Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 10.1s

Alternatives: 14

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, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = t_0 - -1.0f;
	return t_0 / ((t_1 * s) * 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 = t_0 - (-1.0e0)
    code = t_0 / ((t_1 * s) * t_1)
end function

Julia

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

MATLAB

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

Derivation

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. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.3

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

4. Applied rewrites 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Alternative 5: 96.4% 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 99.4%

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

4. Applied rewrites 99.4%

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

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 94.9

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

7. Applied rewrites 94.9%

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

Alternative 6: 94.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{-1}\right)}^{\left(\frac{\left|x\right|}{s}\right)} \cdot 0.25}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (* (pow (exp -1.0) (/ (fabs x) s)) 0.25) s))

C

float code(float x, float s) {
	return (powf(expf(-1.0f), (fabsf(x) / s)) * 0.25f) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = ((exp((-1.0e0)) ** (abs(x) / s)) * 0.25e0) / s
end function

Julia

function code(x, s)
	return Float32(Float32((exp(Float32(-1.0)) ^ Float32(abs(x) / s)) * Float32(0.25)) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = ((exp(single(-1.0)) ^ (abs(x) / s)) * single(0.25)) / s;
end

Derivation

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 93.1%

   \[\leadsto \frac{\color{blue}{0.25} \cdot e^{\frac{-\left|x\right|}{s}}}{s} \]
8. Step-by-step derivation

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 93.1

      \[\leadsto \frac{0.25 \cdot {\left(e^{-1}\right)}^{\color{blue}{\left(\frac{\left|x\right|}{s}\right)}}}{s} \]

9. Applied rewrites 93.1%

   \[\leadsto \frac{0.25 \cdot \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left|x\right|}{s}\right)}}}{s} \]

10. Final simplification 93.1%

    \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\frac{\left|x\right|}{s}\right)} \cdot 0.25}{s} \]
11. Add Preprocessing

Alternative 7: 94.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-1}{\frac{s}{\left|x\right|}}} \cdot 0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (* (exp (/ -1.0 (/ s (fabs x)))) 0.25) s))

C

float code(float x, float s) {
	return (expf((-1.0f / (s / fabsf(x)))) * 0.25f) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp(((-1.0e0) / (s / abs(x)))) * 0.25e0) / s
end function

Julia

function code(x, s)
	return Float32(Float32(exp(Float32(Float32(-1.0) / Float32(s / abs(x)))) * Float32(0.25)) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = (exp((single(-1.0) / (s / abs(x)))) * single(0.25)) / s;
end

Derivation

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 93.1%

   \[\leadsto \frac{\color{blue}{0.25} \cdot e^{\frac{-\left|x\right|}{s}}}{s} \]
8. Step-by-step derivation

   1. *-lft-identity N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

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

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

   6. *-inverses N/A

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

   7. times-frac N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. sqr-abs N/A

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

   11. lift-fabs.f32 N/A

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

   12. lift-fabs.f32 N/A

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

   13. unpow3 N/A

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

   14. lift-pow.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. distribute-frac-neg N/A

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

   17. lift-neg.f32 N/A

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

   18. clear-num N/A

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

   19. frac-2neg N/A

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

   20. metadata-eval N/A

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

9. Applied rewrites 93.1%

   \[\leadsto \frac{0.25 \cdot e^{\color{blue}{\frac{-1}{\frac{s}{\left|x\right|}}}}}{s} \]

10. Final simplification 93.1%

    \[\leadsto \frac{e^{\frac{-1}{\frac{s}{\left|x\right|}}} \cdot 0.25}{s} \]
11. Add Preprocessing

Alternative 8: 94.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-1}{s} \cdot \left|x\right|} \cdot 0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (* (exp (* (/ -1.0 s) (fabs x))) 0.25) s))

C

float code(float x, float s) {
	return (expf(((-1.0f / s) * fabsf(x))) * 0.25f) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp((((-1.0e0) / s) * abs(x))) * 0.25e0) / s
end function

Julia

function code(x, s)
	return Float32(Float32(exp(Float32(Float32(Float32(-1.0) / s) * abs(x))) * Float32(0.25)) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = (exp(((single(-1.0) / s) * abs(x))) * single(0.25)) / s;
end

Derivation

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 93.1%

   \[\leadsto \frac{\color{blue}{0.25} \cdot e^{\frac{-\left|x\right|}{s}}}{s} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f32 93.1

      \[\leadsto \frac{0.25 \cdot e^{\left|x\right| \cdot \color{blue}{\frac{-1}{s}}}}{s} \]

9. Applied rewrites 93.1%

   \[\leadsto \frac{0.25 \cdot e^{\color{blue}{\left|x\right| \cdot \frac{-1}{s}}}}{s} \]

10. Final simplification 93.1%

    \[\leadsto \frac{e^{\frac{-1}{s} \cdot \left|x\right|} \cdot 0.25}{s} \]
11. Add Preprocessing

Alternative 9: 94.8% 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 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 93.1

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

5. Applied rewrites 93.1%

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

Alternative 10: 94.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{e^{\frac{\left|x\right|}{s}} \cdot s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 (* (exp (/ (fabs x) s)) s)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 93.1%

   \[\leadsto \frac{\color{blue}{0.25} \cdot e^{\frac{-\left|x\right|}{s}}}{s} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

   7. div-inv N/A

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

   8. lift-exp.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. distribute-frac-neg N/A

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

   12. exp-neg N/A

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

   13. remove-double-div N/A

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

9. Applied rewrites 93.1%

   \[\leadsto \color{blue}{\frac{0.25}{s \cdot e^{\frac{\left|x\right|}{s}}}} \]

10. Final simplification 93.1%

    \[\leadsto \frac{0.25}{e^{\frac{\left|x\right|}{s}} \cdot s} \]
11. Add Preprocessing

Alternative 11: 82.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\frac{s}{-0.0625 \cdot x}}{x}}}{s} + 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x \cdot x}{s \cdot s} + 4\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.9999999996399175e-23)
   (/ (+ (/ (/ 1.0 (/ (/ s (* -0.0625 x)) x)) s) 0.25) s)
   (/ 1.0 (* (+ (/ (* x x) (* s s)) 4.0) s))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.9999999996399175e-23f) {
		tmp = (((1.0f / ((s / (-0.0625f * x)) / x)) / s) + 0.25f) / s;
	} else {
		tmp = 1.0f / ((((x * x) / (s * s)) + 4.0f) * s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 1.9999999996399175e-23) then
        tmp = (((1.0e0 / ((s / ((-0.0625e0) * x)) / x)) / s) + 0.25e0) / s
    else
        tmp = 1.0e0 / ((((x * x) / (s * s)) + 4.0e0) * s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(s / Float32(Float32(-0.0625) * x)) / x)) / s) + Float32(0.25)) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x * x) / Float32(s * s)) + Float32(4.0)) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(1.9999999996399175e-23))
		tmp = (((single(1.0) / ((s / (single(-0.0625) * x)) / x)) / s) + single(0.25)) / s;
	else
		tmp = single(1.0) / ((((x * x) / (s * s)) + single(4.0)) * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 2e-23

   1. Initial program 97.9%

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in s around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f32 N/A

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

   7. Applied rewrites 72.2%

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

   9. Applied rewrites 73.5%

      \[\leadsto \frac{\frac{\frac{1}{\frac{\frac{s}{-0.0625 \cdot x}}{x}}}{s} + 0.25}{s} \]

   if 2e-23 < (fabs.f32 x)

   1. Initial program 99.8%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

   7. Applied rewrites 81.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\frac{s}{-0.0625 \cdot x}}{x}}}{s} + 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x \cdot x}{s \cdot s} + 4\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.0625}{s} + 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x \cdot x}{s \cdot s} + 4\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.9999999996399175e-23)
   (/ (+ (/ (* (* (/ x s) x) -0.0625) s) 0.25) s)
   (/ 1.0 (* (+ (/ (* x x) (* s s)) 4.0) s))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.9999999996399175e-23f) {
		tmp = (((((x / s) * x) * -0.0625f) / s) + 0.25f) / s;
	} else {
		tmp = 1.0f / ((((x * x) / (s * s)) + 4.0f) * s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 1.9999999996399175e-23) then
        tmp = (((((x / s) * x) * (-0.0625e0)) / s) + 0.25e0) / s
    else
        tmp = 1.0e0 / ((((x * x) / (s * s)) + 4.0e0) * s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(x / s) * x) * Float32(-0.0625)) / s) + Float32(0.25)) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x * x) / Float32(s * s)) + Float32(4.0)) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(1.9999999996399175e-23))
		tmp = (((((x / s) * x) * single(-0.0625)) / s) + single(0.25)) / s;
	else
		tmp = single(1.0) / ((((x * x) / (s * s)) + single(4.0)) * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 2e-23

   1. Initial program 97.9%

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in s around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f32 N/A

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

   7. Applied rewrites 72.2%

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

   9. Applied rewrites 73.5%

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

   if 2e-23 < (fabs.f32 x)

   1. Initial program 99.8%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

   7. Applied rewrites 81.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.0625}{s} + 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x \cdot x}{s \cdot s} + 4\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x \cdot x}{s \cdot s} + 4\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 4.999999999099794e-24)
   (/ 0.25 s)
   (/ 1.0 (* (+ (/ (* x x) (* s s)) 4.0) s))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 4.999999999099794e-24f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / ((((x * x) / (s * s)) + 4.0f) * s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 4.999999999099794e-24) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((((x * x) / (s * s)) + 4.0e0) * s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(4.999999999099794e-24))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x * x) / Float32(s * s)) + Float32(4.0)) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(4.999999999099794e-24))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((((x * x) / (s * s)) + single(4.0)) * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 5e-24

   1. Initial program 97.9%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 72.7

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

   5. Applied rewrites 72.7%

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

   if 5e-24 < (fabs.f32 x)

   1. Initial program 99.8%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

   7. Applied rewrites 81.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x \cdot x}{s \cdot s} + 4\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.3% 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 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 29.0

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

5. Applied rewrites 29.0%

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

Reproduce

herbie shell --seed 2024276 
(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))))))