Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 14.0s

Alternatives: 9

Speedup: 1.1×

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: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{-s}\\ \frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(e^{t\_0}\right), -2, t\_0\right)}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) (- s))))
   (/ (exp (fma (log1p (exp t_0)) -2.0 t_0)) s)))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / -s;
	return expf(fmaf(log1pf(expf(t_0)), -2.0f, t_0)) / s;
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / Float32(-s))
	return Float32(exp(fma(log1p(exp(t_0)), Float32(-2.0), t_0)) / s)
end

Derivation

1. Initial program 99.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 N/A

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

   4. pow2 N/A

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

   5. pow-lowering-pow.f32 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f32 N/A

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

   8. exp-lowering-exp.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. neg-lowering-neg.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. fabs-lowering-fabs.f32 99.5

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

4. Applied egg-rr 99.5%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f32 N/A

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{e^{\left(-\frac{\left|x\right|}{s}\right) - 2 \cdot \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right)}}{s}} \]
7. Step-by-step derivation

   1. exp-lowering-exp.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

8. Applied egg-rr 99.6%

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

Alternative 2: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := \frac{\left|x\right|}{s}\\ t_2 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_2 \cdot \left(s \cdot t\_2\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, t\_1, \mathsf{fma}\left(t\_1, \left(t\_1 + 1\right) \cdot -4, 4\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (/ (fabs x) s)) (t_2 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_2 (* s t_2))) 1.999999987845058e-8)
     (/ t_0 s)
     (/
      1.0
      (*
       s
       (fma
        (/ x s)
        (* (/ x s) 5.0)
        (fma 4.0 t_1 (fma t_1 (* (+ t_1 1.0) -4.0) 4.0))))))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = fabsf(x) / s;
	float t_2 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_2 * (s * t_2))) <= 1.999999987845058e-8f) {
		tmp = t_0 / s;
	} else {
		tmp = 1.0f / (s * fmaf((x / s), ((x / s) * 5.0f), fmaf(4.0f, t_1, fmaf(t_1, ((t_1 + 1.0f) * -4.0f), 4.0f))));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	t_1 = Float32(abs(x) / s)
	t_2 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_2 * Float32(s * t_2))) <= Float32(1.999999987845058e-8))
		tmp = Float32(t_0 / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * fma(Float32(x / s), Float32(Float32(x / s) * Float32(5.0)), fma(Float32(4.0), t_1, fma(t_1, Float32(Float32(t_1 + Float32(1.0)) * Float32(-4.0)), Float32(4.0))))));
	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))))) < 1.99999999e-8

   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. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f32 N/A

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

      4. pow2 N/A

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

      5. pow-lowering-pow.f32 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f32 N/A

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

      8. exp-lowering-exp.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. neg-lowering-neg.f32 N/A

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

      11. /-lowering-/.f32 N/A

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

      12. fabs-lowering-fabs.f32 99.7

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

   4. Applied egg-rr 99.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f32 N/A

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in s around 0

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

      1. neg-mul-1 N/A

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

      2. neg-lowering-neg.f32 N/A

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

      3. /-lowering-/.f32 N/A

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

      4. fabs-lowering-fabs.f32 99.7

         \[\leadsto \frac{e^{-\frac{\color{blue}{\left|x\right|}}{s}}}{s} \]

   9. Simplified 99.7%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{s} \]

   if 1.99999999e-8 < (/.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 99.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f32 N/A

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

      3. div-inv N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in s around inf

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

   7. Simplified 74.7%

      \[\leadsto \frac{1}{\color{blue}{s \cdot \mathsf{fma}\left(-4, \frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}, \mathsf{fma}\left(5, \frac{x \cdot x}{s \cdot s}, \frac{\left|x\right| \cdot 4}{s}\right) + 4\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\left(\left(5 \cdot \frac{x \cdot x}{s \cdot s} + \frac{\left|x\right| \cdot 4}{s}\right) + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)}} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\left(5 \cdot \frac{x \cdot x}{s \cdot s} + \left(\frac{\left|x\right| \cdot 4}{s} + 4\right)\right)} + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{1}{s \cdot \color{blue}{\left(5 \cdot \frac{x \cdot x}{s \cdot s} + \left(\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\frac{x \cdot x}{s \cdot s} \cdot 5} + \left(\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot 5 + \left(\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\frac{x}{s} \cdot \left(\frac{x}{s} \cdot 5\right)} + \left(\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)\right)} \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)}} \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x}{s} \cdot 5, \left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)} \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x}{s} \cdot 5}, \left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)} \]

      10. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x}{s}} \cdot 5, \left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)} \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)}\right)} \]

   9. Applied egg-rr 92.5%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(\left|x\right|, \frac{4}{s}, 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-+l+ N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\left|x\right| \cdot \frac{4}{s} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\frac{4}{s} \cdot \left|x\right|} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       3. div-inv N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\left(4 \cdot \frac{1}{s}\right)} \cdot \left|x\right| + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       4. associate-*l* N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{4 \cdot \left(\frac{1}{s} \cdot \left|x\right|\right)} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       5. associate-/r/ N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, 4 \cdot \color{blue}{\frac{1}{\frac{s}{\left|x\right|}}} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       6. clear-num N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, 4 \cdot \color{blue}{\frac{\left|x\right|}{s}} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       7. accelerator-lowering-fma.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\mathsf{fma}\left(4, \frac{\left|x\right|}{s}, 4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)}\right)} \]

       8. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \color{blue}{\frac{\left|x\right|}{s}}, 4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       9. fabs-lowering-fabs.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\color{blue}{\left|x\right|}}{s}, 4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       10. +-commutative N/A

           \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \color{blue}{-4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right) + 4}\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \color{blue}{\left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right) \cdot -4} + 4\right)\right)} \]

       12. associate-*l* N/A

           \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \color{blue}{\frac{\left|x\right|}{s} \cdot \left(\left(\frac{\left|x\right|}{s} + 1\right) \cdot -4\right)} + 4\right)\right)} \]

       13. accelerator-lowering-fma.f32 N/A

           \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \color{blue}{\mathsf{fma}\left(\frac{\left|x\right|}{s}, \left(\frac{\left|x\right|}{s} + 1\right) \cdot -4, 4\right)}\right)\right)} \]

   11. Applied egg-rr 92.7%

       \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \mathsf{fma}\left(\frac{\left|x\right|}{s}, \left(1 + \frac{\left|x\right|}{s}\right) \cdot -4, 4\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{\frac{\left|x\right|}{-s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \mathsf{fma}\left(\frac{\left|x\right|}{s}, \left(\frac{\left|x\right|}{s} + 1\right) \cdot -4, 4\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := \frac{\left|x\right|}{s}\\ t_2 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_2 \cdot \left(s \cdot t\_2\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, t\_1, \mathsf{fma}\left(t\_1, \left(t\_1 + 1\right) \cdot -4, 4\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (/ (fabs x) s)) (t_2 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_2 (* s t_2))) 1.999999987845058e-8)
     0.0
     (/
      1.0
      (*
       s
       (fma
        (/ x s)
        (* (/ x s) 5.0)
        (fma 4.0 t_1 (fma t_1 (* (+ t_1 1.0) -4.0) 4.0))))))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = fabsf(x) / s;
	float t_2 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_2 * (s * t_2))) <= 1.999999987845058e-8f) {
		tmp = 0.0f;
	} else {
		tmp = 1.0f / (s * fmaf((x / s), ((x / s) * 5.0f), fmaf(4.0f, t_1, fmaf(t_1, ((t_1 + 1.0f) * -4.0f), 4.0f))));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	t_1 = Float32(abs(x) / s)
	t_2 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_2 * Float32(s * t_2))) <= Float32(1.999999987845058e-8))
		tmp = Float32(0.0);
	else
		tmp = Float32(Float32(1.0) / Float32(s * fma(Float32(x / s), Float32(Float32(x / s) * Float32(5.0)), fma(Float32(4.0), t_1, fma(t_1, Float32(Float32(t_1 + Float32(1.0)) * Float32(-4.0)), Float32(4.0))))));
	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))))) < 1.99999999e-8

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Simplified 0.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), \frac{0.041666666666666664}{s \cdot \left(s \cdot s\right)}, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(-s\right)} - \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{s}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right) - \left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right)}{{s}^{3}}}}{s}\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}}{s}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{1}{24}}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      5. +-inverses 99.0

         \[\leadsto -\frac{\color{blue}{0}}{s} \]

   8. Simplified 99.0%

      \[\leadsto -\frac{\color{blue}{0}}{s} \]
   9. Step-by-step derivation

      1. div0 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{0}\right) \]

      2. metadata-eval 99.0

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{0} \]

   if 1.99999999e-8 < (/.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 99.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f32 N/A

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

      3. div-inv N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in s around inf

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

   7. Simplified 74.7%

      \[\leadsto \frac{1}{\color{blue}{s \cdot \mathsf{fma}\left(-4, \frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}, \mathsf{fma}\left(5, \frac{x \cdot x}{s \cdot s}, \frac{\left|x\right| \cdot 4}{s}\right) + 4\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\left(\left(5 \cdot \frac{x \cdot x}{s \cdot s} + \frac{\left|x\right| \cdot 4}{s}\right) + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)}} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\left(5 \cdot \frac{x \cdot x}{s \cdot s} + \left(\frac{\left|x\right| \cdot 4}{s} + 4\right)\right)} + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{1}{s \cdot \color{blue}{\left(5 \cdot \frac{x \cdot x}{s \cdot s} + \left(\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\frac{x \cdot x}{s \cdot s} \cdot 5} + \left(\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot 5 + \left(\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\frac{x}{s} \cdot \left(\frac{x}{s} \cdot 5\right)} + \left(\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)\right)} \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)}} \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x}{s} \cdot 5, \left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)} \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x}{s} \cdot 5}, \left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)} \]

      10. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x}{s}} \cdot 5, \left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)\right)} \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\left(\frac{\left|x\right| \cdot 4}{s} + 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}\right)}\right)} \]

   9. Applied egg-rr 92.5%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(\left|x\right|, \frac{4}{s}, 4\right) + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-+l+ N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\left|x\right| \cdot \frac{4}{s} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\frac{4}{s} \cdot \left|x\right|} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       3. div-inv N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\left(4 \cdot \frac{1}{s}\right)} \cdot \left|x\right| + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       4. associate-*l* N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{4 \cdot \left(\frac{1}{s} \cdot \left|x\right|\right)} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       5. associate-/r/ N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, 4 \cdot \color{blue}{\frac{1}{\frac{s}{\left|x\right|}}} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       6. clear-num N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, 4 \cdot \color{blue}{\frac{\left|x\right|}{s}} + \left(4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       7. accelerator-lowering-fma.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\mathsf{fma}\left(4, \frac{\left|x\right|}{s}, 4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)}\right)} \]

       8. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \color{blue}{\frac{\left|x\right|}{s}}, 4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       9. fabs-lowering-fabs.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\color{blue}{\left|x\right|}}{s}, 4 + -4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right)\right)\right)} \]

       10. +-commutative N/A

           \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \color{blue}{-4 \cdot \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right) + 4}\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \color{blue}{\left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right) \cdot -4} + 4\right)\right)} \]

       12. associate-*l* N/A

           \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \color{blue}{\frac{\left|x\right|}{s} \cdot \left(\left(\frac{\left|x\right|}{s} + 1\right) \cdot -4\right)} + 4\right)\right)} \]

       13. accelerator-lowering-fma.f32 N/A

           \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \color{blue}{\mathsf{fma}\left(\frac{\left|x\right|}{s}, \left(\frac{\left|x\right|}{s} + 1\right) \cdot -4, 4\right)}\right)\right)} \]

   11. Applied egg-rr 92.7%

       \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \color{blue}{\mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \mathsf{fma}\left(\frac{\left|x\right|}{s}, \left(1 + \frac{\left|x\right|}{s}\right) \cdot -4, 4\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s} \cdot 5, \mathsf{fma}\left(4, \frac{\left|x\right|}{s}, \mathsf{fma}\left(\frac{\left|x\right|}{s}, \left(\frac{\left|x\right|}{s} + 1\right) \cdot -4, 4\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := \frac{x \cdot x}{s}\\ t_2 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_2 \cdot \left(s \cdot t\_2\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{\mathsf{fma}\left(t\_1, -4, \mathsf{fma}\left(5, t\_1, 0\right)\right)}{s} - -4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (/ (* x x) s)) (t_2 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_2 (* s t_2))) 1.999999987845058e-8)
     0.0
     (/ 1.0 (* s (- (/ (fma t_1 -4.0 (fma 5.0 t_1 0.0)) s) -4.0))))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = (x * x) / s;
	float t_2 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_2 * (s * t_2))) <= 1.999999987845058e-8f) {
		tmp = 0.0f;
	} else {
		tmp = 1.0f / (s * ((fmaf(t_1, -4.0f, fmaf(5.0f, t_1, 0.0f)) / s) - -4.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	t_1 = Float32(Float32(x * x) / s)
	t_2 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_2 * Float32(s * t_2))) <= Float32(1.999999987845058e-8))
		tmp = Float32(0.0);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(fma(t_1, Float32(-4.0), fma(Float32(5.0), t_1, Float32(0.0))) / s) - Float32(-4.0))));
	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))))) < 1.99999999e-8

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Simplified 0.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), \frac{0.041666666666666664}{s \cdot \left(s \cdot s\right)}, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(-s\right)} - \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{s}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right) - \left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right)}{{s}^{3}}}}{s}\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}}{s}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{1}{24}}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      5. +-inverses 99.0

         \[\leadsto -\frac{\color{blue}{0}}{s} \]

   8. Simplified 99.0%

      \[\leadsto -\frac{\color{blue}{0}}{s} \]
   9. Step-by-step derivation

      1. div0 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{0}\right) \]

      2. metadata-eval 99.0

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{0} \]

   if 1.99999999e-8 < (/.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 99.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f32 N/A

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

      3. div-inv N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

   7. Simplified 90.1%

      \[\leadsto \frac{1}{\color{blue}{-s \cdot \left(\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, -4, \mathsf{fma}\left(5, \frac{x \cdot x}{s}, 0\right)\right)}{-s} + -4\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{\mathsf{fma}\left(\frac{x \cdot x}{s}, -4, \mathsf{fma}\left(5, \frac{x \cdot x}{s}, 0\right)\right)}{s} - -4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}\\ \end{array} \end{array} \]

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))) 1.999999987845058e-8)
     0.0
     (/ 1.0 (fma s 4.0 (/ (* x x) 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))) <= 1.999999987845058e-8f) {
		tmp = 0.0f;
	} else {
		tmp = 1.0f / fmaf(s, 4.0f, ((x * x) / s));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-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(1.999999987845058e-8))
		tmp = Float32(0.0);
	else
		tmp = Float32(Float32(1.0) / fma(s, Float32(4.0), Float32(Float32(x * x) / 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))))) < 1.99999999e-8

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Simplified 0.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), \frac{0.041666666666666664}{s \cdot \left(s \cdot s\right)}, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(-s\right)} - \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{s}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right) - \left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right)}{{s}^{3}}}}{s}\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}}{s}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{1}{24}}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      5. +-inverses 99.0

         \[\leadsto -\frac{\color{blue}{0}}{s} \]

   8. Simplified 99.0%

      \[\leadsto -\frac{\color{blue}{0}}{s} \]
   9. Step-by-step derivation

      1. div0 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{0}\right) \]

      2. metadata-eval 99.0

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{0} \]

   if 1.99999999e-8 < (/.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 99.0%

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

      1. clear-num N/A

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

      2. /-lowering-/.f32 N/A

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

      3. div-inv N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in s around inf

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

   7. Simplified 74.7%

      \[\leadsto \frac{1}{\color{blue}{s \cdot \mathsf{fma}\left(-4, \frac{\left|x\right|}{s} + \frac{x \cdot x}{s \cdot s}, \mathsf{fma}\left(5, \frac{x \cdot x}{s \cdot s}, \frac{\left|x\right| \cdot 4}{s}\right) + 4\right)}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-out N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{s \cdot \left(\frac{\left|x\right|}{s} \cdot \color{blue}{0} + 4\right) + \frac{{x}^{2}}{s}} \]

      4. mul0-rgt N/A

         \[\leadsto \frac{1}{s \cdot \left(\color{blue}{0} + 4\right) + \frac{{x}^{2}}{s}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{s \cdot \color{blue}{4} + \frac{{x}^{2}}{s}} \]

      6. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(s, 4, \frac{{x}^{2}}{s}\right)}} \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \color{blue}{\frac{{x}^{2}}{s}}\right)} \]

      8. unpow2 N/A

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

      9. *-lowering-*.f32 90.1

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

   10. Simplified 90.1%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

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))) 1.999999987845058e-8) 0.0 (/ 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))) <= 1.999999987845058e-8f) {
		tmp = 0.0f;
	} 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))) <= 1.999999987845058e-8) then
        tmp = 0.0e0
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-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(1.999999987845058e-8))
		tmp = Float32(0.0);
	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(1.999999987845058e-8))
		tmp = single(0.0);
	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))))) < 1.99999999e-8

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Simplified 0.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), \frac{0.041666666666666664}{s \cdot \left(s \cdot s\right)}, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(-s\right)} - \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{s}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right) - \left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right)}{{s}^{3}}}}{s}\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}}{s}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{1}{24}}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

      5. +-inverses 99.0

         \[\leadsto -\frac{\color{blue}{0}}{s} \]

   8. Simplified 99.0%

      \[\leadsto -\frac{\color{blue}{0}}{s} \]
   9. Step-by-step derivation

      1. div0 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{0}\right) \]

      2. metadata-eval 99.0

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{0} \]

   if 1.99999999e-8 < (/.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 99.0%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 88.4

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

   5. Simplified 88.4%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ \frac{t\_0}{\left(s \cdot \left(2 - \frac{\left|x\right|}{s}\right)\right) \cdot \left(t\_0 + 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))))
   (/ t_0 (* (* s (- 2.0 (/ (fabs x) s))) (+ t_0 1.0)))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	return t_0 / ((s * (2.0f - (fabsf(x) / s))) * (t_0 + 1.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((abs(x) / -s))
    code = t_0 / ((s * (2.0e0 - (abs(x) / s))) * (t_0 + 1.0e0))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	return Float32(t_0 / Float32(Float32(s * Float32(Float32(2.0) - Float32(abs(x) / s))) * Float32(t_0 + Float32(1.0))))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((abs(x) / -s));
	tmp = t_0 / ((s * (single(2.0) - (abs(x) / s))) * (t_0 + single(1.0)));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-lowering-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. --lowering--.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. fabs-lowering-fabs.f32 95.9

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

5. Simplified 95.9%

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

6. Final simplification 95.9%

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

Alternative 8: 94.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ \frac{t\_0}{\left(t\_0 + 1\right) \cdot \left(s \cdot 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s))))) (/ t_0 (* (+ t_0 1.0) (* s 2.0)))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	return t_0 / ((t_0 + 1.0f) * (s * 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((abs(x) / -s))
    code = t_0 / ((t_0 + 1.0e0) * (s * 2.0e0))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	return Float32(t_0 / Float32(Float32(t_0 + Float32(1.0)) * Float32(s * Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((abs(x) / -s));
	tmp = t_0 / ((t_0 + single(1.0)) * (s * single(2.0)));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 94.8

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

5. Simplified 94.8%

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

6. Final simplification 94.8%

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

Alternative 9: 72.9% accurate, 373.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.0)

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

5. Simplified 16.3%

   \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left|x\right| \cdot \left(x \cdot x\right), \frac{0.041666666666666664}{s \cdot \left(s \cdot s\right)}, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(-s\right)} - \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{s}} \]

6. Taylor expanded in s around 0

   \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right) - \left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right)}{{s}^{3}}}}{s}\right) \]
7. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}}{s}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{1}{24}}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\frac{\frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{3}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{3}}}{s}\right) \]

   5. +-inverses 74.5

      \[\leadsto -\frac{\color{blue}{0}}{s} \]

8. Simplified 74.5%

   \[\leadsto -\frac{\color{blue}{0}}{s} \]
9. Step-by-step derivation

   1. div0 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{0}\right) \]

   2. metadata-eval 74.5

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 74.5%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))))