Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 11.9s

Alternatives: 16

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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(((-1.0e0) / (s / abs(x)))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. associate-/l/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

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

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

   8. distribute-frac-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-frac-neg N/A

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

   11. cosh-undef N/A

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

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

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

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

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

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

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

   15. 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. Step-by-step derivation

   1. frac-2neg N/A

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

   2. distribute-frac-neg N/A

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

   3. cosh-neg N/A

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

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

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

   5. distribute-frac-neg2 N/A

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

   6. clear-num N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. fabs-lowering-fabs.f32 99.5%

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

8. Applied egg-rr 99.5%

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

Alternative 2: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0000000063421537 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{elif}\;\left|x\right| \leq 5.999999941330714 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + x \cdot \frac{x}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\left|x\right|}{-s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 2.0000000063421537e-29)
   (/ 1.0 (* s (+ 4.0 (* (/ x s) (/ x s)))))
   (if (<= (fabs x) 5.999999941330714e-10)
     (/ 1.0 (* s (+ 4.0 (* x (/ x (* s s))))))
     (exp (/ (fabs x) (- s))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 2.0000000063421537e-29f) {
		tmp = 1.0f / (s * (4.0f + ((x / s) * (x / s))));
	} else if (fabsf(x) <= 5.999999941330714e-10f) {
		tmp = 1.0f / (s * (4.0f + (x * (x / (s * s)))));
	} else {
		tmp = expf((fabsf(x) / -s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 2.0000000063421537e-29) then
        tmp = 1.0e0 / (s * (4.0e0 + ((x / s) * (x / s))))
    else if (abs(x) <= 5.999999941330714e-10) then
        tmp = 1.0e0 / (s * (4.0e0 + (x * (x / (s * s)))))
    else
        tmp = exp((abs(x) / -s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(2.0000000063421537e-29))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s)))));
	elseif (abs(x) <= Float32(5.999999941330714e-10))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(x * Float32(x / Float32(s * s))))));
	else
		tmp = exp(Float32(abs(x) / Float32(-s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(2.0000000063421537e-29))
		tmp = single(1.0) / (s * (single(4.0) + ((x / s) * (x / s))));
	elseif (abs(x) <= single(5.999999941330714e-10))
		tmp = single(1.0) / (s * (single(4.0) + (x * (x / (s * s)))));
	else
		tmp = exp((abs(x) / -s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 2.00000001e-29

   1. Initial program 97.2%

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

   3. Simplified 97.4%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-frac-neg N/A

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

      11. cosh-undef N/A

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

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

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

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

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

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

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

      15. fabs-lowering-fabs.f32 97.5%

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

   6. Applied egg-rr 97.5%

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

   7. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 53.7%

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

   9. Simplified 53.7%

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

       1. times-frac N/A

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

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

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

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

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

       4. /-lowering-/.f32 86.7%

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

   11. Applied egg-rr 86.7%

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

   if 2.00000001e-29 < (fabs.f32 x) < 5.99999994e-10

   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 99.9%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-frac-neg N/A

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

      11. cosh-undef N/A

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

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

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

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

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

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

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

      15. fabs-lowering-fabs.f32 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 84.2%

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

   9. Simplified 84.2%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. *-lowering-*.f32 90.6%

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

   11. Applied egg-rr 90.6%

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

   if 5.99999994e-10 < (fabs.f32 x)

   1. Initial program 99.9%

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

   3. 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 98.0%

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

   5. Simplified 98.0%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

      11. neg-lowering-neg.f32 98.0%

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

   7. Applied egg-rr 98.0%

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

      1. clear-num N/A

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

      2. associate-/l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. rem-exp-log N/A

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

      7. div-exp N/A

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

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

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. clear-num N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

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

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

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

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

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

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

      17. rem-exp-log N/A

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

      18. log-lowering-log.f32 N/A

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

      19. rem-exp-log N/A

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

      20. /-lowering-/.f32 98.0%

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

   9. Applied egg-rr 98.0%

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

   10. Taylor expanded in s around 0

       \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{\left|x\right|}{s}\right)}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{\left|x\right|}{s}\right)\right) \]

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

          \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{\left|x\right|}{s}\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right) \]

       5. fabs-lowering-fabs.f32 97.3%

          \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right) \]

   12. Simplified 97.3%

       \[\leadsto e^{\color{blue}{0 - \frac{\left|x\right|}{s}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0000000063421537 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{elif}\;\left|x\right| \leq 5.999999941330714 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + x \cdot \frac{x}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\left|x\right|}{-s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.5%

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

3. Simplified 99.5%

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

   1. associate-/l/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

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

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

   8. distribute-frac-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-frac-neg N/A

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

   11. cosh-undef N/A

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

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

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

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

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

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

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

   15. 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 4: 96.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{3 + e^{\frac{\left|x\right|}{s}}}}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 1.0 (+ 3.0 (exp (/ (fabs x) s)))) s))

C

float code(float x, float s) {
	return (1.0f / (3.0f + expf((fabsf(x) / s)))) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / (3.0e0 + exp((abs(x) / s)))) / s
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(Float32(3.0) + exp(Float32(abs(x) / s)))) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / (single(3.0) + exp((abs(x) / s)))) / s;
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

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

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

6. Applied egg-rr 99.5%

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

   1. +-commutative N/A

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

   2. cosh-undef N/A

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

   3. distribute-frac-neg N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

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

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

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

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

   10. distribute-frac-neg N/A

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

   11. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in s around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\color{blue}{3}, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right), s\right) \]
10. Step-by-step derivation

11. Simplified 96.0%

    \[\leadsto \frac{\frac{1}{\color{blue}{3} + e^{\frac{\left|x\right|}{s}}}}{s} \]
12. Add Preprocessing

Alternative 5: 94.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return expf((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((abs(x) / -s)) / (s * 4.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. 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 94.9%

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

5. Simplified 94.9%

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

6. Final simplification 94.9%

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

Alternative 6: 94.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. 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 94.9%

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

5. Simplified 94.9%

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

   1. div-inv N/A

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

   2. distribute-frac-neg N/A

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

   3. exp-neg N/A

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

   4. associate-*l/ N/A

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

   5. div-inv N/A

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

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

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

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

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

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

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

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

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

   13. fabs-lowering-fabs.f32 94.9%

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

7. Applied egg-rr 94.9%

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

Alternative 7: 83.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{1}{s \cdot s} \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 2.0000000390829628e-25)
   (/ 1.0 (* s (+ 4.0 (* (/ x s) (/ x s)))))
   (/ 1.0 (* s (+ 4.0 (* (/ 1.0 (* s s)) (* x x)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 2.0000000390829628e-25f) {
		tmp = 1.0f / (s * (4.0f + ((x / s) * (x / s))));
	} else {
		tmp = 1.0f / (s * (4.0f + ((1.0f / (s * 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 (abs(x) <= 2.0000000390829628e-25) then
        tmp = 1.0e0 / (s * (4.0e0 + ((x / s) * (x / s))))
    else
        tmp = 1.0e0 / (s * (4.0e0 + ((1.0e0 / (s * s)) * (x * x))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(2.0000000390829628e-25))
		tmp = Float32(Float32(1.0) / Float32(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(Float32(1.0) / Float32(s * s)) * Float32(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(2.0000000390829628e-25))
		tmp = single(1.0) / (s * (single(4.0) + ((x / s) * (x / s))));
	else
		tmp = single(1.0) / (s * (single(4.0) + ((single(1.0) / (s * s)) * (x * x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 2.00000004e-25

   1. Initial program 97.7%

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

   3. Simplified 97.8%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-frac-neg N/A

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

      11. cosh-undef N/A

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

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

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

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

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

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

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

      15. fabs-lowering-fabs.f32 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 52.0%

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

   9. Simplified 52.0%

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

       1. times-frac N/A

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

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

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

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

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

       4. /-lowering-/.f32 80.0%

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

   11. Applied egg-rr 80.0%

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

   if 2.00000004e-25 < (fabs.f32 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-frac-neg N/A

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

      11. cosh-undef N/A

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

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

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

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

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

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

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

      15. fabs-lowering-fabs.f32 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 83.6%

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

   9. Simplified 83.6%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

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

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

       6. *-lowering-*.f32 86.8%

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

   11. Applied egg-rr 86.8%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{1}{s \cdot s} \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.5% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000063421537 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000063421537e-29)
   (/ 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 <= 2.0000000063421537e-29f) {
		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 <= 2.0000000063421537e-29) 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(2.0000000063421537e-29))
		tmp = Float32(Float32(1.0) / Float32(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(x * Float32(x / Float32(s * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.0000000063421537e-29))
		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 < 2.00000001e-29

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-frac-neg N/A

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

      11. cosh-undef N/A

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

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

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

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

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

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

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

      15. fabs-lowering-fabs.f32 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 73.8%

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

   9. Simplified 73.8%

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

       1. times-frac N/A

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

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

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

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

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

       4. /-lowering-/.f32 77.6%

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

   11. Applied egg-rr 77.6%

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

   if 2.00000001e-29 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-frac-neg N/A

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

      11. cosh-undef N/A

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

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

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

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

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

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

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

      15. fabs-lowering-fabs.f32 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 82.2%

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

   9. Simplified 82.2%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. *-lowering-*.f32 83.1%

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

   11. Applied egg-rr 83.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.0000000063421537 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.9% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9000000043279782 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s} - \frac{s}{-0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{x \cdot x}{s \cdot s}}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9000000043279782e-19)
   (/ 1.0 (- (/ (* x x) s) (/ s -0.25)))
   (/ (/ 1.0 (/ (* x x) (* s s))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9000000043279782e-19f) {
		tmp = 1.0f / (((x * x) / s) - (s / -0.25f));
	} else {
		tmp = (1.0f / ((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 <= 1.9000000043279782e-19) then
        tmp = 1.0e0 / (((x * x) / s) - (s / (-0.25e0)))
    else
        tmp = (1.0e0 / ((x * x) / (s * s))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9000000043279782e-19))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * x) / s) - Float32(s / Float32(-0.25))));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(x * x) / Float32(s * s))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9000000043279782e-19))
		tmp = single(1.0) / (((x * x) / s) - (s / single(-0.25)));
	else
		tmp = (single(1.0) / ((x * x) / (s * s))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.9e-19

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-frac-neg N/A

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

      11. cosh-undef N/A

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

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

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

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

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

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

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

      15. fabs-lowering-fabs.f32 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 74.7%

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

   9. Simplified 74.7%

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

       1. distribute-lft-in N/A

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

       2. metadata-eval N/A

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

       3. div-inv N/A

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

       4. fma-define N/A

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

       5. frac-2neg N/A

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

       6. distribute-frac-neg N/A

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

       7. fmm-undef N/A

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

       8. *-commutative N/A

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

       9. div-inv N/A

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

       10. associate-*l* N/A

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

       11. pow2 N/A

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

       12. pow-flip N/A

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

       13. pow-plus N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. inv-pow N/A

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

       17. div-inv N/A

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

       18. --lowering--.f32 N/A

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

       19. /-lowering-/.f32 N/A

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

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

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

       21. /-lowering-/.f32 N/A

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

       22. metadata-eval 67.8%

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

   11. Applied egg-rr 67.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s} - \frac{s}{-0.25}}} \]

   if 1.9e-19 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

   6. Applied egg-rr 99.9%

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

      1. +-commutative N/A

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

      2. cosh-undef N/A

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

      3. distribute-frac-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

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

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

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

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

      10. distribute-frac-neg N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in s around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

   11. Simplified 72.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{-\frac{x \cdot x + 0}{s}}{s}}}}{s} \]

   12. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f32 74.7%

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

   14. Simplified 74.7%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{x \cdot x}{s \cdot s}}}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 68.0% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s} - \frac{s}{-0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \frac{s}{x \cdot x}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9)
   (/ 1.0 (- (/ (* x x) s) (/ s -0.25)))
   (/ (* s (/ s (* x x))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		tmp = 1.0f / (((x * x) / s) - (s / -0.25f));
	} else {
		tmp = (s * (s / (x * x))) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.99999993922529e-9) then
        tmp = 1.0e0 / (((x * x) / s) - (s / (-0.25e0)))
    else
        tmp = (s * (s / (x * x))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * x) / s) - Float32(s / Float32(-0.25))));
	else
		tmp = Float32(Float32(s * Float32(s / Float32(x * x))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.99999993922529e-9))
		tmp = single(1.0) / (((x * x) / s) - (s / single(-0.25)));
	else
		tmp = (s * (s / (x * x))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999994e-9

   1. Initial program 99.2%

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

   3. Simplified 99.3%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

      8. distribute-frac-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-frac-neg N/A

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

      11. cosh-undef N/A

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

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

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

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

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

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

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

      15. fabs-lowering-fabs.f32 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 74.8%

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

   9. Simplified 74.8%

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

       1. distribute-lft-in N/A

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

       2. metadata-eval N/A

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

       3. div-inv N/A

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

       4. fma-define N/A

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

       5. frac-2neg N/A

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

       6. distribute-frac-neg N/A

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

       7. fmm-undef N/A

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

       8. *-commutative N/A

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

       9. div-inv N/A

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

       10. associate-*l* N/A

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

       11. pow2 N/A

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

       12. pow-flip N/A

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

       13. pow-plus N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. inv-pow N/A

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

       17. div-inv N/A

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

       18. --lowering--.f32 N/A

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

       19. /-lowering-/.f32 N/A

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

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

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

       21. /-lowering-/.f32 N/A

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

       22. metadata-eval 62.7%

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

   11. Applied egg-rr 62.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s} - \frac{s}{-0.25}}} \]

   if 9.99999994e-9 < 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 99.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

   6. Applied egg-rr 100.0%

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

      1. +-commutative N/A

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

      2. cosh-undef N/A

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

      3. distribute-frac-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

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

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

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

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

      10. distribute-frac-neg N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in s around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

   11. Simplified 83.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{-\frac{x \cdot x + 0}{s}}{s}}}}{s} \]

   12. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f32 73.4%

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

   14. Simplified 73.4%

       \[\leadsto \frac{\color{blue}{s \cdot \frac{s}{x \cdot x}}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 49.0% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \frac{s}{x \cdot x}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (/ (* s (/ s (* x x))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s * (s / (x * x))) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.99999993922529e-9) then
        tmp = 0.25e0 / s
    else
        tmp = (s * (s / (x * x))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s * Float32(s / Float32(x * x))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.99999993922529e-9))
		tmp = single(0.25) / s;
	else
		tmp = (s * (s / (x * x))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999994e-9

   1. Initial program 99.2%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 37.1%

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

   5. Simplified 37.1%

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

   if 9.99999994e-9 < 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 99.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

   6. Applied egg-rr 100.0%

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

      1. +-commutative N/A

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

      2. cosh-undef N/A

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

      3. distribute-frac-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

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

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

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

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

      10. distribute-frac-neg N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in s around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

   11. Simplified 83.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{-\frac{x \cdot x + 0}{s}}{s}}}}{s} \]

   12. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f32 73.4%

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

   14. Simplified 73.4%

       \[\leadsto \frac{\color{blue}{s \cdot \frac{s}{x \cdot x}}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 45.8% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (/ 1.0 (* 2.0 (/ (* x x) s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (2.0f * ((x * x) / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.99999993922529e-9) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (2.0e0 * ((x * x) / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(Float32(x * x) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.99999993922529e-9))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (single(2.0) * ((x * x) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999994e-9

   1. Initial program 99.2%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 37.1%

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

   5. Simplified 37.1%

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

   if 9.99999994e-9 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 98.3%

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

   5. Simplified 98.3%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

      11. neg-lowering-neg.f32 98.3%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in s around inf

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

   10. Simplified 82.9%

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

   11. Taylor expanded in s around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f32 58.1%

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

   13. Simplified 58.1%

       \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x \cdot x}{s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 76.9% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. associate-/l/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

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

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

   8. distribute-frac-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-frac-neg N/A

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

   11. cosh-undef N/A

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

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

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

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

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

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

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

   15. 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. Taylor expanded in s around inf

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

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

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

   2. +-commutative N/A

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

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

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

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

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f32 77.5%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right)\right) \]

9. Simplified 77.5%

   \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\frac{x \cdot x}{s \cdot s} + 4\right)}} \]
10. Step-by-step derivation

    1. times-frac N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right), 4\right)\right)\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\frac{x}{s}\right), \left(\frac{x}{s}\right)\right), 4\right)\right)\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(\frac{x}{s}\right)\right), 4\right)\right)\right) \]

    4. /-lowering-/.f32 74.8%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \mathsf{/.f32}\left(x, s\right)\right), 4\right)\right)\right) \]

11. Applied egg-rr 74.8%

    \[\leadsto \frac{1}{s \cdot \left(\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4\right)} \]

12. Final simplification 74.8%

    \[\leadsto \frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)} \]
13. Add Preprocessing

Alternative 14: 45.7% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (/ 1.0 (/ (* x x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / ((x * x) / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.99999993922529e-9) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((x * x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.99999993922529e-9))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((x * x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999994e-9

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 37.1%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   5. Simplified 37.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   if 9.99999994e-9 < 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 99.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right) \cdot s}} \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right) \cdot s\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(s \cdot \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 100.0%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}} \]

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right)\right) \]

      6. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right)\right) \]

      9. *-lowering-*.f32 83.5%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right)\right) \]

   9. Simplified 83.5%

      \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\frac{x \cdot x}{s \cdot s} + 4\right)}} \]

   10. Taylor expanded in s around 0

       \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{{x}^{2}}{s}\right)}\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left({x}^{2}\right), \color{blue}{s}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(x \cdot x\right), s\right)\right) \]

       3. *-lowering-*.f32 57.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), s\right)\right) \]

   12. Simplified 57.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.0% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		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 <= 9.99999993922529e-9) 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(9.99999993922529e-9))
		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(9.99999993922529e-9))
		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 < 9.99999994e-9

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 37.1%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   5. Simplified 37.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   if 9.99999994e-9 < 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 99.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right) \cdot s}} \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right) \cdot s\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(s \cdot \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(2 + \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right)\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right)\right) \]

      13. cosh-lowering-cosh.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right)\right)\right) \]

      15. fabs-lowering-fabs.f32 100.0%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)\right)}} \]

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}\right)\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({\left(\left|x\right|\right)}^{2}\right), \left({s}^{2}\right)\right), 4\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right|\right), \left({s}^{2}\right)\right), 4\right)\right)\right) \]

      6. sqr-abs N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), \left({s}^{2}\right)\right), 4\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({s}^{2}\right)\right), 4\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot s\right)\right), 4\right)\right)\right) \]

      9. *-lowering-*.f32 83.5%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, s\right)\right), 4\right)\right)\right) \]

   9. Simplified 83.5%

      \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\frac{x \cdot x}{s \cdot s} + 4\right)}} \]

   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 54.2%

          \[\leadsto \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 54.2%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 27.1% 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 99.5%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s}} \]
4. Step-by-step derivation

   1. /-lowering-/.f32 27.6%

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

5. Simplified 27.6%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(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))))))