot s}\right)}{s}} \]

       8. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, \color{blue}{\left|x\right| \cdot s}\right)}{s}} \]

       9. lower-fabs.f32 63.1

          \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, \color{blue}{\left|x\right|} \cdot s\right)}{s}} \]

   13. Simplified 63.1%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right| \cdot s\right)}{s}}} \]

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 89.3

         \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right| \cdot s\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\frac{0.5 \cdot \left(x \cdot x\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0)
     (/ 0.5 (/ (* 0.5 (* x x)) s))
     (/ 0.25 s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = 0.5f / ((0.5f * (x * x)) / s);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp(-(abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / (t_1 * (s * t_1))) <= 0.0e0) then
        tmp = 0.5e0 / ((0.5e0 * (x * x)) / s)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(Float32(0.5) / Float32(Float32(Float32(0.5) * Float32(x * x)) / s));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp(-(abs(x) / s));
	t_1 = t_0 + single(1.0);
	tmp = single(0.0);
	if ((t_0 / (t_1 * (s * t_1))) <= single(0.0))
		tmp = single(0.5) / ((single(0.5) * (x * x)) / s);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

      1. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      4. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      5. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      8. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      9. lift-+.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left|x\right| + \left(-1 \cdot \frac{-1 \cdot {\left(\left|x\right|\right)}^{2} + \left(\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + 2 \cdot \left|x\right|\right)}{s} - 2\right)\right)}} \]

   10. Simplified 73.7%

       \[\leadsto \frac{0.5}{\color{blue}{\left(\frac{\left|x\right| + \left(-\frac{-0.5 \cdot \left(x \cdot x\right)}{s}\right)}{-s} + -2\right) \cdot \left(-s\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{s}}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{2}}{s}}} \]

       2. lower-/.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{2}}{s}}} \]

       3. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\frac{1}{2} \cdot {x}^{2}}}{s}} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}}{s}} \]

       5. lower-*.f32 63.1

          \[\leadsto \frac{0.5}{\frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s}} \]

   13. Simplified 63.1%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{0.5 \cdot \left(x \cdot x\right)}{s}}} \]

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 89.3

         \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\frac{0.5 \cdot \left(x \cdot x\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0) (/ s (* x x)) (/ 0.25 s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = s / (x * x);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp(-(abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / (t_1 * (s * t_1))) <= 0.0e0) then
        tmp = s / (x * x)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(s / Float32(x * x));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp(-(abs(x) / s));
	t_1 = t_0 + single(1.0);
	tmp = single(0.0);
	if ((t_0 / (t_1 * (s * t_1))) <= single(0.0))
		tmp = s / (x * x);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

      1. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      4. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      5. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      8. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      9. lift-+.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left|x\right| + \left(-1 \cdot \frac{-1 \cdot {\left(\left|x\right|\right)}^{2} + \left(\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + 2 \cdot \left|x\right|\right)}{s} - 2\right)\right)}} \]

   10. Simplified 73.7%

       \[\leadsto \frac{0.5}{\color{blue}{\left(\frac{\left|x\right| + \left(-\frac{-0.5 \cdot \left(x \cdot x\right)}{s}\right)}{-s} + -2\right) \cdot \left(-s\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{s}{\color{blue}{x \cdot x}} \]

       3. lower-*.f32 60.4

          \[\leadsto \frac{s}{\color{blue}{x \cdot x}} \]

   13. Simplified 60.4%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 89.3

         \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0) (/ 0.5 (fabs x)) (/ 0.25 s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = 0.5f / fabsf(x);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp(-(abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / (t_1 * (s * t_1))) <= 0.0e0) then
        tmp = 0.5e0 / abs(x)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(Float32(0.5) / abs(x));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp(-(abs(x) / s));
	t_1 = t_0 + single(1.0);
	tmp = single(0.0);
	if ((t_0 / (t_1 * (s * t_1))) <= single(0.0))
		tmp = single(0.5) / abs(x);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

      1. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      4. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      5. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      8. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      9. lift-+.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

   8. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{s \cdot \left(2 + \left(-1 \cdot \frac{\left|x\right|}{s} + 2 \cdot \frac{\left|x\right|}{s}\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{s \cdot \left(2 + \left(-1 \cdot \frac{\left|x\right|}{s} + 2 \cdot \frac{\left|x\right|}{s}\right)\right)}} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{\frac{1}{2}}{s \cdot \left(2 + \color{blue}{\frac{\left|x\right|}{s} \cdot \left(-1 + 2\right)}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2}}{s \cdot \left(2 + \frac{\left|x\right|}{s} \cdot \color{blue}{1}\right)} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{\frac{1}{2}}{s \cdot \left(2 + \color{blue}{\frac{\left|x\right|}{s}}\right)} \]

      5. lower-+.f32 N/A

         \[\leadsto \frac{\frac{1}{2}}{s \cdot \color{blue}{\left(2 + \frac{\left|x\right|}{s}\right)}} \]

      6. lower-/.f32 N/A

         \[\leadsto \frac{\frac{1}{2}}{s \cdot \left(2 + \color{blue}{\frac{\left|x\right|}{s}}\right)} \]

      7. lower-fabs.f32 42.9

         \[\leadsto \frac{0.5}{s \cdot \left(2 + \frac{\color{blue}{\left|x\right|}}{s}\right)} \]

   10. Simplified 42.9%

       \[\leadsto \frac{0.5}{\color{blue}{s \cdot \left(2 + \frac{\left|x\right|}{s}\right)}} \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left|x\right|}} \]
   12. Step-by-step derivation

       1. lower-fabs.f32 9.9

          \[\leadsto \frac{0.5}{\color{blue}{\left|x\right|}} \]

   13. Simplified 9.9%

       \[\leadsto \frac{0.5}{\color{blue}{\left|x\right|}} \]

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 89.3

         \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ \frac{t\_0}{\mathsf{fma}\left(t\_0, s, s\right) \cdot \left(t\_0 + 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s))))) (/ t_0 (* (fma t_0 s s) (+ t_0 1.0)))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	return t_0 / (fmaf(t_0, s, s) * (t_0 + 1.0f));
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	return Float32(t_0 / Float32(fma(t_0, s, s) * Float32(t_0 + Float32(1.0))))
end

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-fabs.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   2. remove-double-neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)}\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   3. lift-neg.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left|x\right|\right)\right)}\right)\right)\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   4. remove-double-neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(s\right)\right)\right)}}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   5. frac-2neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)}{\mathsf{neg}\left(s\right)}}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   6. frac-2neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   8. lift-exp.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(1 \cdot s + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} \cdot s\right)} \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   10. *-lft-identity N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(\color{blue}{s} + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} \cdot s\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   11. +-commutative N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} \cdot s + s\right)} \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   12. lower-fma.f32 99.8

       \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\mathsf{fma}\left(e^{\frac{-\left|x\right|}{s}}, s, s\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

   13. lift-/.f32 N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\mathsf{fma}\left(e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}, s, s\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   14. lift-neg.f32 N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\mathsf{fma}\left(e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}, s, s\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   15. distribute-frac-neg N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\mathsf{fma}\left(e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}, s, s\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   16. lower-neg.f32 N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\mathsf{fma}\left(e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}, s, s\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   17. lower-/.f32 99.8

       \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\mathsf{fma}\left(e^{-\color{blue}{\frac{\left|x\right|}{s}}}, s, s\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\mathsf{fma}\left(e^{-\frac{\left|x\right|}{s}}, s, s\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

5. Final simplification 99.8%

   \[\leadsto \frac{e^{-\frac{\left|x\right|}{s}}}{\mathsf{fma}\left(e^{-\frac{\left|x\right|}{s}}, s, s\right) \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)} \]
6. Add Preprocessing

Alternative 9: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{-1}{\mathsf{fma}\left(s, e^{t\_0}, s\right) \cdot \left(-1 - e^{-t\_0}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)))
   (/ -1.0 (* (fma s (exp t_0) s) (- -1.0 (exp (- t_0)))))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	return -1.0f / (fmaf(s, expf(t_0), s) * (-1.0f - expf(-t_0)));
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	return Float32(Float32(-1.0) / Float32(fma(s, exp(t_0), s) * Float32(Float32(-1.0) - exp(Float32(-t_0)))))
end

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-fabs.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   2. remove-double-neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   3. lift-neg.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left|x\right|\right)\right)}\right)\right)\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   4. remove-double-neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(s\right)\right)\right)}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   5. frac-2neg N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)}{\mathsf{neg}\left(s\right)}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   6. frac-2neg N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   8. lift-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{\left|x\right|}{s}}}{1 + e^{-\frac{\left|x\right|}{s}}}}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right)}} \]
5. Step-by-step derivation

   1. lift-fabs.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\color{blue}{\left|x\right|}}{s}\right)}}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\color{blue}{\frac{\left|x\right|}{s}}\right)}}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   3. lift-neg.f32 N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   4. lift-exp.f32 N/A

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   5. lift-fabs.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{1 + e^{\mathsf{neg}\left(\frac{\color{blue}{\left|x\right|}}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{1 + e^{\mathsf{neg}\left(\color{blue}{\frac{\left|x\right|}{s}}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   7. lift-neg.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   8. lift-exp.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{1 + \color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   9. lift-+.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{\color{blue}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\mathsf{fma}\left(2, \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right), \frac{\left|x\right|}{s}\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{2 \cdot \log \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) + \frac{\left|x\right|}{s}}}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{\left|x\right|}{s} + 2 \cdot \log \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)}}} \]

   2. exp-sum N/A

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{\left|x\right|}{s}} \cdot e^{2 \cdot \log \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot e^{\color{blue}{\log \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) \cdot 2}}} \]

   4. neg-mul-1 N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot e^{\log \left(1 + e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}}\right) \cdot 2}} \]

   5. exp-to-pow N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot \color{blue}{{\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}}} \]

   6. neg-mul-1 N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot {\left(1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}\right)}^{2}} \]

   7. neg-mul-1 N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot {\left(1 + e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}}\right)}^{2}} \]

9. Simplified 99.8%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + e^{\frac{\left|x\right|}{s}}\right) \cdot \left(1 + e^{-\frac{\left|x\right|}{s}}\right)}} \]

10. Taylor expanded in s around 0

    \[\leadsto \color{blue}{\frac{1}{s \cdot \left(\left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)\right)}} \]
11. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{1}{s \cdot \left(\left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)\right)}} \]

    2. associate-*r* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]

    3. distribute-rgt-in N/A

       \[\leadsto \frac{1}{\color{blue}{1 \cdot \left(s \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right) + e^{\frac{\left|x\right|}{s}} \cdot \left(s \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)}} \]

    4. *-lft-identity N/A

       \[\leadsto \frac{1}{\color{blue}{s \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)} + e^{\frac{\left|x\right|}{s}} \cdot \left(s \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)} \]

    5. associate-*l* N/A

       \[\leadsto \frac{1}{s \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) + \color{blue}{\left(e^{\frac{\left|x\right|}{s}} \cdot s\right) \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)}} \]

    6. *-commutative N/A

       \[\leadsto \frac{1}{s \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) + \color{blue}{\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)} \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)} \]

    7. distribute-rgt-in N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

    8. lower-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

12. Simplified 99.8%

    \[\leadsto \color{blue}{\frac{1}{\left(1 + e^{-\frac{\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

13. Final simplification 99.8%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right) \cdot \left(-1 - e^{-\frac{\left|x\right|}{s}}\right)} \]
14. Add Preprocessing

Alternative 10: 94.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{\left(e^{\frac{\left|x\right|}{s}} + 1\right) \cdot 2} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 s) (* (+ (exp (/ (fabs x) s)) 1.0) 2.0)))

C

float code(float x, float s) {
	return (1.0f / s) / ((expf((fabsf(x) / s)) + 1.0f) * 2.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / ((exp((abs(x) / s)) + 1.0e0) * 2.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(exp(Float32(abs(x) / s)) + Float32(1.0)) * Float32(2.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / s) / ((exp((abs(x) / s)) + single(1.0)) * single(2.0));
end

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-fabs.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   2. remove-double-neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   3. lift-neg.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left|x\right|\right)\right)}\right)\right)\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   4. remove-double-neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(s\right)\right)\right)}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   5. frac-2neg N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)}{\mathsf{neg}\left(s\right)}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   6. frac-2neg N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

   8. lift-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{\left|x\right|}{s}}}{1 + e^{-\frac{\left|x\right|}{s}}}}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right)}} \]
5. Step-by-step derivation

   1. lift-fabs.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\color{blue}{\left|x\right|}}{s}\right)}}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\color{blue}{\frac{\left|x\right|}{s}}\right)}}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   3. lift-neg.f32 N/A

      \[\leadsto \frac{\frac{e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   4. lift-exp.f32 N/A

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   5. lift-fabs.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{1 + e^{\mathsf{neg}\left(\frac{\color{blue}{\left|x\right|}}{s}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{1 + e^{\mathsf{neg}\left(\color{blue}{\frac{\left|x\right|}{s}}\right)}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   7. lift-neg.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   8. lift-exp.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{1 + \color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

   9. lift-+.f32 N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{\color{blue}{1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}}{s \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + s} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\mathsf{fma}\left(2, \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right), \frac{\left|x\right|}{s}\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{2 \cdot \log \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) + \frac{\left|x\right|}{s}}}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{\left|x\right|}{s} + 2 \cdot \log \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)}}} \]

   2. exp-sum N/A

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{\left|x\right|}{s}} \cdot e^{2 \cdot \log \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot e^{\color{blue}{\log \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right) \cdot 2}}} \]

   4. neg-mul-1 N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot e^{\log \left(1 + e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}}\right) \cdot 2}} \]

   5. exp-to-pow N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot \color{blue}{{\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}}} \]

   6. neg-mul-1 N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot {\left(1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}\right)}^{2}} \]

   7. neg-mul-1 N/A

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot {\left(1 + e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}}\right)}^{2}} \]

9. Simplified 99.8%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + e^{\frac{\left|x\right|}{s}}\right) \cdot \left(1 + e^{-\frac{\left|x\right|}{s}}\right)}} \]

10. Taylor expanded in s around inf

    \[\leadsto \frac{\frac{1}{s}}{\left(1 + e^{\frac{\left|x\right|}{s}}\right) \cdot \color{blue}{2}} \]
11. Step-by-step derivation

12. Simplified 95.4%

    \[\leadsto \frac{\frac{1}{s}}{\left(1 + e^{\frac{\left|x\right|}{s}}\right) \cdot \color{blue}{2}} \]

13. Final simplification 95.4%

    \[\leadsto \frac{\frac{1}{s}}{\left(e^{\frac{\left|x\right|}{s}} + 1\right) \cdot 2} \]
14. Add Preprocessing

Alternative 11: 95.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (fma s (exp (/ (fabs x) s)) s)))

C

float code(float x, float s) {
	return 0.5f / fmaf(s, expf((fabsf(x) / s)), s);
}

Julia

function code(x, s)
	return Float32(Float32(0.5) / fma(s, exp(Float32(abs(x) / s)), s))
end

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
4. Step-by-step derivation

5. Simplified 95.4%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
6. Step-by-step derivation

   1. lift-fabs.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

   2. lift-neg.f32 N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

   4. lift-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

   5. lift-fabs.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

   6. lift-neg.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

   8. lift-exp.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

   9. lift-+.f32 N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

   11. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

   12. div-inv N/A

       \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

7. Applied egg-rr 95.4%

   \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

8. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{2}}{\color{blue}{s \cdot \left(e^{\frac{\left|x\right|}{s}} \cdot \left(1 + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)}} \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{1}{2}}{s \cdot \color{blue}{\left(1 \cdot e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   2. exp-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{s \cdot \left(1 \cdot e^{\frac{\left|x\right|}{s}} + \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   3. lft-mult-inverse N/A

      \[\leadsto \frac{\frac{1}{2}}{s \cdot \left(1 \cdot e^{\frac{\left|x\right|}{s}} + \color{blue}{1}\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{\frac{1}{2}}{s \cdot \left(\color{blue}{e^{\frac{\left|x\right|}{s}}} + 1\right)} \]

   5. distribute-lft-in N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{s \cdot e^{\frac{\left|x\right|}{s}} + s \cdot 1}} \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{\frac{1}{2}}{s \cdot e^{\frac{\left|x\right|}{s}} + \color{blue}{s}} \]

   7. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   8. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{\mathsf{fma}\left(s, \color{blue}{e^{\frac{\left|x\right|}{s}}}, s\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{\mathsf{fma}\left(s, e^{\color{blue}{\frac{\left|x\right|}{s}}}, s\right)} \]

   10. lower-fabs.f32 95.4

       \[\leadsto \frac{0.5}{\mathsf{fma}\left(s, e^{\frac{\color{blue}{\left|x\right|}}{s}}, s\right)} \]

10. Simplified 95.4%

    \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]
11. Add Preprocessing

Alternative 12: 86.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \mathbf{elif}\;\left|x\right| \leq 0.05000000074505806:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\left(--2\right) - \left(x \cdot x\right) \cdot \frac{-0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\frac{\left|x\right| + \frac{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left|x\right|\right), 0.16666666666666666, 0\right)}{s} - \left(x \cdot x\right) \cdot -0.5}{s}}{s} - -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 2.0000000390829628e-24)
   (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s)
   (if (<= (fabs x) 0.05000000074505806)
     (/ 0.5 (* s (- (- -2.0) (* (* x x) (/ -0.5 (* s s))))))
     (/
      0.5
      (*
       s
       (-
        (/
         (+
          (fabs x)
          (/
           (-
            (/ (fma (* x (* x (fabs x))) 0.16666666666666666 0.0) s)
            (* (* x x) -0.5))
           s))
         s)
        -2.0))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 2.0000000390829628e-24f) {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	} else if (fabsf(x) <= 0.05000000074505806f) {
		tmp = 0.5f / (s * (-(-2.0f) - ((x * x) * (-0.5f / (s * s)))));
	} else {
		tmp = 0.5f / (s * (((fabsf(x) + (((fmaf((x * (x * fabsf(x))), 0.16666666666666666f, 0.0f) / s) - ((x * x) * -0.5f)) / s)) / s) - -2.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(2.0000000390829628e-24))
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	elseif (abs(x) <= Float32(0.05000000074505806))
		tmp = Float32(Float32(0.5) / Float32(s * Float32(Float32(-Float32(-2.0)) - Float32(Float32(x * x) * Float32(Float32(-0.5) / Float32(s * s))))));
	else
		tmp = Float32(Float32(0.5) / Float32(s * Float32(Float32(Float32(abs(x) + Float32(Float32(Float32(fma(Float32(x * Float32(x * abs(x))), Float32(0.16666666666666666), Float32(0.0)) / s) - Float32(Float32(x * x) * Float32(-0.5))) / s)) / s) - Float32(-2.0))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 2.00000004e-24

   1. Initial program 99.3%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   5. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s}}{s}} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s}}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s}}{s} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{\color{blue}{s \cdot s}}}{s} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}}}{s} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}}{s} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}} + \frac{1}{4}}{s} \]

      7. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s} + \frac{1}{4}}{s} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}{s} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \frac{-1}{16}\right)}}{s \cdot s} + \frac{1}{4}}{s} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{x \cdot \left(x \cdot \frac{-1}{16}\right)}{\color{blue}{s \cdot s}} + \frac{1}{4}}{s} \]

      11. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{s} \cdot \frac{x \cdot \frac{-1}{16}}{s}} + \frac{1}{4}}{s} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}}{s} \]

      13. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}{s} \]

      14. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x \cdot \frac{-1}{16}}{s}}, \frac{1}{4}\right)}{s} \]

      15. lower-*.f32 75.4

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{\color{blue}{x \cdot -0.0625}}{s}, 0.25\right)}{s} \]

   7. Applied egg-rr 75.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}}{s} \]

   if 2.00000004e-24 < (fabs.f32 x) < 0.0500000007

   1. Initial program 99.7%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Simplified 92.8%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

      1. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      4. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      5. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      8. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      9. lift-+.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

   7. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left|x\right| + \left(-1 \cdot \frac{-1 \cdot {\left(\left|x\right|\right)}^{2} + \left(\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + 2 \cdot \left|x\right|\right)}{s} - 2\right)\right)}} \]

   10. Simplified 52.1%

       \[\leadsto \frac{0.5}{\color{blue}{\left(\frac{\left|x\right| + \left(-\frac{-0.5 \cdot \left(x \cdot x\right)}{s}\right)}{-s} + -2\right) \cdot \left(-s\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\frac{-1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\frac{\frac{-1}{2} \cdot {x}^{2}}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\frac{\color{blue}{{x}^{2} \cdot \frac{-1}{2}}}{{s}^{2}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       3. associate-/l* N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{{x}^{2} \cdot \frac{\frac{-1}{2}}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{{s}^{2}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       5. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{s}^{2}}\right)\right)} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot 1}}{{s}^{2}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       7. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{{s}^{2}}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       8. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{{x}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right)} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       10. lower-*.f32 N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       11. associate-*r/ N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{{s}^{2}}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{{s}^{2}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       13. distribute-neg-frac N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \frac{\color{blue}{\frac{-1}{2}}}{{s}^{2}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       15. lower-/.f32 N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       16. unpow2 N/A

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

       17. lower-*.f32 84.3

           \[\leadsto \frac{0.5}{\left(\left(x \cdot x\right) \cdot \frac{-0.5}{\color{blue}{s \cdot s}} + -2\right) \cdot \left(-s\right)} \]

   13. Simplified 84.3%

       \[\leadsto \frac{0.5}{\left(\color{blue}{\left(x \cdot x\right) \cdot \frac{-0.5}{s \cdot s}} + -2\right) \cdot \left(-s\right)} \]

   if 0.0500000007 < (fabs.f32 x)

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

      1. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      4. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      5. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      8. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      9. lift-+.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left|x\right| + \left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{-1}{6} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{1}{3} \cdot {\left(\left|x\right|\right)}^{3} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{3}\right)\right)}{s} + \left(-1 \cdot {\left(\left|x\right|\right)}^{2} + \left(\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{s} + 2 \cdot \left|x\right|\right)}{s} - 2\right)\right)}} \]

   10. Simplified 94.2%

       \[\leadsto \frac{0.5}{\color{blue}{-s \cdot \left(\left(-\frac{\left|x\right| + \frac{-0.5 \cdot \left(x \cdot x\right) - \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left|x\right|\right), 0.16666666666666666, 0\right)}{s}}{-s}}{s}\right) + -2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \mathbf{elif}\;\left|x\right| \leq 0.05000000074505806:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\left(--2\right) - \left(x \cdot x\right) \cdot \frac{-0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\frac{\left|x\right| + \frac{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left|x\right|\right), 0.16666666666666666, 0\right)}{s} - \left(x \cdot x\right) \cdot -0.5}{s}}{s} - -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\left|x\right| \leq -1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\left(--2\right) - \left(x \cdot x\right) \cdot \frac{-0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (fabs x)) -1.9999999996399175e-23)
   (/ 0.5 (* s (- (- -2.0) (* (* x x) (/ -0.5 (* s s))))))
   (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s)))

C

float code(float x, float s) {
	float tmp;
	if (-fabsf(x) <= -1.9999999996399175e-23f) {
		tmp = 0.5f / (s * (-(-2.0f) - ((x * x) * (-0.5f / (s * s)))));
	} else {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-abs(x)) <= Float32(-1.9999999996399175e-23))
		tmp = Float32(Float32(0.5) / Float32(s * Float32(Float32(-Float32(-2.0)) - Float32(Float32(x * x) * Float32(Float32(-0.5) / Float32(s * s))))));
	else
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (fabs.f32 x)) < -2e-23

   1. Initial program 99.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Simplified 97.8%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

      1. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      2. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      3. lift-/.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      4. lift-exp.f32 N/A

         \[\leadsto \frac{\color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right) \cdot 2} \]

      5. lift-fabs.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left|x\right|}\right)}{s}}\right)\right) \cdot 2} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left|x\right|\right)}}{s}}\right)\right) \cdot 2} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      8. lift-exp.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \left(1 + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right) \cdot 2} \]

      9. lift-+.f32 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\left(s \cdot \color{blue}{\left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}\right) \cdot 2} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right)} \cdot 2} \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)}}{2}} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{s \cdot \left(1 + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)} \cdot \frac{1}{2}} \]

   7. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{\frac{0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right) \cdot e^{\frac{\left|x\right|}{s}}}} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{-1 \cdot \left(s \cdot \left(-1 \cdot \frac{-1 \cdot \left|x\right| + \left(-1 \cdot \frac{-1 \cdot {\left(\left|x\right|\right)}^{2} + \left(\frac{-1}{2} \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{s} + 2 \cdot \left|x\right|\right)}{s} - 2\right)\right)}} \]

   10. Simplified 77.6%

       \[\leadsto \frac{0.5}{\color{blue}{\left(\frac{\left|x\right| + \left(-\frac{-0.5 \cdot \left(x \cdot x\right)}{s}\right)}{-s} + -2\right) \cdot \left(-s\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\frac{-1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\frac{\frac{-1}{2} \cdot {x}^{2}}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\frac{\color{blue}{{x}^{2} \cdot \frac{-1}{2}}}{{s}^{2}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       3. associate-/l* N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{{x}^{2} \cdot \frac{\frac{-1}{2}}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{{s}^{2}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       5. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{s}^{2}}\right)\right)} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot 1}}{{s}^{2}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       7. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{{s}^{2}}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       8. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{{x}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right)} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       10. lower-*.f32 N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       11. associate-*r/ N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{{s}^{2}}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{{s}^{2}}\right)\right) + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       13. distribute-neg-frac N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \frac{\color{blue}{\frac{-1}{2}}}{{s}^{2}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       15. lower-/.f32 N/A

           \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{{s}^{2}}} + -2\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} \]

       16. unpow2 N/A

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

       17. lower-*.f32 87.3

           \[\leadsto \frac{0.5}{\left(\left(x \cdot x\right) \cdot \frac{-0.5}{\color{blue}{s \cdot s}} + -2\right) \cdot \left(-s\right)} \]

   13. Simplified 87.3%

       \[\leadsto \frac{0.5}{\left(\color{blue}{\left(x \cdot x\right) \cdot \frac{-0.5}{s \cdot s}} + -2\right) \cdot \left(-s\right)} \]

   if -2e-23 < (neg.f32 (fabs.f32 x))

   1. Initial program 99.3%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   5. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s}}{s}} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s}}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s}}{s} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{\color{blue}{s \cdot s}}}{s} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}}}{s} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}}{s} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}} + \frac{1}{4}}{s} \]

      7. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s} + \frac{1}{4}}{s} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}{s} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \frac{-1}{16}\right)}}{s \cdot s} + \frac{1}{4}}{s} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{x \cdot \left(x \cdot \frac{-1}{16}\right)}{\color{blue}{s \cdot s}} + \frac{1}{4}}{s} \]

      11. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{s} \cdot \frac{x \cdot \frac{-1}{16}}{s}} + \frac{1}{4}}{s} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}}{s} \]

      13. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}{s} \]

      14. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x \cdot \frac{-1}{16}}{s}}, \frac{1}{4}\right)}{s} \]

      15. lower-*.f32 75.9

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{\color{blue}{x \cdot -0.0625}}{s}, 0.25\right)}{s} \]

   7. Applied egg-rr 75.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\left|x\right| \leq -1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\left(--2\right) - \left(x \cdot x\right) \cdot \frac{-0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.4% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

float code(float x, float s) {
	return 0.25f / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / s;
end

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s}} \]
4. Step-by-step derivation

   1. lower-/.f32 27.3

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

5. Simplified 27.3%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))