Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 18.6s

Alternatives: 21

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{0 - \left|x\right|}{s}}\\ \frac{t\_0}{\left(s \cdot \left(1 + \frac{1}{e^{\frac{\left|x\right|}{s}}}\right)\right) \cdot \left(t\_0 + 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- 0.0 (fabs x)) s))))
   (/ t_0 (* (* s (+ 1.0 (/ 1.0 (exp (/ (fabs x) s))))) (+ t_0 1.0)))))

C

float code(float x, float s) {
	float t_0 = expf(((0.0f - fabsf(x)) / s));
	return t_0 / ((s * (1.0f + (1.0f / expf((fabsf(x) / s))))) * (t_0 + 1.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp(((0.0e0 - abs(x)) / s))
    code = t_0 / ((s * (1.0e0 + (1.0e0 / exp((abs(x) / s))))) * (t_0 + 1.0e0))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(Float32(0.0) - abs(x)) / s))
	return Float32(t_0 / Float32(Float32(s * Float32(Float32(1.0) + Float32(Float32(1.0) / exp(Float32(abs(x) / s))))) * Float32(t_0 + Float32(1.0))))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp(((single(0.0) - abs(x)) / s));
	tmp = t_0 / ((s * (single(1.0) + (single(1.0) / exp((abs(x) / s))))) * (t_0 + single(1.0)));
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(1, \left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right)\right)\right)\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(1, \left(\frac{1}{e^{\frac{\left|x\right|}{s}}}\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(1, \left(e^{\frac{\left|x\right|}{s}}\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right)\right)\right)\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right)\right)\right)\right) \]

   6. fabs-lowering-fabs.f32 99.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \frac{e^{\frac{0 - \left|x\right|}{s}}}{\left(s \cdot \left(1 + \frac{1}{e^{\frac{\left|x\right|}{s}}}\right)\right) \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + 1\right)} \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{0 - \left|x\right|}{s}}\\ \frac{\frac{t\_0}{s}}{{\left(t\_0 + 1\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- 0.0 (fabs x)) s))))
   (/ (/ t_0 s) (pow (+ t_0 1.0) 2.0))))

C

float code(float x, float s) {
	float t_0 = expf(((0.0f - fabsf(x)) / s));
	return (t_0 / s) / powf((t_0 + 1.0f), 2.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp(((0.0e0 - abs(x)) / s))
    code = (t_0 / s) / ((t_0 + 1.0e0) ** 2.0e0)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(Float32(0.0) - abs(x)) / s))
	return Float32(Float32(t_0 / s) / (Float32(t_0 + Float32(1.0)) ^ Float32(2.0)))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp(((single(0.0) - abs(x)) / s));
	tmp = (t_0 / s) / ((t_0 + single(1.0)) ^ single(2.0));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{s}\right), \color{blue}{\left({\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}\right)}\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(e^{-1 \cdot \frac{\left|x\right|}{s}}\right), s\right), \left({\color{blue}{\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}}^{2}\right)\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(-1 \cdot \frac{\left|x\right|}{s}\right)\right), s\right), \left({\left(\color{blue}{1} + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}\right)\right) \]

   5. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)\right)\right), s\right), \left({\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}\right)\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}\right)\right), s\right), \left({\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}\right)\right) \]

   7. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\frac{\left|x\right|}{-1 \cdot s}\right)\right), s\right), \left({\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), \left(-1 \cdot s\right)\right)\right), s\right), \left({\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}\right)\right) \]

   9. fabs-lowering-fabs.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(-1 \cdot s\right)\right)\right), s\right), \left({\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right), s\right), \left({\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}\right)\right) \]

   11. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{neg.f32}\left(s\right)\right)\right), s\right), \left({\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}\right)\right) \]

   12. pow-lowering-pow.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{neg.f32}\left(s\right)\right)\right), s\right), \mathsf{pow.f32}\left(\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right), \color{blue}{2}\right)\right) \]

5. Simplified 99.8%

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

6. Final simplification 99.8%

   \[\leadsto \frac{\frac{e^{\frac{0 - \left|x\right|}{s}}}{s}}{{\left(e^{\frac{0 - \left|x\right|}{s}} + 1\right)}^{2}} \]
7. Add Preprocessing

Alternative 3: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. inv-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\left({\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{-1}\right), s\right) \]

   2. sqr-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\left({\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{\left(\frac{-1}{2}\right)}\right), s\right) \]

   3. pow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\left({\left({\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}\right), s\right) \]

   4. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left({\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{\left(\frac{-1}{2}\right)}\right), 2\right), s\right) \]

6. Applied egg-rr 99.5%

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

   1. unpow1 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left({\left({\left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}^{\frac{-1}{2}}\right)}^{1}\right), 2\right), s\right) \]

   2. sqr-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left({\left({\left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{1}{2}\right)}\right), 2\right), s\right) \]

   3. pow-prod-down N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left({\left({\left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}^{\frac{-1}{2}} \cdot {\left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{1}{2}\right)}\right), 2\right), s\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left({\left({\left({\left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}^{\frac{-1}{2}}\right)}^{2}\right)}^{\left(\frac{1}{2}\right)}\right), 2\right), s\right) \]

   5. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\left({\left({\left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}^{\frac{-1}{2}}\right)}^{2}\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

   6. pow-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\left({\left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}^{\left(\frac{-1}{2} \cdot 2\right)}\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\left({\left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}^{-1}\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

   8. inv-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)}\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

   10. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \left(2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)\right)\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \cosh \left(\frac{\left|x\right|}{s}\right)\right)\right)\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

   12. cosh-lowering-cosh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

   14. fabs-lowering-fabs.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right), \left(\frac{1}{2}\right)\right), 2\right), s\right) \]

8. Applied egg-rr 99.5%

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

   1. unpow1/2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left(\sqrt{\frac{1}{2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)}}\right), 2\right), s\right) \]

   2. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right), 2\right), s\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)\right)\right), 2\right), s\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \left(2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right), 2\right), s\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \cosh \left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right), 2\right), s\right) \]

   6. cosh-lowering-cosh.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right)\right), 2\right), s\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right)\right), 2\right), s\right) \]

   8. fabs-lowering-fabs.f32 99.5%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right)\right), 2\right), s\right) \]

10. Applied egg-rr 99.5%

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

Alternative 4: 99.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ (pow (pow (+ 2.0 (* 2.0 (cosh (/ (fabs x) s)))) -0.5) 2.0) s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. inv-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\left({\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{-1}\right), s\right) \]

   2. sqr-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\left({\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{\left(\frac{-1}{2}\right)}\right), s\right) \]

   3. pow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\left({\left({\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}\right), s\right) \]

   4. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left({\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}^{\left(\frac{-1}{2}\right)}\right), 2\right), s\right) \]

6. Applied egg-rr 99.5%

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

Alternative 5: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right)}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right)\right) \]

   4. associate-+r+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right)\right) \]

   9. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right)\right) \]

   10. cosh-undef N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right)\right) \]

   12. cosh-lowering-cosh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right)\right) \]

   14. fabs-lowering-fabs.f32 99.5%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.5%

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

Alternative 6: 95.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \color{blue}{\left(4 \cdot s\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \left(s \cdot \color{blue}{4}\right)\right) \]

   2. *-lowering-*.f32 95.7%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(s, \color{blue}{4}\right)\right) \]

5. Simplified 95.7%

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

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)\right)\right), \mathsf{*.f32}\left(s, 4\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{1}{\frac{s}{\left|x\right|}}\right)\right)\right), \mathsf{*.f32}\left(s, 4\right)\right) \]

   3. associate-/r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{1}{s} \cdot \left|x\right|\right)\right)\right), \mathsf{*.f32}\left(s, 4\right)\right) \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\left(\mathsf{neg}\left(\frac{1}{s}\right)\right) \cdot \left|x\right|\right)\right), \mathsf{*.f32}\left(s, 4\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{1}{s}\right)\right), \left(\left|x\right|\right)\right)\right), \mathsf{*.f32}\left(s, 4\right)\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\left(\frac{\mathsf{neg}\left(1\right)}{s}\right), \left(\left|x\right|\right)\right)\right), \mathsf{*.f32}\left(s, 4\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\left(\frac{-1}{s}\right), \left(\left|x\right|\right)\right)\right), \mathsf{*.f32}\left(s, 4\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, s\right), \left(\left|x\right|\right)\right)\right), \mathsf{*.f32}\left(s, 4\right)\right) \]

   9. fabs-lowering-fabs.f32 95.7%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{*.f32}\left(s, 4\right)\right) \]

7. Applied egg-rr 95.7%

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

8. Final simplification 95.7%

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

Alternative 7: 95.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{0 - \left|x\right|}{s}}}{s \cdot 4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (exp (/ (- 0.0 (fabs x)) s)) (* s 4.0)))

C

float code(float x, float s) {
	return expf(((0.0f - fabsf(x)) / s)) / (s * 4.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp(((0.0e0 - abs(x)) / s)) / (s * 4.0e0)
end function

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(Float32(0.0) - abs(x)) / s)) / Float32(s * Float32(4.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = exp(((single(0.0) - abs(x)) / s)) / (s * single(4.0));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \color{blue}{\left(4 \cdot s\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \left(s \cdot \color{blue}{4}\right)\right) \]

   2. *-lowering-*.f32 95.7%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(s, \color{blue}{4}\right)\right) \]

5. Simplified 95.7%

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

6. Final simplification 95.7%

   \[\leadsto \frac{e^{\frac{0 - \left|x\right|}{s}}}{s \cdot 4} \]
7. Add Preprocessing

Alternative 8: 95.1% accurate, 3.0× speedup

Math

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

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(Float32(0.25) / 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 99.8%

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

3. Taylor expanded in s around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \color{blue}{\left(4 \cdot s\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \left(s \cdot \color{blue}{4}\right)\right) \]

   2. *-lowering-*.f32 95.7%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(s, \color{blue}{4}\right)\right) \]

5. Simplified 95.7%

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

6. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{4} \cdot e^{-1 \cdot \frac{\left|x\right|}{s}}\right), \color{blue}{s}\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right), s\right) \]

   4. rec-exp N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{4} \cdot \frac{1}{e^{\frac{\left|x\right|}{s}}}\right), s\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{4} \cdot 1}{e^{\frac{\left|x\right|}{s}}}\right), s\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{4}}{e^{\frac{\left|x\right|}{s}}}\right), s\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, \left(e^{\frac{\left|x\right|}{s}}\right)\right), s\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, \mathsf{exp.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right), s\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right), s\right) \]

   10. fabs-lowering-fabs.f32 95.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), s\right) \]

8. Simplified 95.6%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{e^{\frac{\left|x\right|}{s}}}}{s}} \]
9. Add Preprocessing

Alternative 9: 82.9% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{\left(x \cdot x\right) \cdot 0.041666666666666664}{s \cdot \left(s \cdot \left(s \cdot s\right)\right)} + \frac{0.5}{s \cdot s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999936531045e-21)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (/
    (/ 1.0 s)
    (+
     2.0
     (*
      2.0
      (+
       1.0
       (*
        (* x x)
        (+
         (/ (* (* x x) 0.041666666666666664) (* s (* s (* s s))))
         (/ 0.5 (* s s))))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999936531045e-21f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else {
		tmp = (1.0f / s) / (2.0f + (2.0f * (1.0f + ((x * x) * ((((x * x) * 0.041666666666666664f) / (s * (s * (s * s)))) + (0.5f / (s * s)))))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.999999936531045e-21) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else
        tmp = (1.0e0 / s) / (2.0e0 + (2.0e0 * (1.0e0 + ((x * x) * ((((x * x) * 0.041666666666666664e0) / (s * (s * (s * s)))) + (0.5e0 / (s * s)))))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999936531045e-21))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(Float32(Float32(x * x) * Float32(0.041666666666666664)) / Float32(s * Float32(s * Float32(s * s)))) + Float32(Float32(0.5) / Float32(s * s))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999936531045e-21))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	else
		tmp = (single(1.0) / s) / (single(2.0) + (single(2.0) * (single(1.0) + ((x * x) * ((((x * x) * single(0.041666666666666664)) / (s * (s * (s * s)))) + (single(0.5) / (s * s)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999e-21

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 74.4%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 74.4%

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

       1. times-frac N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right), 4\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\frac{x}{s}\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       4. /-lowering-/.f32 76.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \mathsf{/.f32}\left(x, s\right)\right), 4\right)\right) \]

   11. Applied egg-rr 76.6%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   if 1.9999999e-21 < x

   1. Initial program 99.7%

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

   3. Simplified 99.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.1%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in s around -inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\left(1 + \left(-1 \cdot \frac{\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot {\left(\left|x\right|\right)}^{3} + \left|x\right| \cdot \left(-1 \cdot {\left(\left|x\right|\right)}^{2} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}\right)\right) + \frac{1}{2} \cdot \frac{\frac{-1}{2} \cdot \left({\left(\left|x\right|\right)}^{2} \cdot \left(-1 \cdot {\left(\left|x\right|\right)}^{2} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}\right)\right) + \left(\frac{-1}{6} \cdot {\left(\left|x\right|\right)}^{4} + \left|x\right| \cdot \left(\frac{-1}{6} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{1}{2} \cdot {\left(\left|x\right|\right)}^{3} + \left|x\right| \cdot \left(-1 \cdot {\left(\left|x\right|\right)}^{2} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}\right)\right)\right)\right)}{s}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right)\right)\right) \]

   9. Simplified 33.9%

      \[\leadsto \frac{\frac{1}{s}}{2 + 2 \cdot \color{blue}{\left(\left(1 + \frac{\left(x \cdot x\right) \cdot 0.5}{s \cdot s}\right) - \frac{\frac{0.5 \cdot \left(0.25 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.16666666666666666 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.16666666666666666\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.5\right)\right)\right)\right)}{s} + 0}{s \cdot \left(s \cdot s\right)}\right)}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2}}{{s}^{4}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{+.f32}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{x}^{2}}{{s}^{4}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right)}\right)\right)\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{{x}^{2}}{{s}^{4}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)}\right)\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24} \cdot \frac{{x}^{2}}{{s}^{4}}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\color{blue}{\frac{1}{24} \cdot \frac{{x}^{2}}{{s}^{4}}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)\right)\right)\right)\right)\right) \]

       5. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\left(\frac{1}{24} \cdot \frac{{x}^{2}}{{s}^{4}}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)}\right)\right)\right)\right)\right)\right) \]

   12. Simplified 93.9%

       \[\leadsto \frac{\frac{1}{s}}{2 + 2 \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(\frac{\left(x \cdot x\right) \cdot 0.041666666666666664}{s \cdot \left(s \cdot \left(s \cdot s\right)\right)} + \frac{0.5}{s \cdot s}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{\left(x \cdot x\right) \cdot 0.041666666666666664}{s \cdot \left(s \cdot \left(s \cdot s\right)\right)} + \frac{0.5}{s \cdot s}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.3% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{0.041666666666666664}{s \cdot \left(s \cdot \left(s \cdot s\right)\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.000000136226006e-28)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (if (<= x 4.999999980020986e-12)
     (/ (/ 1.0 (+ 4.0 (* x (/ x (* s s))))) s)
     (/
      (/ 1.0 s)
      (+
       2.0
       (*
        2.0
        (*
         (* (* x x) (* x x))
         (/ 0.041666666666666664 (* s (* s (* s s)))))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.000000136226006e-28f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else if (x <= 4.999999980020986e-12f) {
		tmp = (1.0f / (4.0f + (x * (x / (s * s))))) / s;
	} else {
		tmp = (1.0f / s) / (2.0f + (2.0f * (((x * x) * (x * x)) * (0.041666666666666664f / (s * (s * (s * s)))))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 5.000000136226006e-28) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else if (x <= 4.999999980020986e-12) then
        tmp = (1.0e0 / (4.0e0 + (x * (x / (s * s))))) / s
    else
        tmp = (1.0e0 / s) / (2.0e0 + (2.0e0 * (((x * x) * (x * x)) * (0.041666666666666664e0 / (s * (s * (s * s)))))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000136226006e-28))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	elseif (x <= Float32(4.999999980020986e-12))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(x * Float32(x / Float32(s * s))))) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(Float32(x * x) * Float32(x * x)) * Float32(Float32(0.041666666666666664) / Float32(s * Float32(s * Float32(s * s))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000136226006e-28))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	elseif (x <= single(4.999999980020986e-12))
		tmp = (single(1.0) / (single(4.0) + (x * (x / (s * s))))) / s;
	else
		tmp = (single(1.0) / s) / (single(2.0) + (single(2.0) * (((x * x) * (x * x)) * (single(0.041666666666666664) / (s * (s * (s * s)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < 5.00000014e-28

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 74.5%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 74.5%

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

       1. times-frac N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right), 4\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\frac{x}{s}\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       4. /-lowering-/.f32 76.9%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \mathsf{/.f32}\left(x, s\right)\right), 4\right)\right) \]

   11. Applied egg-rr 76.9%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   if 5.00000014e-28 < x < 4.99999998e-12

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

      1. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + 2\right)\right), s\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right), 2\right)\right), s\right) \]

      3. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\left|x\right|}{s}}\right), 2\right)\right), s\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\left|x\right|}{s}} + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right), 2\right)\right), s\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right), 2\right)\right), s\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right), 2\right)\right), s\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \cosh \left(\frac{\left|x\right|}{s}\right)\right), 2\right)\right), s\right) \]

      8. cosh-lowering-cosh.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right), 2\right)\right), s\right) \]

      9. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right), 2\right)\right), s\right) \]

      10. fabs-lowering-fabs.f32 100.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), 2\right)\right), s\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right), s\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right), s\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), 4\right)\right), s\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\left|x\right| \cdot \left|x\right|}{{s}^{2}}\right), 4\right)\right), s\right) \]

      4. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{x \cdot x}{{s}^{2}}\right), 4\right)\right), s\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(x \cdot \frac{x}{{s}^{2}}\right), 4\right)\right), s\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{x}{{s}^{2}}\right)\right), 4\right)\right), s\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left({s}^{2}\right)\right)\right), 4\right)\right), s\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left(s \cdot s\right)\right)\right), 4\right)\right), s\right) \]

      9. *-lowering-*.f32 88.8%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \mathsf{*.f32}\left(s, s\right)\right)\right), 4\right)\right), s\right) \]

   9. Simplified 88.8%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot \frac{x}{s \cdot s} + 4}}}{s} \]

   if 4.99999998e-12 < x

   1. Initial program 99.7%

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

   3. Simplified 98.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in s around -inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\left(1 + \left(-1 \cdot \frac{\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot {\left(\left|x\right|\right)}^{3} + \left|x\right| \cdot \left(-1 \cdot {\left(\left|x\right|\right)}^{2} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}\right)\right) + \frac{1}{2} \cdot \frac{\frac{-1}{2} \cdot \left({\left(\left|x\right|\right)}^{2} \cdot \left(-1 \cdot {\left(\left|x\right|\right)}^{2} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}\right)\right) + \left(\frac{-1}{6} \cdot {\left(\left|x\right|\right)}^{4} + \left|x\right| \cdot \left(\frac{-1}{6} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{1}{2} \cdot {\left(\left|x\right|\right)}^{3} + \left|x\right| \cdot \left(-1 \cdot {\left(\left|x\right|\right)}^{2} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}\right)\right)\right)\right)}{s}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right)\right)\right) \]

   9. Simplified 33.0%

      \[\leadsto \frac{\frac{1}{s}}{2 + 2 \cdot \color{blue}{\left(\left(1 + \frac{\left(x \cdot x\right) \cdot 0.5}{s \cdot s}\right) - \frac{\frac{0.5 \cdot \left(0.25 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.16666666666666666 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.16666666666666666\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.5\right)\right)\right)\right)}{s} + 0}{s \cdot \left(s \cdot s\right)}\right)}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\left(\frac{1}{24} \cdot \frac{{x}^{4}}{{s}^{4}}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \left(\frac{{x}^{4}}{{s}^{4}} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

       2. associate-*l/ N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \left(\frac{{x}^{4} \cdot \frac{1}{24}}{\color{blue}{{s}^{4}}}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \left({x}^{4} \cdot \color{blue}{\frac{\frac{1}{24}}{{s}^{4}}}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \left({x}^{4} \cdot \frac{\frac{1}{24} \cdot 1}{{\color{blue}{s}}^{4}}\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \left({x}^{4} \cdot \left(\frac{1}{24} \cdot \color{blue}{\frac{1}{{s}^{4}}}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{s}^{4}}\right)}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \frac{1}{{s}^{4}}\right)\right)\right)\right)\right) \]

       8. pow-sqr N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{1}{{s}^{4}}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{1}{{s}^{4}}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \frac{1}{{s}^{4}}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \frac{1}{{s}^{4}}\right)\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(x \cdot x\right)\right), \left(\frac{1}{24} \cdot \frac{1}{{s}^{4}}\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \left(\frac{1}{24} \cdot \frac{1}{{s}^{4}}\right)\right)\right)\right)\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \left(\frac{\frac{1}{24} \cdot 1}{\color{blue}{{s}^{4}}}\right)\right)\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \left(\frac{\frac{1}{24}}{{\color{blue}{s}}^{4}}\right)\right)\right)\right)\right) \]

       16. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \color{blue}{\left({s}^{4}\right)}\right)\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \left({s}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right)\right)\right) \]

       18. pow-sqr N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \left({s}^{2} \cdot \color{blue}{{s}^{2}}\right)\right)\right)\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \left(\left(s \cdot s\right) \cdot {\color{blue}{s}}^{2}\right)\right)\right)\right)\right)\right) \]

       20. associate-*l* N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \left(s \cdot \color{blue}{\left(s \cdot {s}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \left(s \cdot \left(s \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right)\right)\right) \]

       22. cube-mult N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \left(s \cdot {s}^{\color{blue}{3}}\right)\right)\right)\right)\right)\right) \]

       23. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(s, \color{blue}{\left({s}^{3}\right)}\right)\right)\right)\right)\right)\right) \]

       24. cube-mult N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(s, \left(s \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right)\right)\right)\right) \]

       25. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(s, \left(s \cdot {s}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

       26. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right)\right)\right) \]

       28. *-lowering-*.f32 91.5%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 91.5%

       \[\leadsto \frac{\frac{1}{s}}{2 + 2 \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{0.041666666666666664}{s \cdot \left(s \cdot \left(s \cdot s\right)\right)}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{0.041666666666666664}{s \cdot \left(s \cdot \left(s \cdot s\right)\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.7% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.4999999397961872 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(\left(s \cdot s\right) \cdot -4\right)}{x \cdot x}}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.000000136226006e-28)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (if (<= x 1.4999999397961872e-13)
     (/ (/ 1.0 (+ 4.0 (* x (/ x (* s s))))) s)
     (/ (/ (* s (* (* s s) -4.0)) (* x x)) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.000000136226006e-28f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else if (x <= 1.4999999397961872e-13f) {
		tmp = (1.0f / (4.0f + (x * (x / (s * s))))) / s;
	} else {
		tmp = ((s * ((s * s) * -4.0f)) / (x * x)) / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 5.000000136226006e-28) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else if (x <= 1.4999999397961872e-13) then
        tmp = (1.0e0 / (4.0e0 + (x * (x / (s * s))))) / s
    else
        tmp = ((s * ((s * s) * (-4.0e0))) / (x * x)) / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000136226006e-28))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	elseif (x <= Float32(1.4999999397961872e-13))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(x * Float32(x / Float32(s * s))))) / s);
	else
		tmp = Float32(Float32(Float32(s * Float32(Float32(s * s) * Float32(-4.0))) / Float32(x * x)) / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000136226006e-28))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	elseif (x <= single(1.4999999397961872e-13))
		tmp = (single(1.0) / (single(4.0) + (x * (x / (s * s))))) / s;
	else
		tmp = ((s * ((s * s) * single(-4.0))) / (x * x)) / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < 5.00000014e-28

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 74.5%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 74.5%

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

       1. times-frac N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right), 4\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\frac{x}{s}\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       4. /-lowering-/.f32 76.9%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \mathsf{/.f32}\left(x, s\right)\right), 4\right)\right) \]

   11. Applied egg-rr 76.9%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   if 5.00000014e-28 < x < 1.49999994e-13

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

      1. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + 2\right)\right), s\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right), 2\right)\right), s\right) \]

      3. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\left|x\right|}{s}}\right), 2\right)\right), s\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\left|x\right|}{s}} + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right), 2\right)\right), s\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right), 2\right)\right), s\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right), 2\right)\right), s\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \cosh \left(\frac{\left|x\right|}{s}\right)\right), 2\right)\right), s\right) \]

      8. cosh-lowering-cosh.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right), 2\right)\right), s\right) \]

      9. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right), 2\right)\right), s\right) \]

      10. fabs-lowering-fabs.f32 100.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), 2\right)\right), s\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right), s\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right), s\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), 4\right)\right), s\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\left|x\right| \cdot \left|x\right|}{{s}^{2}}\right), 4\right)\right), s\right) \]

      4. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{x \cdot x}{{s}^{2}}\right), 4\right)\right), s\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(x \cdot \frac{x}{{s}^{2}}\right), 4\right)\right), s\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{x}{{s}^{2}}\right)\right), 4\right)\right), s\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left({s}^{2}\right)\right)\right), 4\right)\right), s\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left(s \cdot s\right)\right)\right), 4\right)\right), s\right) \]

      9. *-lowering-*.f32 95.6%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \mathsf{*.f32}\left(s, s\right)\right)\right), 4\right)\right), s\right) \]

   9. Simplified 95.6%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot \frac{x}{s \cdot s} + 4}}}{s} \]

   if 1.49999994e-13 < x

   1. Initial program 99.7%

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

   3. Simplified 99.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 78.1%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 78.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x \cdot x}{s \cdot s} + 4}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{s + -4 \cdot \frac{{s}^{3}}{{x}^{2}}}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(s + -4 \cdot \frac{{s}^{3}}{{x}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       2. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \left(-4 \cdot \frac{{s}^{3}}{{x}^{2}}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \left(\frac{-4 \cdot {s}^{3}}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right) \]

       4. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\left(-4 \cdot {s}^{3}\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\left({s}^{3} \cdot -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({s}^{3}\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot \left(s \cdot s\right)\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot {s}^{2}\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \left({s}^{2}\right)\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \left(s \cdot s\right)\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \left(x \cdot x\right)\right)\right), \left({x}^{2}\right)\right) \]

       13. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \mathsf{*.f32}\left(x, x\right)\right)\right), \left({x}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \mathsf{*.f32}\left(x, x\right)\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       15. *-lowering-*.f32 56.5%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 56.5%

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

   13. Taylor expanded in s around inf

       \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(-4 \cdot \frac{{s}^{3}}{{x}^{2}}\right)}, \mathsf{*.f32}\left(x, x\right)\right) \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\frac{-4 \cdot {s}^{3}}{{x}^{2}}\right), \mathsf{*.f32}\left(\color{blue}{x}, x\right)\right) \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-4 \cdot {s}^{3}\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(\color{blue}{x}, x\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({s}^{3} \cdot -4\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\left(s \cdot \left(s \cdot s\right)\right) \cdot -4\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\left(s \cdot {s}^{2}\right) \cdot -4\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(s \cdot \left({s}^{2} \cdot -4\right)\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(s \cdot \left(-4 \cdot {s}^{2}\right)\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left(-4 \cdot {s}^{2}\right)\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left({s}^{2} \cdot -4\right)\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left({s}^{2}\right), -4\right)\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left(s \cdot s\right), -4\right)\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       12. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \left({x}^{2}\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \left(x \cdot x\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

       14. *-lowering-*.f32 87.0%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{*.f32}\left(x, x\right)\right) \]

   15. Simplified 87.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.4999999397961872 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot \left(\left(s \cdot s\right) \cdot -4\right)}{x \cdot x}}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.8% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -4\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.000000136226006e-28)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (if (<= x 4.999999980020986e-12)
     (/ (/ 1.0 (+ 4.0 (* x (/ x (* s s))))) s)
     (/ (* s (* (* s s) -4.0)) (* (* x x) (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.000000136226006e-28f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else if (x <= 4.999999980020986e-12f) {
		tmp = (1.0f / (4.0f + (x * (x / (s * s))))) / s;
	} else {
		tmp = (s * ((s * s) * -4.0f)) / ((x * x) * (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 5.000000136226006e-28) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else if (x <= 4.999999980020986e-12) then
        tmp = (1.0e0 / (4.0e0 + (x * (x / (s * s))))) / s
    else
        tmp = (s * ((s * s) * (-4.0e0))) / ((x * x) * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000136226006e-28))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	elseif (x <= Float32(4.999999980020986e-12))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(x * Float32(x / Float32(s * s))))) / s);
	else
		tmp = Float32(Float32(s * Float32(Float32(s * s) * Float32(-4.0))) / Float32(Float32(x * x) * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000136226006e-28))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	elseif (x <= single(4.999999980020986e-12))
		tmp = (single(1.0) / (single(4.0) + (x * (x / (s * s))))) / s;
	else
		tmp = (s * ((s * s) * single(-4.0))) / ((x * x) * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < 5.00000014e-28

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 74.5%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 74.5%

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

       1. times-frac N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right), 4\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\frac{x}{s}\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       4. /-lowering-/.f32 76.9%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \mathsf{/.f32}\left(x, s\right)\right), 4\right)\right) \]

   11. Applied egg-rr 76.9%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   if 5.00000014e-28 < x < 4.99999998e-12

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

      1. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + 2\right)\right), s\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right), 2\right)\right), s\right) \]

      3. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\left|x\right|}{s}}\right), 2\right)\right), s\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\left|x\right|}{s}} + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right), 2\right)\right), s\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right), 2\right)\right), s\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right), 2\right)\right), s\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \cosh \left(\frac{\left|x\right|}{s}\right)\right), 2\right)\right), s\right) \]

      8. cosh-lowering-cosh.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right), 2\right)\right), s\right) \]

      9. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right), 2\right)\right), s\right) \]

      10. fabs-lowering-fabs.f32 100.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), 2\right)\right), s\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right), s\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right), s\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), 4\right)\right), s\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\left|x\right| \cdot \left|x\right|}{{s}^{2}}\right), 4\right)\right), s\right) \]

      4. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{x \cdot x}{{s}^{2}}\right), 4\right)\right), s\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(x \cdot \frac{x}{{s}^{2}}\right), 4\right)\right), s\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{x}{{s}^{2}}\right)\right), 4\right)\right), s\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left({s}^{2}\right)\right)\right), 4\right)\right), s\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left(s \cdot s\right)\right)\right), 4\right)\right), s\right) \]

      9. *-lowering-*.f32 88.8%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \mathsf{*.f32}\left(s, s\right)\right)\right), 4\right)\right), s\right) \]

   9. Simplified 88.8%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot \frac{x}{s \cdot s} + 4}}}{s} \]

   if 4.99999998e-12 < x

   1. Initial program 99.7%

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

   3. Simplified 98.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 79.4%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 79.4%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x \cdot x}{s \cdot s} + 4}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{s + -4 \cdot \frac{{s}^{3}}{{x}^{2}}}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(s + -4 \cdot \frac{{s}^{3}}{{x}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       2. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \left(-4 \cdot \frac{{s}^{3}}{{x}^{2}}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \left(\frac{-4 \cdot {s}^{3}}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right) \]

       4. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\left(-4 \cdot {s}^{3}\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\left({s}^{3} \cdot -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({s}^{3}\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot \left(s \cdot s\right)\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot {s}^{2}\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \left({s}^{2}\right)\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \left(s \cdot s\right)\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \left(x \cdot x\right)\right)\right), \left({x}^{2}\right)\right) \]

       13. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \mathsf{*.f32}\left(x, x\right)\right)\right), \left({x}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \mathsf{*.f32}\left(x, x\right)\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       15. *-lowering-*.f32 58.1%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -4\right), \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 58.1%

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

   13. Taylor expanded in s around inf

       \[\leadsto \color{blue}{-4 \cdot \frac{{s}^{3}}{{x}^{4}}} \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-4 \cdot {s}^{3}}{\color{blue}{{x}^{4}}} \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(-4 \cdot {s}^{3}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\left({s}^{3} \cdot -4\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\left(s \cdot \left(s \cdot s\right)\right) \cdot -4\right), \left({x}^{4}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\left(s \cdot {s}^{2}\right) \cdot -4\right), \left({x}^{4}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \left({s}^{2} \cdot -4\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \left(-4 \cdot {s}^{2}\right)\right), \left({x}^{4}\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left(-4 \cdot {s}^{2}\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left({s}^{2} \cdot -4\right)\right), \left({x}^{4}\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left({s}^{2}\right), -4\right)\right), \left({x}^{4}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left(s \cdot s\right), -4\right)\right), \left({x}^{4}\right)\right) \]

       12. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \left({x}^{4}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       14. pow-sqr N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right) \]

       15. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

       17. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       19. *-lowering-*.f32 87.6%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right) \]

   15. Simplified 87.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -4\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.1% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.000000136226006e-28)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (/ (/ 1.0 (+ 4.0 (* x (/ x (* s s))))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.000000136226006e-28f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else {
		tmp = (1.0f / (4.0f + (x * (x / (s * s))))) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 5.000000136226006e-28) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else
        tmp = (1.0e0 / (4.0e0 + (x * (x / (s * s))))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000136226006e-28))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(x * Float32(x / Float32(s * s))))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000136226006e-28))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	else
		tmp = (single(1.0) / (single(4.0) + (x * (x / (s * s))))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000014e-28

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 74.5%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 74.5%

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

       1. times-frac N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right), 4\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\frac{x}{s}\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       4. /-lowering-/.f32 76.9%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \mathsf{/.f32}\left(x, s\right)\right), 4\right)\right) \]

   11. Applied egg-rr 76.9%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   if 5.00000014e-28 < x

   1. Initial program 99.7%

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

   3. Simplified 99.2%

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

      1. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + 2\right)\right), s\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right), 2\right)\right), s\right) \]

      3. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\left|x\right|}{s}}\right), 2\right)\right), s\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\left|x\right|}{s}} + e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right), 2\right)\right), s\right) \]

      5. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right), 2\right)\right), s\right) \]

      6. cosh-undef N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right), 2\right)\right), s\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \cosh \left(\frac{\left|x\right|}{s}\right)\right), 2\right)\right), s\right) \]

      8. cosh-lowering-cosh.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right), 2\right)\right), s\right) \]

      9. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right), 2\right)\right), s\right) \]

      10. fabs-lowering-fabs.f32 99.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), 2\right)\right), s\right) \]

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right), s\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right), s\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), 4\right)\right), s\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\left|x\right| \cdot \left|x\right|}{{s}^{2}}\right), 4\right)\right), s\right) \]

      4. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{x \cdot x}{{s}^{2}}\right), 4\right)\right), s\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(x \cdot \frac{x}{{s}^{2}}\right), 4\right)\right), s\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{x}{{s}^{2}}\right)\right), 4\right)\right), s\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left({s}^{2}\right)\right)\right), 4\right)\right), s\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left(s \cdot s\right)\right)\right), 4\right)\right), s\right) \]

      9. *-lowering-*.f32 81.4%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \mathsf{*.f32}\left(s, s\right)\right)\right), 4\right)\right), s\right) \]

   9. Simplified 81.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.4% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999998199587e-24)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (/ 1.0 (* s (+ 4.0 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.999999998199587e-24f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else {
		tmp = 1.0f / (s * (4.0f + ((x * x) / (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.999999998199587e-24) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else
        tmp = 1.0e0 / (s * (4.0e0 + ((x * x) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.999999998199587e-24))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.999999998199587e-24))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	else
		tmp = single(1.0) / (s * (single(4.0) + ((x * x) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1e-23

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 74.0%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 74.0%

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

       1. times-frac N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right), 4\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\frac{x}{s}\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(\frac{x}{s}\right)\right), 4\right)\right) \]

       4. /-lowering-/.f32 76.4%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \mathsf{/.f32}\left(x, s\right)\right), 4\right)\right) \]

   11. Applied egg-rr 76.4%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   if 1e-23 < x

   1. Initial program 99.7%

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

   3. Simplified 99.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.1%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 79.9%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 79.9%

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

       1. associate-/l/ N/A

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

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(\frac{x \cdot x}{s \cdot s} + 4\right) \cdot s\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(s \cdot \color{blue}{\left(\frac{x \cdot x}{s \cdot s} + 4\right)}\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(\frac{x \cdot x}{s \cdot s} + 4\right)}\right)\right) \]

       5. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{x \cdot x}{s \cdot s}\right), \color{blue}{4}\right)\right)\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left(s \cdot s\right)\right), 4\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right)\right) \]

       8. *-lowering-*.f32 79.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right)\right) \]

   11. Applied egg-rr 79.9%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.6% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999998199587e-24)
   (/ 0.25 s)
   (/ 1.0 (* s (+ 4.0 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.999999998199587e-24f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (s * (4.0f + ((x * x) / (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.999999998199587e-24) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (s * (4.0e0 + ((x * x) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.999999998199587e-24))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.999999998199587e-24))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (s * (single(4.0) + ((x * x) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1e-23

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 38.1%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   5. Simplified 38.1%

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

   if 1e-23 < x

   1. Initial program 99.7%

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

   3. Simplified 99.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.1%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 79.9%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 79.9%

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

       1. associate-/l/ N/A

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

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(\frac{x \cdot x}{s \cdot s} + 4\right) \cdot s\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(s \cdot \color{blue}{\left(\frac{x \cdot x}{s \cdot s} + 4\right)}\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(\frac{x \cdot x}{s \cdot s} + 4\right)}\right)\right) \]

       5. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{x \cdot x}{s \cdot s}\right), \color{blue}{4}\right)\right)\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left(s \cdot s\right)\right), 4\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right)\right) \]

       8. *-lowering-*.f32 79.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right)\right) \]

   11. Applied egg-rr 79.9%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.199999961918627 \cdot 10^{-19}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{x \cdot \frac{x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.199999961918627e-19)
   (/ 0.25 s)
   (/ (/ 1.0 s) (* x (/ x (* s s))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.199999961918627e-19f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / s) / (x * (x / (s * s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.199999961918627e-19) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / s) / (x * (x / (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.199999961918627e-19))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(x * Float32(x / Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.199999961918627e-19))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / s) / (x * (x / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.19999996e-19

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 39.7%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   5. Simplified 39.7%

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

   if 1.19999996e-19 < x

   1. Initial program 99.7%

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

   3. Simplified 99.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 99.1%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 79.7%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 79.7%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x \cdot x}{s \cdot s} + 4}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(\frac{{x}^{2}}{{s}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{x \cdot x}{{\color{blue}{s}}^{2}}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(x \cdot \color{blue}{\frac{x}{{s}^{2}}}\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{x}{{s}^{2}}\right)}\right)\right) \]

       4. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \color{blue}{\left({s}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left(s \cdot \color{blue}{s}\right)\right)\right)\right) \]

       6. *-lowering-*.f32 75.1%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   12. Simplified 75.1%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{x \cdot \frac{x}{s \cdot s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.3% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000467443897e-7) (/ 0.25 s) (/ 1.0 (/ x (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000467443897e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x / (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.0000000467443897e-7) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x / (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000467443897e-7))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x / (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000005e-7

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 38.3%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   5. Simplified 38.3%

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

   if 4.00000005e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 100.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 81.1%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 81.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x \cdot x}{s \cdot s} + 4}} \]

   10. Taylor expanded in s around 0

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

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(s, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(s, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f32 64.7%

          \[\leadsto \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   13. Step-by-step derivation

       1. associate-/r* N/A

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

       2. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{s}{x}}}} \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{x}{\frac{s}{x}}\right)}\right) \]

       4. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(x, \color{blue}{\left(\frac{s}{x}\right)}\right)\right) \]

       5. /-lowering-/.f32 68.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(x, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right)\right) \]

   14. Applied egg-rr 68.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{s}{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 45.6% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000467443897e-7) (/ 0.25 s) (* (/ s x) (/ 1.0 x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000467443897e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s / x) * (1.0f / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.0000000467443897e-7) then
        tmp = 0.25e0 / s
    else
        tmp = (s / x) * (1.0e0 / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s / x) * Float32(Float32(1.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000467443897e-7))
		tmp = single(0.25) / s;
	else
		tmp = (s / x) * (single(1.0) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000005e-7

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 38.3%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   5. Simplified 38.3%

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

   if 4.00000005e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 100.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 81.1%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 81.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x \cdot x}{s \cdot s} + 4}} \]

   10. Taylor expanded in s around 0

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

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(s, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(s, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f32 64.7%

          \[\leadsto \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   13. Step-by-step derivation

       1. associate-/r* N/A

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

       2. div-inv N/A

          \[\leadsto \frac{s}{x} \cdot \color{blue}{\frac{1}{x}} \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{s}{x}\right), \color{blue}{\left(\frac{1}{x}\right)}\right) \]

       4. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, x\right), \left(\frac{\color{blue}{1}}{x}\right)\right) \]

       5. /-lowering-/.f32 64.7%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, x\right), \mathsf{/.f32}\left(1, \color{blue}{x}\right)\right) \]

   14. Applied egg-rr 64.7%

       \[\leadsto \color{blue}{\frac{s}{x} \cdot \frac{1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 45.6% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000467443897e-7) (/ 0.25 s) (* s (/ 1.0 (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000467443897e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = s * (1.0f / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.0000000467443897e-7) then
        tmp = 0.25e0 / s
    else
        tmp = s * (1.0e0 / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s * Float32(Float32(1.0) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000467443897e-7))
		tmp = single(0.25) / s;
	else
		tmp = s * (single(1.0) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000005e-7

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 38.3%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   5. Simplified 38.3%

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

   if 4.00000005e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 100.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 81.1%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 81.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x \cdot x}{s \cdot s} + 4}} \]

   10. Taylor expanded in s around 0

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

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(s, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(s, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f32 64.7%

          \[\leadsto \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   13. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{x \cdot x} \cdot \color{blue}{s} \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{1}{x \cdot x}\right), \color{blue}{s}\right) \]

       4. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot x\right)\right), s\right) \]

       5. *-lowering-*.f32 64.7%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, x\right)\right), s\right) \]

   14. Applied egg-rr 64.7%

       \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.6% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000467443897e-7) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000467443897e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.0000000467443897e-7) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000467443897e-7))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000005e-7

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 38.3%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   5. Simplified 38.3%

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

   if 4.00000005e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 100.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      5. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right) \]

      8. *-lowering-*.f32 81.1%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right) \]

   9. Simplified 81.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x \cdot x}{s \cdot s} + 4}} \]

   10. Taylor expanded in s around 0

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

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(s, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(s, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f32 64.7%

          \[\leadsto \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 27.0% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. /-lowering-/.f32 26.9%

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

5. Simplified 26.9%

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

Reproduce

herbie shell --seed 2024191 
(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))))))