Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 9.7s

Alternatives: 10

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.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. lower-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. pow2 N/A

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

   9. lower-pow.f32 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x \cdot x}{s}}{s}\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 9.999999747378752e-6)
     (/ t_0 (* 4.0 s))
     (/ 1.0 (* (+ 4.0 (/ (/ (* x x) s) s)) s)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 9.999999747378752e-6f) {
		tmp = t_0 / (4.0f * s);
	} else {
		tmp = 1.0f / ((4.0f + (((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 = 1.0e0 + t_0
    if ((t_0 / ((s * t_1) * t_1)) <= 9.999999747378752e-6) then
        tmp = t_0 / (4.0e0 * s)
    else
        tmp = 1.0e0 / ((4.0e0 + (((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(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(9.999999747378752e-6))
		tmp = Float32(t_0 / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) + Float32(Float32(Float32(x * x) / s) / s)) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((s * t_1) * t_1)) <= single(9.999999747378752e-6))
		tmp = t_0 / (single(4.0) * s);
	else
		tmp = single(1.0) / ((single(4.0) + (((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))))) < 9.99999975e-6

   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

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

      1. lower-*.f32 99.2

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

   5. Applied rewrites 99.2%

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

   if 9.99999975e-6 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.3%

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

      1. lift-/.f32 N/A

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

      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 99.4%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

   7. Applied rewrites 94.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(4 + \frac{\frac{x \cdot x}{s}}{s}\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + t_0) ** (-2.0e0)) * (1.0e0 / s)) * t_0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.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. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.7%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
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 99.7

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

4. Applied rewrites 99.7%

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

Alternative 5: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{\left(\left(\frac{\left(\frac{x}{s} \cdot x\right) \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))))
   (/
    t_0
    (* (* (+ (/ (- (* (* (/ x s) x) 0.5) (fabs x)) s) 2.0) s) (+ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / ((((((((x / s) * x) * 0.5f) - fabsf(x)) / s) + 2.0f) * s) * (1.0f + t_0));
}

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 / ((((((((x / s) * x) * 0.5e0) - abs(x)) / s) + 2.0e0) * s) * (1.0e0 + t_0))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(x / s) * x) * Float32(0.5)) - abs(x)) / s) + Float32(2.0)) * s) * Float32(Float32(1.0) + t_0)))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / ((((((((x / s) * x) * single(0.5)) - abs(x)) / s) + single(2.0)) * s) * (single(1.0) + t_0));
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 97.4%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 97.8%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(\frac{\left(\frac{x}{s} \cdot x\right) \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
8. Add Preprocessing

Alternative 6: 96.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\\ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(t\_0 \cdot s\right) \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (+ (/ (- (* (/ (* x x) s) 0.5) (fabs x)) s) 2.0)))
   (/ (exp (/ (- (fabs x)) s)) (* (* t_0 s) t_0))))

C

float code(float x, float s) {
	float t_0 = (((((x * x) / s) * 0.5f) - fabsf(x)) / s) + 2.0f;
	return expf((-fabsf(x) / s)) / ((t_0 * s) * t_0);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = (((((x * x) / s) * 0.5e0) - abs(x)) / s) + 2.0e0
    code = exp((-abs(x) / s)) / ((t_0 * s) * t_0)
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(Float32(Float32(Float32(Float32(x * x) / s) * Float32(0.5)) - abs(x)) / s) + Float32(2.0))
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(t_0 * s) * t_0))
end

MATLAB

function tmp = code(x, s)
	t_0 = (((((x * x) / s) * single(0.5)) - abs(x)) / s) + single(2.0);
	tmp = exp((-abs(x) / s)) / ((t_0 * s) * t_0);
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 97.4%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

6. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

8. Applied rewrites 97.5%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \color{blue}{\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right)}} \]
9. Add Preprocessing

Alternative 7: 96.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \left(1 - \frac{\left|x\right|}{s}\right)\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (exp (/ (- (fabs x)) s))
  (*
   (* (+ (/ (- (* (/ (* x x) s) 0.5) (fabs x)) s) 2.0) s)
   (+ 1.0 (- 1.0 (/ (fabs x) s))))))

C

float code(float x, float s) {
	return expf((-fabsf(x) / s)) / ((((((((x * x) / s) * 0.5f) - fabsf(x)) / s) + 2.0f) * s) * (1.0f + (1.0f - (fabsf(x) / s))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((-abs(x) / s)) / ((((((((x * x) / s) * 0.5e0) - abs(x)) / s) + 2.0e0) * s) * (1.0e0 + (1.0e0 - (abs(x) / s))))
end function

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(x * x) / s) * Float32(0.5)) - abs(x)) / s) + Float32(2.0)) * s) * Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(abs(x) / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = exp((-abs(x) / s)) / ((((((((x * x) / s) * single(0.5)) - abs(x)) / s) + single(2.0)) * s) * (single(1.0) + (single(1.0) - (abs(x) / s))));
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 97.4%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

6. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 97.0

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \left(1 - \frac{\color{blue}{\left|x\right|}}{s}\right)\right)} \]

8. Applied rewrites 97.0%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \color{blue}{\left(1 - \frac{\left|x\right|}{s}\right)}\right)} \]
9. Add Preprocessing

Alternative 8: 79.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left|x\right|}{s}\\ t_1 := \frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s}\\ t_2 := \left(t\_1 + 2\right) \cdot s\\ \mathbf{if}\;-\left|x\right| \leq -15000:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(1 + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_2 \cdot \left(1 + \left(t\_1 + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- 1.0 (/ (fabs x) s)))
        (t_1 (/ (- (* (/ (* x x) s) 0.5) (fabs x)) s))
        (t_2 (* (+ t_1 2.0) s)))
   (if (<= (- (fabs x)) -15000.0)
     (/ 1.0 (* t_2 (+ 1.0 t_0)))
     (/ t_0 (* t_2 (+ 1.0 (+ t_1 1.0)))))))

C

float code(float x, float s) {
	float t_0 = 1.0f - (fabsf(x) / s);
	float t_1 = ((((x * x) / s) * 0.5f) - fabsf(x)) / s;
	float t_2 = (t_1 + 2.0f) * s;
	float tmp;
	if (-fabsf(x) <= -15000.0f) {
		tmp = 1.0f / (t_2 * (1.0f + t_0));
	} else {
		tmp = t_0 / (t_2 * (1.0f + (t_1 + 1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: t_2
    real(4) :: tmp
    t_0 = 1.0e0 - (abs(x) / s)
    t_1 = ((((x * x) / s) * 0.5e0) - abs(x)) / s
    t_2 = (t_1 + 2.0e0) * s
    if (-abs(x) <= (-15000.0e0)) then
        tmp = 1.0e0 / (t_2 * (1.0e0 + t_0))
    else
        tmp = t_0 / (t_2 * (1.0e0 + (t_1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) - Float32(abs(x) / s))
	t_1 = Float32(Float32(Float32(Float32(Float32(x * x) / s) * Float32(0.5)) - abs(x)) / s)
	t_2 = Float32(Float32(t_1 + Float32(2.0)) * s)
	tmp = Float32(0.0)
	if (Float32(-abs(x)) <= Float32(-15000.0))
		tmp = Float32(Float32(1.0) / Float32(t_2 * Float32(Float32(1.0) + t_0)));
	else
		tmp = Float32(t_0 / Float32(t_2 * Float32(Float32(1.0) + Float32(t_1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(1.0) - (abs(x) / s);
	t_1 = ((((x * x) / s) * single(0.5)) - abs(x)) / s;
	t_2 = (t_1 + single(2.0)) * s;
	tmp = single(0.0);
	if (-abs(x) <= single(-15000.0))
		tmp = single(1.0) / (t_2 * (single(1.0) + t_0));
	else
		tmp = t_0 / (t_2 * (single(1.0) + (t_1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (fabs.f32 x)) < -15000

   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

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

   6. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 100.0

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \left(1 - \frac{\color{blue}{\left|x\right|}}{s}\right)\right)} \]

   8. Applied rewrites 100.0%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \color{blue}{\left(1 - \frac{\left|x\right|}{s}\right)}\right)} \]

   9. Taylor expanded in s around inf

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

   11. Applied rewrites 93.8%

       \[\leadsto \frac{\color{blue}{1}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \left(1 - \frac{\left|x\right|}{s}\right)\right)} \]

   if -15000 < (neg.f32 (fabs.f32 x))

   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

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 95.2%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

   6. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 94.4

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \left(1 - \frac{\color{blue}{\left|x\right|}}{s}\right)\right)} \]

   8. Applied rewrites 94.4%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \color{blue}{\left(1 - \frac{\left|x\right|}{s}\right)}\right)} \]

   9. Taylor expanded in s around inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f32 N/A

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

       4. lower-/.f32 N/A

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

       5. lower-fabs.f32 64.4

          \[\leadsto \frac{1 - \frac{\color{blue}{\left|x\right|}}{s}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \left(1 - \frac{\left|x\right|}{s}\right)\right)} \]

   11. Applied rewrites 64.4%

       \[\leadsto \frac{\color{blue}{1 - \frac{\left|x\right|}{s}}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \left(1 - \frac{\left|x\right|}{s}\right)\right)} \]

   12. Taylor expanded in s around inf

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

       1. +-commutative N/A

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

       2. lower-+.f32 N/A

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

   14. Applied rewrites 69.7%

       \[\leadsto \frac{1 - \frac{\left|x\right|}{s}}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right) \cdot \left(1 + \color{blue}{\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 1\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(4 + \frac{\frac{x \cdot x}{s}}{s}\right) \cdot s} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. 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 99.4%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

7. Applied rewrites 77.0%

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

8. Final simplification 77.0%

   \[\leadsto \frac{1}{\left(4 + \frac{\frac{x \cdot x}{s}}{s}\right) \cdot s} \]
9. Add Preprocessing

Alternative 10: 27.5% accurate, 31.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 29.2

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

5. Applied rewrites 29.2%

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

Reproduce

herbie shell --seed 2024318 
(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))))))