Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 7.3s

Alternatives: 15

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

Alternative 2: 97.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / (((s / (1.0f - (((-0.5f * ((x * x) / s)) - fabsf(x)) / s))) + 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 / (((s / (1.0e0 - ((((-0.5e0) * ((x * x) / s)) - abs(x)) / s))) + 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(s / Float32(Float32(1.0) - Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(x * x) / s)) - abs(x)) / s))) + s) * Float32(Float32(1.0) + t_0)))
end

MATLAB

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 97.5%

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

8. Final simplification 97.5%

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

Alternative 3: 96.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / (((1.0f / ((1.0f + (fabsf(x) / s)) / s)) + 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 / (((1.0e0 / ((1.0e0 + (abs(x) / s)) / s)) + 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(1.0) / Float32(Float32(Float32(1.0) + Float32(abs(x) / s)) / s)) + s) * Float32(Float32(1.0) + t_0)))
end

MATLAB

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-fabs.f32 96.9

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

7. Applied rewrites 96.9%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 97.1

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

9. Applied rewrites 97.1%

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

10. Final simplification 97.1%

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

Alternative 4: 96.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-fabs.f32 96.9

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

7. Applied rewrites 96.9%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-exp.f32 N/A

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

   6. rec-exp N/A

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

   7. lift-/.f32 N/A

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

   8. distribute-frac-neg2 N/A

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

   9. lift-neg.f32 N/A

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

   10. frac-2neg N/A

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

9. Applied rewrites 96.9%

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

10. Final simplification 96.9%

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

Alternative 5: 96.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / (((s / (1.0f + (fabsf(x) / s))) + 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 / (((s / (1.0e0 + (abs(x) / s))) + 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(s / Float32(Float32(1.0) + Float32(abs(x) / s))) + s) * Float32(Float32(1.0) + t_0)))
end

MATLAB

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-fabs.f32 96.9

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

7. Applied rewrites 96.9%

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

8. Final simplification 96.9%

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

Alternative 6: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return (powf((2.0f - (fabsf(x) / s)), -2.0f) * expf((-fabsf(x) / s))) / 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)) * exp((-abs(x) / s))) / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 96.1

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

7. Applied rewrites 96.1%

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

Alternative 7: 94.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-fabs.f32 96.9

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

7. Applied rewrites 96.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. exp-neg N/A

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

   7. lift-exp.f32 N/A

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

   8. lower-/.f32 96.9

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

9. Applied rewrites 96.9%

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

10. Taylor expanded in s around inf

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

12. Applied rewrites 95.4%

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

13. Final simplification 95.4%

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

Alternative 8: 94.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-fabs.f32 96.9

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

7. Applied rewrites 96.9%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 95.4%

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

11. Final simplification 95.4%

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

Alternative 9: 94.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return ((1.0f / 2.0f) / (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 = ((1.0e0 / 2.0e0) / (2.0e0 * s)) / exp((abs(x) / s))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 95.4%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. lift-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lift-exp.f32 N/A

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

7. Applied rewrites 75.1%

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

8. Taylor expanded in s around inf

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

   1. lower-*.f32 95.1

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

10. Applied rewrites 95.1%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f32 N/A

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

12. Applied rewrites 95.1%

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{2 \cdot s}}{e^{\frac{\left|x\right|}{s}}}} \]
13. Add Preprocessing

Alternative 10: 94.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 95.4%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. lift-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lift-exp.f32 N/A

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

7. Applied rewrites 75.1%

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

8. Taylor expanded in s around inf

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

   1. lower-*.f32 95.1

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

10. Applied rewrites 95.1%

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

    1. lift-/.f32 N/A

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

    2. clear-num N/A

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

    3. associate-/r/ N/A

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

    4. lower-*.f32 N/A

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

    5. lower-/.f32 95.1

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

12. Applied rewrites 95.1%

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

13. Final simplification 95.1%

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

Alternative 11: 94.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 95.4%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. lift-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lift-exp.f32 N/A

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

7. Applied rewrites 75.1%

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

8. Taylor expanded in s around inf

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

   1. lower-*.f32 95.1

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

10. Applied rewrites 95.1%

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

11. Final simplification 95.1%

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

Alternative 12: 94.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = exp((-abs(x) / s)) / (single(4.0) * 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. 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 95.1

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

5. Applied rewrites 95.1%

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

Alternative 13: 73.5% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 95.4%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. lift-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lift-exp.f32 N/A

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

7. Applied rewrites 75.1%

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

8. Taylor expanded in s around inf

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

   1. lower-*.f32 95.1

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

10. Applied rewrites 95.1%

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

11. Taylor expanded in s around -inf

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 N/A

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

    5. +-commutative N/A

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

    6. mul-1-neg N/A

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

    7. unsub-neg N/A

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

    8. lower--.f32 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f32 N/A

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

    11. lower-/.f32 N/A

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

    12. unpow2 N/A

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

    13. sqr-abs N/A

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

    14. lower-*.f32 N/A

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

    15. lower-fabs.f32 74.5

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

13. Applied rewrites 74.5%

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

14. Final simplification 74.5%

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

Alternative 14: 49.9% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 95.4%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. lift-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lift-exp.f32 N/A

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

7. Applied rewrites 75.1%

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

8. Taylor expanded in s around inf

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

   1. lower-*.f32 95.1

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

10. Applied rewrites 95.1%

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

11. Taylor expanded in s around inf

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-fabs.f32 53.9

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

13. Applied rewrites 53.9%

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

14. Final simplification 53.9%

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

Alternative 15: 27.6% 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 26.2

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

5. Applied rewrites 26.2%

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

Reproduce

herbie shell --seed 2024332 
(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))))))