Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 10.9s

Alternatives: 12

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

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 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{t\_0}{\left(2 \cdot s\right) \cdot \left(1 + \mathsf{fma}\left(\left(-0.5\right) \cdot \frac{x \cdot x}{s} + \left|x\right|, \frac{-1}{s}, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-0.0625 \cdot x\right) \cdot \frac{x}{s}}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.0020000000949949026)
     (/
      t_0
      (*
       (* 2.0 s)
       (+ 1.0 (fma (+ (* (- 0.5) (/ (* x x) s)) (fabs x)) (/ -1.0 s) 1.0))))
     (/ (+ (/ (* (* -0.0625 x) (/ x s)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.0020000000949949026f) {
		tmp = t_0 / ((2.0f * s) * (1.0f + fmaf(((-0.5f * ((x * x) / s)) + fabsf(x)), (-1.0f / s), 1.0f)));
	} else {
		tmp = ((((-0.0625f * x) * (x / s)) / s) + 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(0.0020000000949949026))
		tmp = Float32(t_0 / Float32(Float32(Float32(2.0) * s) * Float32(Float32(1.0) + fma(Float32(Float32(Float32(-Float32(0.5)) * Float32(Float32(x * x) / s)) + abs(x)), Float32(Float32(-1.0) / s), Float32(1.0)))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * x) * Float32(x / s)) / s) + Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.00200000009

   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. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in s around inf

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

      1. lower-*.f32 99.1

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

   10. Applied rewrites 99.1%

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

   if 0.00200000009 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 89.5%

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

   9. Applied rewrites 92.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(2 \cdot s\right) \cdot \left(1 + \mathsf{fma}\left(\left(-0.5\right) \cdot \frac{x \cdot x}{s} + \left|x\right|, \frac{-1}{s}, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-0.0625 \cdot x\right) \cdot \frac{x}{s}}{s} + 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\left|x\right|}{s}\\ t_1 := e^{t\_0}\\ t_2 := 1 + t\_1\\ \mathbf{if}\;\frac{t\_1}{\left(s \cdot t\_2\right) \cdot t\_2} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{{\mathsf{E}\left(\right)}^{t\_0}}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-0.0625 \cdot x\right) \cdot \frac{x}{s}}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- (fabs x)) s)) (t_1 (exp t_0)) (t_2 (+ 1.0 t_1)))
   (if (<= (/ t_1 (* (* s t_2) t_2)) 0.0020000000949949026)
     (/ (pow (E) t_0) (* 4.0 s))
     (/ (+ (/ (* (* -0.0625 x) (/ x s)) s) 0.25) s))))

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))))) < 0.00200000009

   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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f32 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-exp.f32 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-exp.f32 N/A

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

      2. exp-1-e N/A

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

      3. lower-E.f32 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in s around inf

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

   11. Applied rewrites 99.2%

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

   if 0.00200000009 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 89.5%

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

   9. Applied rewrites 92.7%

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

Alternative 4: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{t\_0}{4 \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-0.0625 \cdot x\right) \cdot \frac{x}{s}}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.0020000000949949026)
     (/ t_0 (* 4.0 s))
     (/ (+ (/ (* (* -0.0625 x) (/ x s)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.0020000000949949026f) {
		tmp = t_0 / (4.0f * s);
	} else {
		tmp = ((((-0.0625f * x) * (x / s)) / s) + 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 / ((s * t_1) * t_1)) <= 0.0020000000949949026e0) then
        tmp = t_0 / (4.0e0 * s)
    else
        tmp = (((((-0.0625e0) * x) * (x / s)) / s) + 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(s * t_1) * t_1)) <= Float32(0.0020000000949949026))
		tmp = Float32(t_0 / Float32(Float32(4.0) * s));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * x) * Float32(x / s)) / s) + 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 / ((s * t_1) * t_1)) <= single(0.0020000000949949026))
		tmp = t_0 / (single(4.0) * s);
	else
		tmp = ((((single(-0.0625) * x) * (x / s)) / s) + 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))))) < 0.00200000009

   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. 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 0.00200000009 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 89.5%

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

   9. Applied rewrites 92.7%

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

Alternative 5: 47.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-0.0625 \cdot x\right) \cdot \frac{x}{s}}{s} + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 0.0020000000949949026)
     (/ (/ (/ (fma (* x x) -0.0625 (* (* s s) 0.25)) s) s) s)
     (/ (+ (/ (* (* -0.0625 x) (/ x s)) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 0.0020000000949949026f) {
		tmp = ((fmaf((x * x), -0.0625f, ((s * s) * 0.25f)) / s) / s) / s;
	} else {
		tmp = ((((-0.0625f * x) * (x / s)) / s) + 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(Float32(fma(Float32(x * x), Float32(-0.0625), Float32(Float32(s * s) * Float32(0.25))) / s) / s) / s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * x) * Float32(x / s)) / s) + Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.00200000009

   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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in s around 0

      \[\leadsto \frac{\frac{\frac{-1}{16} \cdot {x}^{2} + \frac{1}{4} \cdot {s}^{2}}{{s}^{2}}}{s} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.6%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(x \cdot x, -0.0625, \left(s \cdot s\right) \cdot 0.25\right)}{s}}{s}}{s} \]

   if 0.00200000009 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.4%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 89.5%

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

   9. Applied rewrites 92.7%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

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

   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%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. pow-flip N/A

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

   10. lower-pow.f32 N/A

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

   11. metadata-eval 99.3

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

6. Applied rewrites 99.3%

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

Alternative 7: 97.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = t_0 / ((s * (1.0e0 + t_0)) * (1.0e0 + ((((((x / s) * x) * 0.5e0) - abs(x)) / s) + 1.0e0)))
end function

Julia

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

MATLAB

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

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

5. Applied rewrites 96.7%

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

7. Applied rewrites 97.1%

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

Alternative 8: 96.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))))
   (/
    t_0
    (* (* (+ (/ (- (* (/ (* x x) s) 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 * x) / s) * 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 * x) / s) * 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 * x) / s) * 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 * x) / s) * 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 96.7%

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

Alternative 9: 96.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = ((((x * x) / s) * 0.5e0) - abs(x)) / s
    code = exp((-abs(x) / s)) / ((s * (t_0 + 2.0e0)) * (1.0e0 + (t_0 + 1.0e0)))
end function

Julia

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

MATLAB

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

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

8. Applied rewrites 96.4%

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

Alternative 10: 96.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. sub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 95.7

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

8. Applied rewrites 95.7%

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

Alternative 11: 27.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(-0.0625 \cdot x\right) \cdot \frac{x}{s}}{s} + 0.25}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (+ (/ (* (* -0.0625 x) (/ x s)) s) 0.25) s))

C

float code(float x, float s) {
	return ((((-0.0625f * x) * (x / s)) / s) + 0.25f) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (((((-0.0625e0) * x) * (x / s)) / s) + 0.25e0) / s
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(Float32(Float32(Float32(-0.0625) * x) * Float32(x / s)) / s) + Float32(0.25)) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = ((((single(-0.0625) * x) * (x / s)) / s) + 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. 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%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 25.5%

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

9. Applied rewrites 26.3%

   \[\leadsto \frac{\frac{\left(-0.0625 \cdot x\right) \cdot \frac{x}{s}}{s} + 0.25}{s} \]
10. Add Preprocessing

Alternative 12: 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 25.8

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

5. Applied rewrites 25.8%

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

6. Final simplification 25.8%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024299 
(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))))))