Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.5s

Alternatives: 15

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ t_1 := e^{-t\_0}\\ t_2 := t\_1 + 1\\ \mathbf{if}\;\frac{t\_1}{t\_2 \cdot \left(s \cdot t\_2\right)} \leq 0.5:\\ \;\;\;\;\frac{1}{s \cdot e^{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{s \cdot \left(\frac{\left(x \cdot \frac{x \cdot 0.25}{s}\right) \cdot -4}{s} - 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)) (t_1 (exp (- t_0))) (t_2 (+ t_1 1.0)))
   (if (<= (/ t_1 (* t_2 (* s t_2))) 0.5)
     (/ 1.0 (* s (exp t_0)))
     (/ -1.0 (* s (- (/ (* (* x (/ (* x 0.25) s)) -4.0) s) 4.0))))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	float t_1 = expf(-t_0);
	float t_2 = t_1 + 1.0f;
	float tmp;
	if ((t_1 / (t_2 * (s * t_2))) <= 0.5f) {
		tmp = 1.0f / (s * expf(t_0));
	} else {
		tmp = -1.0f / (s * ((((x * ((x * 0.25f) / s)) * -4.0f) / s) - 4.0f));
	}
	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) :: t_2
    real(4) :: tmp
    t_0 = abs(x) / s
    t_1 = exp(-t_0)
    t_2 = t_1 + 1.0e0
    if ((t_1 / (t_2 * (s * t_2))) <= 0.5e0) then
        tmp = 1.0e0 / (s * exp(t_0))
    else
        tmp = (-1.0e0) / (s * ((((x * ((x * 0.25e0) / s)) * (-4.0e0)) / s) - 4.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	t_1 = exp(Float32(-t_0))
	t_2 = Float32(t_1 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_1 / Float32(t_2 * Float32(s * t_2))) <= Float32(0.5))
		tmp = Float32(Float32(1.0) / Float32(s * exp(t_0)));
	else
		tmp = Float32(Float32(-1.0) / Float32(s * Float32(Float32(Float32(Float32(x * Float32(Float32(x * Float32(0.25)) / s)) * Float32(-4.0)) / s) - Float32(4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = abs(x) / s;
	t_1 = exp(-t_0);
	t_2 = t_1 + single(1.0);
	tmp = single(0.0);
	if ((t_1 / (t_2 * (s * t_2))) <= single(0.5))
		tmp = single(1.0) / (s * exp(t_0));
	else
		tmp = single(-1.0) / (s * ((((x * ((x * single(0.25)) / s)) * single(-4.0)) / s) - single(4.0)));
	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))))) < 0.5

   1. Initial program 99.9%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fabs.f32 99.6

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

   9. Applied rewrites 99.6%

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

   if 0.5 < (/.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 98.8%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 98.4%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 24.1%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 11.7%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-fma.f32 N/A

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

       3. lift-/.f32 N/A

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

       4. +-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 80.0

          \[\leadsto \frac{1}{s \cdot \left(4 - \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s} \cdot -4}}{s}\right)} \]

       7. lift-/.f32 N/A

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

       8. lift-fma.f32 N/A

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

       9. lift-*.f32 N/A

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

       10. +-rgt-identity N/A

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

       11. lift-*.f32 N/A

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

       12. lift-*.f32 N/A

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

       13. associate-*l* N/A

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

       14. associate-/l* N/A

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

       15. lower-*.f32 N/A

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

       16. lower-/.f32 N/A

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

       17. lower-*.f32 85.8

           \[\leadsto \frac{1}{s \cdot \left(4 - \frac{\left(x \cdot \frac{\color{blue}{x \cdot 0.25}}{s}\right) \cdot -4}{s}\right)} \]

   11. Applied rewrites 85.8%

       \[\leadsto \frac{1}{s \cdot \left(4 - \frac{\color{blue}{\left(x \cdot \frac{x \cdot 0.25}{s}\right) \cdot -4}}{s}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0.5:\\ \;\;\;\;\frac{1}{s \cdot e^{\frac{\left|x\right|}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{s \cdot \left(\frac{\left(x \cdot \frac{x \cdot 0.25}{s}\right) \cdot -4}{s} - 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.5)
     (/ 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 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.5f) {
		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 = t_0 + 1.0e0
    if ((t_0 / (t_1 * (s * t_1))) <= 0.5e0) 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(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.5))
		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 = t_0 + single(1.0);
	tmp = single(0.0);
	if ((t_0 / (t_1 * (s * t_1))) <= single(0.5))
		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))))) < 0.5

   1. Initial program 99.9%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.4%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   10. Taylor expanded in x around inf

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

       1. lower-/.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f32 79.5

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

   12. Applied rewrites 79.5%

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

   if 0.5 < (/.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 98.8%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 79.9%

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

Alternative 4: 64.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.5

   1. Initial program 99.9%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.4%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   10. Taylor expanded in s around 0

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

       1. lower-/.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 58.8

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

   12. Applied rewrites 58.8%

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

   if 0.5 < (/.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 98.8%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 63.8%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-fabs.f32 N/A

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

   12. remove-double-neg N/A

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

   13. lift-neg.f32 N/A

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

   14. remove-double-neg N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 6: 96.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. neg-mul-1 N/A

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

   2. sub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 95.2

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

7. Applied rewrites 95.2%

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

8. Final simplification 95.2%

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

Alternative 7: 94.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{{\mathsf{E}\left(\right)}^{\left(-\frac{\left|x\right|}{s}\right)}}{s \cdot 4} \end{array} \]

FPCore

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

Derivation

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-fabs.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. *-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.7

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

   10. lift-/.f32 N/A

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

   11. lift-neg.f32 N/A

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

   12. distribute-frac-neg N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.0

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

7. Applied rewrites 94.0%

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

8. Final simplification 94.0%

   \[\leadsto \frac{{\mathsf{E}\left(\right)}^{\left(-\frac{\left|x\right|}{s}\right)}}{s \cdot 4} \]
9. Add Preprocessing

Alternative 8: 94.8% 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 99.7%

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

3. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{4 \cdot s}} \]
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 94.0

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

5. Applied rewrites 94.0%

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

6. Final simplification 94.0%

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

Alternative 9: 67.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{s \cdot \left(\frac{\left(x \cdot \frac{x \cdot 0.25}{s}\right) \cdot -4}{s} - 4\right)}\\ \mathbf{elif}\;\left|x\right| \leq 1:\\ \;\;\;\;\frac{-1}{s \cdot \left(4 \cdot \left(-1 - \frac{\left|x\right| + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\left|x\right|}{s} \cdot \left(x \cdot x\right), 0.5 \cdot \left(x \cdot x\right)\right)}{s}}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(s, \frac{4}{x \cdot x}, \frac{1}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 3.999999935100636e-17)
   (/ -1.0 (* s (- (/ (* (* x (/ (* x 0.25) s)) -4.0) s) 4.0)))
   (if (<= (fabs x) 1.0)
     (/
      -1.0
      (*
       s
       (*
        4.0
        (-
         -1.0
         (/
          (+
           (fabs x)
           (/
            (fma
             0.16666666666666666
             (* (/ (fabs x) s) (* x x))
             (* 0.5 (* x x)))
            s))
          s)))))
     (/ 1.0 (* (* x x) (fma s (/ 4.0 (* x x)) (/ 1.0 s)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 3.999999935100636e-17f) {
		tmp = -1.0f / (s * ((((x * ((x * 0.25f) / s)) * -4.0f) / s) - 4.0f));
	} else if (fabsf(x) <= 1.0f) {
		tmp = -1.0f / (s * (4.0f * (-1.0f - ((fabsf(x) + (fmaf(0.16666666666666666f, ((fabsf(x) / s) * (x * x)), (0.5f * (x * x))) / s)) / s))));
	} else {
		tmp = 1.0f / ((x * x) * fmaf(s, (4.0f / (x * x)), (1.0f / s)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(3.999999935100636e-17))
		tmp = Float32(Float32(-1.0) / Float32(s * Float32(Float32(Float32(Float32(x * Float32(Float32(x * Float32(0.25)) / s)) * Float32(-4.0)) / s) - Float32(4.0))));
	elseif (abs(x) <= Float32(1.0))
		tmp = Float32(Float32(-1.0) / Float32(s * Float32(Float32(4.0) * Float32(Float32(-1.0) - Float32(Float32(abs(x) + Float32(fma(Float32(0.16666666666666666), Float32(Float32(abs(x) / s) * Float32(x * x)), Float32(Float32(0.5) * Float32(x * x))) / s)) / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) * fma(s, Float32(Float32(4.0) / Float32(x * x)), Float32(Float32(1.0) / s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 3.99999994e-17

   1. Initial program 98.9%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 98.8%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 42.5%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 11.6%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-fma.f32 N/A

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

       3. lift-/.f32 N/A

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

       4. +-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 65.9

          \[\leadsto \frac{1}{s \cdot \left(4 - \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s} \cdot -4}}{s}\right)} \]

       7. lift-/.f32 N/A

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

       8. lift-fma.f32 N/A

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

       9. lift-*.f32 N/A

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

       10. +-rgt-identity N/A

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

       11. lift-*.f32 N/A

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

       12. lift-*.f32 N/A

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

       13. associate-*l* N/A

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

       14. associate-/l* N/A

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

       15. lower-*.f32 N/A

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

       16. lower-/.f32 N/A

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

       17. lower-*.f32 69.7

           \[\leadsto \frac{1}{s \cdot \left(4 - \frac{\left(x \cdot \frac{\color{blue}{x \cdot 0.25}}{s}\right) \cdot -4}{s}\right)} \]

   11. Applied rewrites 69.7%

       \[\leadsto \frac{1}{s \cdot \left(4 - \frac{\color{blue}{\left(x \cdot \frac{x \cdot 0.25}{s}\right) \cdot -4}}{s}\right)} \]

   if 3.99999994e-17 < (fabs.f32 x) < 1

   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-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 88.7%

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

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{s \cdot \left(4 \cdot \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}\right)}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{s \cdot \left(4 \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}\right)\right)}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \frac{1}{s \cdot \left(4 \cdot \color{blue}{\left(1 - \frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}\right)}\right)} \]

      3. lower--.f32 N/A

         \[\leadsto \frac{1}{s \cdot \left(4 \cdot \color{blue}{\left(1 - \frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}\right)}\right)} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{1}{s \cdot \left(4 \cdot \left(1 - \color{blue}{\frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}}\right)\right)} \]

   10. Applied rewrites 46.2%

       \[\leadsto \frac{1}{s \cdot \left(4 \cdot \color{blue}{\left(1 - \frac{\left(-\left|x\right|\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \left(x \cdot x\right) \cdot \frac{\left|x\right|}{s}, 0.5 \cdot \left(x \cdot x\right)\right)}{s}}{s}\right)}\right)} \]

   if 1 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.3%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   10. Taylor expanded in x around inf

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

       7. metadata-eval N/A

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

       8. associate-*r/ N/A

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

       9. lower-fma.f32 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 N/A

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

       15. lower-/.f32 53.3

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

   12. Applied rewrites 75.9%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{s \cdot \left(\frac{\left(x \cdot \frac{x \cdot 0.25}{s}\right) \cdot -4}{s} - 4\right)}\\ \mathbf{elif}\;\left|x\right| \leq 1:\\ \;\;\;\;\frac{-1}{s \cdot \left(4 \cdot \left(-1 - \frac{\left|x\right| + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\left|x\right|}{s} \cdot \left(x \cdot x\right), 0.5 \cdot \left(x \cdot x\right)\right)}{s}}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(s, \frac{4}{x \cdot x}, \frac{1}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.8400000122904986 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{s \cdot \left(\frac{\left(x \cdot \frac{x \cdot 0.25}{s}\right) \cdot -4}{s} - 4\right)}\\ \mathbf{elif}\;\left|x\right| \leq 50:\\ \;\;\;\;\frac{-1}{s \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot s} - 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(s, \frac{4}{x \cdot x}, \frac{1}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.8400000122904986e-22)
   (/ -1.0 (* s (- (/ (* (* x (/ (* x 0.25) s)) -4.0) s) 4.0)))
   (if (<= (fabs x) 50.0)
     (/ -1.0 (* s (- (* (* x x) (/ -1.0 (* s s))) 4.0)))
     (/ 1.0 (* (* x x) (fma s (/ 4.0 (* x x)) (/ 1.0 s)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.8400000122904986e-22f) {
		tmp = -1.0f / (s * ((((x * ((x * 0.25f) / s)) * -4.0f) / s) - 4.0f));
	} else if (fabsf(x) <= 50.0f) {
		tmp = -1.0f / (s * (((x * x) * (-1.0f / (s * s))) - 4.0f));
	} else {
		tmp = 1.0f / ((x * x) * fmaf(s, (4.0f / (x * x)), (1.0f / s)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.8400000122904986e-22))
		tmp = Float32(Float32(-1.0) / Float32(s * Float32(Float32(Float32(Float32(x * Float32(Float32(x * Float32(0.25)) / s)) * Float32(-4.0)) / s) - Float32(4.0))));
	elseif (abs(x) <= Float32(50.0))
		tmp = Float32(Float32(-1.0) / Float32(s * Float32(Float32(Float32(x * x) * Float32(Float32(-1.0) / Float32(s * s))) - Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) * fma(s, Float32(Float32(4.0) / Float32(x * x)), Float32(Float32(1.0) / s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 1.84000001e-22

   1. Initial program 99.0%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 98.8%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 39.9%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 12.0%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-fma.f32 N/A

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

       3. lift-/.f32 N/A

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

       4. +-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 68.5

          \[\leadsto \frac{1}{s \cdot \left(4 - \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s} \cdot -4}}{s}\right)} \]

       7. lift-/.f32 N/A

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

       8. lift-fma.f32 N/A

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

       9. lift-*.f32 N/A

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

       10. +-rgt-identity N/A

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

       11. lift-*.f32 N/A

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

       12. lift-*.f32 N/A

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

       13. associate-*l* N/A

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

       14. associate-/l* N/A

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

       15. lower-*.f32 N/A

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

       16. lower-/.f32 N/A

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

       17. lower-*.f32 73.4

           \[\leadsto \frac{1}{s \cdot \left(4 - \frac{\left(x \cdot \frac{\color{blue}{x \cdot 0.25}}{s}\right) \cdot -4}{s}\right)} \]

   11. Applied rewrites 73.4%

       \[\leadsto \frac{1}{s \cdot \left(4 - \frac{\color{blue}{\left(x \cdot \frac{x \cdot 0.25}{s}\right) \cdot -4}}{s}\right)} \]

   if 1.84000001e-22 < (fabs.f32 x) < 50

   1. Initial program 99.5%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied 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. Step-by-step derivation

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 79.2%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.6%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   11. Applied rewrites 74.2%

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

   if 50 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.3%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   10. Taylor expanded in x around inf

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

       7. metadata-eval N/A

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

       8. associate-*r/ N/A

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

       9. lower-fma.f32 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 N/A

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

       15. lower-/.f32 55.1

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

   12. Applied rewrites 78.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.8400000122904986 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{s \cdot \left(\frac{\left(x \cdot \frac{x \cdot 0.25}{s}\right) \cdot -4}{s} - 4\right)}\\ \mathbf{elif}\;\left|x\right| \leq 50:\\ \;\;\;\;\frac{-1}{s \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot s} - 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(s, \frac{4}{x \cdot x}, \frac{1}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.8400000122904986 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)}\\ \mathbf{elif}\;\left|x\right| \leq 50:\\ \;\;\;\;\frac{-1}{s \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot s} - 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(s, \frac{4}{x \cdot x}, \frac{1}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.8400000122904986e-22)
   (/ 1.0 (* s (+ 4.0 (/ (/ (* x x) s) s))))
   (if (<= (fabs x) 50.0)
     (/ -1.0 (* s (- (* (* x x) (/ -1.0 (* s s))) 4.0)))
     (/ 1.0 (* (* x x) (fma s (/ 4.0 (* x x)) (/ 1.0 s)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.8400000122904986e-22f) {
		tmp = 1.0f / (s * (4.0f + (((x * x) / s) / s)));
	} else if (fabsf(x) <= 50.0f) {
		tmp = -1.0f / (s * (((x * x) * (-1.0f / (s * s))) - 4.0f));
	} else {
		tmp = 1.0f / ((x * x) * fmaf(s, (4.0f / (x * x)), (1.0f / s)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.8400000122904986e-22))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(Float32(x * x) / s) / s))));
	elseif (abs(x) <= Float32(50.0))
		tmp = Float32(Float32(-1.0) / Float32(s * Float32(Float32(Float32(x * x) * Float32(Float32(-1.0) / Float32(s * s))) - Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) * fma(s, Float32(Float32(4.0) / Float32(x * x)), Float32(Float32(1.0) / s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 1.84000001e-22

   1. Initial program 99.0%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 98.8%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 39.9%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 12.0%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   10. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. lower-/.f32 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-neg.f32 70.6

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

   12. Applied rewrites 70.6%

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

   if 1.84000001e-22 < (fabs.f32 x) < 50

   1. Initial program 99.5%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied 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. Step-by-step derivation

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 79.2%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.6%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   11. Applied rewrites 74.2%

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

   if 50 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.3%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   10. Taylor expanded in x around inf

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

       7. metadata-eval N/A

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

       8. associate-*r/ N/A

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

       9. lower-fma.f32 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 N/A

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

       15. lower-/.f32 55.1

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

   12. Applied rewrites 78.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.8400000122904986 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)}\\ \mathbf{elif}\;\left|x\right| \leq 50:\\ \;\;\;\;\frac{-1}{s \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot s} - 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(s, \frac{4}{x \cdot x}, \frac{1}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.8400000122904986 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{s \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot s} - 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.8400000122904986e-22)
   (/ 1.0 (* s (+ 4.0 (/ (/ (* x x) s) s))))
   (/ -1.0 (* s (- (* (* x x) (/ -1.0 (* s s))) 4.0)))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.8400000122904986e-22f) {
		tmp = 1.0f / (s * (4.0f + (((x * x) / s) / s)));
	} else {
		tmp = -1.0f / (s * (((x * x) * (-1.0f / (s * s))) - 4.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 1.8400000122904986e-22) then
        tmp = 1.0e0 / (s * (4.0e0 + (((x * x) / s) / s)))
    else
        tmp = (-1.0e0) / (s * (((x * x) * ((-1.0e0) / (s * s))) - 4.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.8400000122904986e-22))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(Float32(x * x) / s) / s))));
	else
		tmp = Float32(Float32(-1.0) / Float32(s * Float32(Float32(Float32(x * x) * Float32(Float32(-1.0) / Float32(s * s))) - Float32(4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(1.8400000122904986e-22))
		tmp = single(1.0) / (s * (single(4.0) + (((x * x) / s) / s)));
	else
		tmp = single(-1.0) / (s * (((x * x) * (single(-1.0) / (s * s))) - single(4.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1.84000001e-22

   1. Initial program 99.0%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 98.8%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 39.9%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 12.0%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   10. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. lower-/.f32 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-neg.f32 70.6

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

   12. Applied rewrites 70.6%

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

   if 1.84000001e-22 < (fabs.f32 x)

   1. Initial program 99.9%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied 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. Step-by-step derivation

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 93.8%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.4%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   11. Applied rewrites 84.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.8400000122904986 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{s \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot s} - 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25 + \frac{x}{s} \cdot \frac{x \cdot -0.0625}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 3.999999999279835e-23)
   (/ (+ 0.25 (* (/ x s) (/ (* x -0.0625) s))) s)
   (/ 1.0 (* s (+ 4.0 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 3.999999999279835e-23f) {
		tmp = (0.25f + ((x / s) * ((x * -0.0625f) / s))) / s;
	} else {
		tmp = 1.0f / (s * (4.0f + ((x * x) / (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 3.999999999279835e-23) then
        tmp = (0.25e0 + ((x / s) * ((x * (-0.0625e0)) / s))) / s
    else
        tmp = 1.0e0 / (s * (4.0e0 + ((x * x) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(3.999999999279835e-23))
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(x / s) * Float32(Float32(x * Float32(-0.0625)) / s))) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(3.999999999279835e-23))
		tmp = (single(0.25) + ((x / s) * ((x * single(-0.0625)) / s))) / s;
	else
		tmp = single(1.0) / (s * (single(4.0) + ((x * x) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 4e-23

   1. Initial program 99.0%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 42.9%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 69.8

         \[\leadsto \frac{0.25 + \frac{x}{s} \cdot \frac{\color{blue}{x \cdot -0.0625}}{s}}{s} \]

   7. Applied rewrites 69.8%

      \[\leadsto \frac{0.25 + \color{blue}{\frac{x}{s} \cdot \frac{x \cdot -0.0625}{s}}}{s} \]

   if 4e-23 < (fabs.f32 x)

   1. Initial program 99.8%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied 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. Step-by-step derivation

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 92.7%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.5%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   10. Taylor expanded in x around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 83.1

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

   12. Applied rewrites 83.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25 + \frac{x}{s} \cdot \frac{x \cdot -0.0625}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 3.999999999279835e-23)
   (/ 0.25 s)
   (/ 1.0 (* s (+ 4.0 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 3.999999999279835e-23f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (s * (4.0f + ((x * x) / (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 3.999999999279835e-23) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (s * (4.0e0 + ((x * x) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(3.999999999279835e-23))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(3.999999999279835e-23))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (s * (single(4.0) + ((x * x) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 4e-23

   1. Initial program 99.0%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 69.8

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

   5. Applied rewrites 69.8%

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

   if 4e-23 < (fabs.f32 x)

   1. Initial program 99.8%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied 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. Step-by-step derivation

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-to-exp N/A

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

      14. prod-exp N/A

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

   6. Applied rewrites 92.7%

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

   7. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

   9. Applied rewrites 8.5%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(x \cdot x, 0.25, 0\right)}{s}, 0\right)}{s}\right)}} \]

   10. Taylor expanded in x around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 83.1

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

   12. Applied rewrites 83.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.1% 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 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 21.7

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

5. Applied rewrites 21.7%

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

Reproduce

herbie shell --seed 2024216 
(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))))))