Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 10.6s

Alternatives: 12

Speedup: 1.0×

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, 1.0× speedup

Math

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

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(Float32(t_0 / (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 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 99.9%

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

Alternative 2: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 0.004999999888241291:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 0.004999999888241291)
     (/ 1.0 (* (* (/ (/ x s) s) x) 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 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 0.004999999888241291f) {
		tmp = 1.0f / ((((x / s) / s) * x) * 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 = 1.0e0 + t_0
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.004999999888241291e0) then
        tmp = 1.0e0 / ((((x / s) / s) * x) * 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(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.004999999888241291))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x / s) / s) * x) * s));
	else
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(Float32(Float32(-0.0625) * 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 / ((t_1 * s) * t_1)) <= single(0.004999999888241291))
		tmp = single(1.0) / ((((x / s) / s) * x) * 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))))) < 0.00499999989

   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. 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 100.0%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

   7. Applied rewrites 38.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 38.7%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 80.8%

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

   if 0.00499999989 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 90.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot s\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{\frac{-0.0625 \cdot \left(x \cdot x\right)}{s}}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 0.004999999888241291:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 0.004999999888241291)
     (/ 1.0 (* (* (/ (/ x s) s) x) s))
     (/ 0.25 s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((t_1 * s) * t_1)) <= 0.004999999888241291f) {
		tmp = 1.0f / ((((x / s) / s) * 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 = 1.0e0 + t_0
    if ((t_0 / ((t_1 * s) * t_1)) <= 0.004999999888241291e0) then
        tmp = 1.0e0 / ((((x / s) / s) * 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(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(0.004999999888241291))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x / s) / s) * 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 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((t_1 * s) * t_1)) <= single(0.004999999888241291))
		tmp = single(1.0) / ((((x / s) / s) * 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))))) < 0.00499999989

   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. 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 100.0%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

   7. Applied rewrites 38.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 38.7%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 80.8%

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

   if 0.00499999989 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.2%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot s\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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 99.8

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 64.7%

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

7. Final simplification 64.7%

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

Alternative 5: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (((single(2.0) - (abs(x) / s)) ^ single(-2.0)) / s) * exp((-abs(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. 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 99.9%

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

5. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 96.8

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

7. Applied rewrites 96.8%

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

Alternative 6: 94.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{\frac{1}{s}}{\frac{-1}{\left|x\right|}}}}{4 \cdot s} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = exp(((single(1.0) / s) / (single(-1.0) / abs(x)))) / (single(4.0) * 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

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

   1. lower-*.f32 95.6

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

5. Applied rewrites 95.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lift-neg.f32 N/A

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

   10. remove-double-neg N/A

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

   11. lower-/.f32 95.6

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

7. Applied rewrites 95.6%

   \[\leadsto \frac{e^{\color{blue}{\frac{\frac{1}{s}}{\frac{-1}{\left|x\right|}}}}}{4 \cdot s} \]
8. Add Preprocessing

Alternative 7: 94.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-1}{s} \cdot \left|x\right|}}{4 \cdot s} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = exp(((single(-1.0) / s) * abs(x))) / (single(4.0) * 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

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

   1. lower-*.f32 95.6

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

5. Applied rewrites 95.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. distribute-frac-neg2 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 95.6

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

7. Applied rewrites 95.6%

   \[\leadsto \frac{e^{\color{blue}{\frac{-1}{s} \cdot \left|x\right|}}}{4 \cdot s} \]
8. Add Preprocessing

Alternative 8: 59.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   1. lower-*.f32 95.6

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

5. Applied rewrites 95.6%

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

   1. +-lft-identity N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f32 N/A

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

   4. neg-sub0 N/A

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

   5. flip3-- N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\frac{{0}^{3} - {\left(\left|x\right|\right)}^{3}}{0 \cdot 0 + \left(\left|x\right| \cdot \left|x\right| + 0 \cdot \left|x\right|\right)}} + 0}{s}}}{4 \cdot s} \]

   6. div-inv N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\left({0}^{3} - {\left(\left|x\right|\right)}^{3}\right) \cdot \frac{1}{0 \cdot 0 + \left(\left|x\right| \cdot \left|x\right| + 0 \cdot \left|x\right|\right)}} + 0}{s}}}{4 \cdot s} \]

   7. lower-fma.f32 N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{fma}\left({0}^{3} - {\left(\left|x\right|\right)}^{3}, \frac{1}{0 \cdot 0 + \left(\left|x\right| \cdot \left|x\right| + 0 \cdot \left|x\right|\right)}, 0\right)}}{s}}}{4 \cdot s} \]

7. Applied rewrites 32.5%

   \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{fma}\left(-{\left(\left|x\right|\right)}^{3}, \frac{1}{x \cdot x}, 0\right)}}{s}}}{4 \cdot s} \]

9. Applied rewrites 61.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{-x}{s}}}{4 \cdot s}} \]
10. Add Preprocessing

Alternative 9: 84.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x + 4\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{1}{s \cdot s} + \frac{4}{x \cdot x}\right) \cdot \left(x \cdot x\right)\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.999999936531045e-20)
   (/ 1.0 (* (+ (* (/ (/ x s) s) x) 4.0) s))
   (/ 1.0 (* (* (+ (/ 1.0 (* s s)) (/ 4.0 (* x x))) (* x x)) s))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.999999936531045e-20f) {
		tmp = 1.0f / (((((x / s) / s) * x) + 4.0f) * s);
	} else {
		tmp = 1.0f / ((((1.0f / (s * s)) + (4.0f / (x * x))) * (x * x)) * s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 1.999999936531045e-20) then
        tmp = 1.0e0 / (((((x / s) / s) * x) + 4.0e0) * s)
    else
        tmp = 1.0e0 / ((((1.0e0 / (s * s)) + (4.0e0 / (x * x))) * (x * x)) * s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.999999936531045e-20))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(x / s) / s) * x) + Float32(4.0)) * s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(1.0) / Float32(s * s)) + Float32(Float32(4.0) / Float32(x * x))) * Float32(x * x)) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(1.999999936531045e-20))
		tmp = single(1.0) / (((((x / s) / s) * x) + single(4.0)) * s);
	else
		tmp = single(1.0) / ((((single(1.0) / (s * s)) + (single(4.0) / (x * x))) * (x * x)) * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1.99999994e-20

   1. Initial program 99.5%

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

      1. lift-/.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 99.5%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

   7. Applied rewrites 67.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 67.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 87.8%

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

   if 1.99999994e-20 < (fabs.f32 x)

   1. Initial program 99.9%

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

      1. lift-/.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. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

   7. Applied rewrites 45.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x + 4\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{1}{s \cdot s} + \frac{4}{x \cdot x}\right) \cdot \left(x \cdot x\right)\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.1% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (((((x / s) / s) * x) + single(4.0)) * 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 99.8%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

7. Applied rewrites 49.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 49.8%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 83.8%

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

Alternative 11: 50.6% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (((single(4.0) * (abs(x) / s)) + single(4.0)) * 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 99.8%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

7. Applied rewrites 49.8%

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

9. Applied rewrites 49.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 49.8%

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

Alternative 12: 27.8% 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 25.6

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

5. Applied rewrites 25.6%

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

Reproduce

herbie shell --seed 2024283 
(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))))))