Logistic distribution

Percentage Accurate: 99.4% → 99.5%

Time: 13.6s

Alternatives: 11

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: 99.4% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. associate-*l/ N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-pow.f32 N/A

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

   5. pow-to-exp N/A

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

   6. exp-prod N/A

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

   7. pow-flip N/A

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

   8. metadata-eval N/A

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

   9. exp-prod N/A

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

   10. pow-to-exp N/A

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

   11. lift-pow.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-/.f32 N/A

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

   15. div-inv N/A

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

6. Applied rewrites 99.7%

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

Alternative 2: 97.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

   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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around 0

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fabs.f32 99.8

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

   7. Applied rewrites 99.8%

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

   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 99.3%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 93.0%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 94.1%

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

4. Final simplification 98.0%

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

Alternative 3: 87.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

   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-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 69.4%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 82.7%

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

   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 99.3%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 93.0%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 94.1%

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

4. Final simplification 86.1%

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

Alternative 4: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 50:\\ \;\;\;\;\frac{\frac{1}{s}}{\mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s}, 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 50.0)
     (/ (/ 1.0 s) (fma x (/ x (* s s)) 4.0))
     (/ 1.0 (* s (fma (/ x s) (/ x s) 4.0))))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 50.0f) {
		tmp = (1.0f / s) / fmaf(x, (x / (s * s)), 4.0f);
	} else {
		tmp = 1.0f / (s * fmaf((x / s), (x / s), 4.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(50.0))
		tmp = Float32(Float32(Float32(1.0) / s) / fma(x, Float32(x / Float32(s * s)), Float32(4.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(s * fma(Float32(x / s), Float32(x / 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))))) < 50

   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-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 70.9%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{s \cdot \left(4 - \frac{\left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right)\right) + 0}{s}\right)}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{s \cdot \left(4 - \frac{\left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right)\right) + 0}{s}\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{4 - \frac{\left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right)\right) + 0}{s}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{4 - \frac{\left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right)\right) + 0}{s}}} \]

      5. lower-/.f32 70.9

         \[\leadsto \frac{\color{blue}{\frac{1}{s}}}{4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}} \]

   9. Applied rewrites 81.0%

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

   if 50 < (/.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. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 92.3%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(1 + e^{-\frac{\left|x\right|}{s}}\right) \cdot \left(s \cdot \left(1 + e^{-\frac{\left|x\right|}{s}}\right)\right)} \leq 50:\\ \;\;\;\;\frac{\frac{1}{s}}{\mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(\frac{x}{s}, \frac{x}{s}, 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

   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-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 69.4%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]

   8. Taylor expanded in s around -inf

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

   10. Applied rewrites 80.0%

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

   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 99.3%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 93.0%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 94.1%

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

4. Final simplification 84.3%

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

Alternative 6: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 7999999895928832:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 7999999895928832.0)
     (/ 1.0 (* s (fma x (/ x (* s s)) 4.0)))
     (/ (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 = 1.0f + t_0;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 7999999895928832.0f) {
		tmp = 1.0f / (s * fmaf(x, (x / (s * s)), 4.0f));
	} 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(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(7999999895928832.0))
		tmp = Float32(Float32(1.0) / Float32(s * fma(x, Float32(x / Float32(s * s)), Float32(4.0))));
	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))))) < 7.9999999e15

   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. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 75.0%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]

   8. Taylor expanded in s around -inf

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

   10. Applied rewrites 83.1%

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

   if 7.9999999e15 < (/.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. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 55.0%

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

   9. Applied rewrites 95.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(1 + e^{-\frac{\left|x\right|}{s}}\right) \cdot \left(s \cdot \left(1 + e^{-\frac{\left|x\right|}{s}}\right)\right)} \leq 7999999895928832:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 1.0000000200408773 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 1.0000000200408773e+21)
     (/ 1.0 (* s (fma x (/ x (* s s)) 4.0)))
     (/ 0.25 s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 1.0000000200408773e+21f) {
		tmp = 1.0f / (s * fmaf(x, (x / (s * s)), 4.0f));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(1.0000000200408773e+21))
		tmp = Float32(Float32(1.0) / Float32(s * fma(x, Float32(x / Float32(s * s)), Float32(4.0))));
	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))))) < 1.00000002e21

   1. Initial program 99.6%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 76.1%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]

   8. Taylor expanded in s around -inf

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

   10. Applied rewrites 83.9%

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

   if 1.00000002e21 < (/.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 99.1%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(1 + e^{-\frac{\left|x\right|}{s}}\right) \cdot \left(s \cdot \left(1 + e^{-\frac{\left|x\right|}{s}}\right)\right)} \leq 1.0000000200408773 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = 1.0f / (s * ((x * x) / (s * 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 = 1.0e0 + t_0
    if ((t_0 / (t_1 * (s * t_1))) <= 0.0e0) then
        tmp = 1.0e0 / (s * ((x * x) / (s * 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(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(x * x) / Float32(s * 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 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / (t_1 * (s * t_1))) <= single(0.0))
		tmp = single(1.0) / (s * ((x * x) / (s * 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 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-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 69.4%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\left(-\frac{\mathsf{fma}\left(x, x, 0\right)}{s}\right) + 0}{s}\right)}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 77.2%

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

   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 99.3%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 89.7

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

   5. Applied rewrites 89.7%

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

4. Final simplification 81.0%

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

Alternative 9: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\mathsf{fma}\left(-2, \log 2, \frac{x \cdot 0.25}{s} \cdot \frac{x}{-s}\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (fma -2.0 (log 2.0) (* (/ (* x 0.25) s) (/ x (- s))))) s))

C

float code(float x, float s) {
	return expf(fmaf(-2.0f, logf(2.0f), (((x * 0.25f) / s) * (x / -s)))) / s;
}

Julia

function code(x, s)
	return Float32(exp(fma(Float32(-2.0), log(Float32(2.0)), Float32(Float32(Float32(x * Float32(0.25)) / s) * Float32(x / Float32(-s))))) / s)
end

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. associate-*l/ N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-pow.f32 N/A

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

   5. pow-to-exp N/A

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

   6. exp-prod N/A

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

   7. pow-flip N/A

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

   8. metadata-eval N/A

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

   9. exp-prod N/A

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

   10. pow-to-exp N/A

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

   11. lift-pow.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-/.f32 N/A

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

   15. div-inv N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around inf

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

   1. lower-fma.f32 N/A

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

   2. lower-log.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. unpow2 N/A

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

   8. sqr-abs N/A

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

   9. unpow2 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f32 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f32 N/A

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

   21. unpow2 N/A

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

   22. lower-*.f32 92.0

       \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, -\frac{0.25 \cdot \left(x \cdot x\right)}{\color{blue}{s \cdot s}}\right)}}{s} \]

9. Applied rewrites 92.0%

   \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(-2, \log 2, -\frac{0.25 \cdot \left(x \cdot x\right)}{s \cdot s}\right)}}}{s} \]
10. Step-by-step derivation

11. Applied rewrites 98.1%

    \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, -\frac{x \cdot 0.25}{s} \cdot \frac{x}{s}\right)}}{s} \]

12. Final simplification 98.1%

    \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \frac{x \cdot 0.25}{s} \cdot \frac{x}{-s}\right)}}{s} \]
13. Add Preprocessing

Alternative 10: 94.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{\left|x\right|}{s}}}{s \cdot 4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (exp (- (/ (fabs x) s))) (* s 4.0)))

C

float code(float x, float s) {
	return expf(-(fabsf(x) / s)) / (s * 4.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp(-(abs(x) / s)) / (s * 4.0e0)
end function

Julia

function code(x, s)
	return Float32(exp(Float32(-Float32(abs(x) / s))) / Float32(s * Float32(4.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = exp(-(abs(x) / s)) / (s * single(4.0));
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 95.0

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot 4}} \]

5. Applied rewrites 95.0%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot 4}} \]

6. Final simplification 95.0%

   \[\leadsto \frac{e^{-\frac{\left|x\right|}{s}}}{s \cdot 4} \]
7. Add Preprocessing

Alternative 11: 28.1% 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 30.4

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

5. Applied rewrites 30.4%

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

Reproduce

herbie shell --seed 2024221 
(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))))))