Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 11.0s

Alternatives: 17

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 97.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.0e0) then
        tmp = t_0 / (4.0e0 * s)
    else
        tmp = (-1.0e0) / (((-4.0e0) - ((x / s) * (x / s))) * 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(Float32(t_1 * s) * t_1)) <= Float32(0.0))
		tmp = Float32(t_0 / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(-4.0) - Float32(Float32(x / s) * Float32(x / s))) * s));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. lower-*.f32 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 99.4%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 95.6%

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

4. Final simplification 98.8%

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

Alternative 3: 84.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.0e0) then
        tmp = (-1.0e0) / ((((((-1.0e0) / s) / s) - (4.0e0 / (x * x))) * (x * x)) * s)
    else
        tmp = (-1.0e0) / (((-4.0e0) - ((x / s) * (x / s))) * 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(Float32(t_1 * s) * t_1)) <= Float32(0.0))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(Float32(Float32(-1.0) / s) / s) - Float32(Float32(4.0) / Float32(x * x))) * Float32(x * x)) * s));
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(-4.0) - Float32(Float32(x / s) * Float32(x / s))) * s));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.4%

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

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

   1. Initial program 99.4%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 95.6%

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

4. Final simplification 88.9%

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

Alternative 4: 82.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.0e0) then
        tmp = (-1.0e0) / (((-x * x) / (s * s)) * s)
    else
        tmp = (-1.0e0) / (((-4.0e0) - ((x / s) * (x / s))) * 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(Float32(t_1 * s) * t_1)) <= Float32(0.0))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(-x) * x) / Float32(s * s)) * s));
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(-4.0) - Float32(Float32(x / s) * Float32(x / s))) * s));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 84.9%

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

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

   1. Initial program 99.4%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 95.6%

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

4. Final simplification 87.8%

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

Alternative 5: 81.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.0e0) then
        tmp = (-1.0e0) / (((-x * x) / (s * s)) * s)
    else
        tmp = (0.25e0 + ((((-0.0625e0) * x) * (x / s)) / s)) / 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(Float32(t_1 * s) * t_1)) <= Float32(0.0))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(-x) * x) / Float32(s * s)) * s));
	else
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(Float32(Float32(-0.0625) * x) * Float32(x / s)) / s)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = t_0 + single(1.0);
	tmp = single(0.0);
	if ((t_0 / ((t_1 * s) * t_1)) <= single(0.0))
		tmp = single(-1.0) / (((-x * x) / (s * s)) * s);
	else
		tmp = (single(0.25) + (((single(-0.0625) * x) * (x / s)) / s)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 84.9%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 93.4%

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

   9. Applied rewrites 94.9%

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

4. Final simplification 87.7%

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

Alternative 6: 81.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.0e0) then
        tmp = (-1.0e0) / (((-x * x) / (s * s)) * s)
    else
        tmp = (-1.0e0) / (((-4.0e0) * s) - ((x * x) / 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(Float32(t_1 * s) * t_1)) <= Float32(0.0))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(-x) * x) / Float32(s * s)) * s));
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(-4.0) * s) - Float32(Float32(x * x) / s)));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 84.9%

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

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

   1. Initial program 99.4%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 93.8%

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

4. Final simplification 87.3%

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

Alternative 7: 65.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 4.4%

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

   10. Taylor expanded in s around 0

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

   12. Applied rewrites 57.4%

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

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

   1. Initial program 99.4%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

4. Final simplification 66.7%

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

Alternative 8: 65.0% accurate, 0.9× speedup

Math

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

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)) 0.0)
     (/ -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)) <= 0.0f) {
		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)) <= 0.0e0) 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(Float32(t_1 * s) * t_1)) <= Float32(0.0))
		tmp = Float32(Float32(-1.0) / Float32(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(0.0))
		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))))) < 0.0

   1. Initial program 100.0%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-exp.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 57.4%

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

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

   1. Initial program 99.4%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

4. Final simplification 66.6%

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

Alternative 9: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return (powf((t_0 + 1.0f), -2.0f) * t_0) / 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 + 1.0e0) ** (-2.0e0)) * t_0) / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 10: 99.5% 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(Float32(t_0 / 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 99.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. associate-/l/ N/A

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

   7. *-lft-identity N/A

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

   8. times-frac N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 11: 95.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return (-1.0f / (-1.0f - t_0)) * (t_0 / ((2.0f - (fabsf(x) / s)) * 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 = ((-1.0e0) / ((-1.0e0) - t_0)) * (t_0 / ((2.0e0 - (abs(x) / s)) * s))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 97.3

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

5. Applied rewrites 97.3%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

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

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

   7. times-frac N/A

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

7. Applied rewrites 97.3%

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

8. Final simplification 97.3%

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

Alternative 12: 95.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 97.3

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

5. Applied rewrites 97.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 97.3%

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

Alternative 13: 95.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 97.3

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

5. Applied rewrites 97.3%

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

6. Final simplification 97.3%

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

Alternative 14: 95.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / (2.0f * ((t_0 + 1.0f) * 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 / (2.0e0 * ((t_0 + 1.0e0) * s))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 95.9%

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

6. Final simplification 95.9%

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

Alternative 15: 94.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 97.3

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

5. Applied rewrites 97.3%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 95.9%

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

9. Final simplification 95.9%

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

Alternative 16: 65.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{-4 \cdot s - \frac{x \cdot x}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ -1.0 (- (* -4.0 s) (/ (* x x) s))))

C

float code(float x, float s) {
	return -1.0f / ((-4.0f * s) - ((x * x) / s));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (-1.0e0) / (((-4.0e0) * s) - ((x * x) / s))
end function

Julia

function code(x, s)
	return Float32(Float32(-1.0) / Float32(Float32(Float32(-4.0) * s) - Float32(Float32(x * x) / s)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(-1.0) / ((single(-4.0) * s) - ((x * x) / s));
end

Derivation

1. Initial program 99.8%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-exp.f32 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 28.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 67.5%

    \[\leadsto \frac{-1}{-4 \cdot s - \color{blue}{\frac{x \cdot x}{s}}} \]
11. Add Preprocessing

Alternative 17: 27.4% accurate, 31.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 28.3

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

5. Applied rewrites 28.3%

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

Reproduce

herbie shell --seed 2024249 
(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))))))