Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 14.8s

Alternatives: 17

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{\left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right) \cdot \color{blue}{s}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{\color{blue}{s}} \]

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

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}\right), \color{blue}{s}\right) \]

6. Applied egg-rr 99.8%

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

   1. cosh-def N/A

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

   2. cosh-undef N/A

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

   3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

   9. fabs-lowering-fabs.f32 99.8%

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

8. Applied egg-rr 99.8%

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

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

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

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

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. remove-double-div N/A

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

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

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

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

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

   10. fabs-lowering-fabs.f32 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 2: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{\left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right) \cdot \color{blue}{s}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{\color{blue}{s}} \]

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

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}\right), \color{blue}{s}\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)}}{s}} \]
7. 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 99.8%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right) \cdot \color{blue}{s}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right), \color{blue}{s}\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(s \cdot \left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) \cdot s + \color{blue}{2 \cdot s}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) \cdot s\right), \color{blue}{\left(2 \cdot s\right)}\right)\right) \]

6. Applied egg-rr 99.8%

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

   1. associate-*l* N/A

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

   2. distribute-lft-out N/A

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

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

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

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

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

   5. *-commutative N/A

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

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

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

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

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

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

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

   9. fabs-lowering-fabs.f32 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 94.4% 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 99.8%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 94.6%

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

5. Simplified 94.6%

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

6. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

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

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

   3. mul-1-neg N/A

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

   4. rec-exp N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

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

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

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

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

   10. fabs-lowering-fabs.f32 94.6%

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

8. Simplified 94.6%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   7. fabs-lowering-fabs.f32 94.6%

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

10. Applied egg-rr 94.6%

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

Alternative 6: 85.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \frac{x}{s \cdot s} + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{1}{s \cdot \left(s \cdot s\right)} \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 4.99999991225835e-14)
   (/ 1.0 (* s (+ (* x (/ x (* s s))) 4.0)))
   (/
    1.0
    (* s (* (/ 1.0 (* s (* s s))) (* -0.16666666666666666 (* x (* x x))))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 4.99999991225835e-14f) {
		tmp = 1.0f / (s * ((x * (x / (s * s))) + 4.0f));
	} else {
		tmp = 1.0f / (s * ((1.0f / (s * (s * s))) * (-0.16666666666666666f * (x * (x * x)))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 4.99999991225835e-14) then
        tmp = 1.0e0 / (s * ((x * (x / (s * s))) + 4.0e0))
    else
        tmp = 1.0e0 / (s * ((1.0e0 / (s * (s * s))) * ((-0.16666666666666666e0) * (x * (x * x)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(4.99999991225835e-14))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(x * Float32(x / Float32(s * s))) + Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(1.0) / Float32(s * Float32(s * s))) * Float32(Float32(-0.16666666666666666) * Float32(x * Float32(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(4.99999991225835e-14))
		tmp = single(1.0) / (s * ((x * (x / (s * s))) + single(4.0)));
	else
		tmp = single(1.0) / (s * ((single(1.0) / (s * (s * s))) * (single(-0.16666666666666666) * (x * (x * x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 4.99999991e-14

   1. Initial program 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 64.8%

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

   7. Simplified 64.8%

      \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\frac{x \cdot x}{s \cdot s} + 4\right)}} \]
   8. 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 80.5%

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

   9. Applied egg-rr 80.5%

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

   if 4.99999991e-14 < (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 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   7. Simplified 14.4%

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

   8. Taylor expanded in s around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left({s}^{3}\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(s \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f32 89.2%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   10. Simplified 89.2%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\frac{\left|x\right| \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left(\frac{1}{\left(s \cdot s\right) \cdot s}\right), \left(\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       5. associate-/r* N/A

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, s\right)\right), s\right), \left(\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. sqr-abs N/A

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

       12. unpow3 N/A

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

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

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

       14. sqr-pow N/A

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

       15. pow-prod-down N/A

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

       16. sqr-abs N/A

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

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

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

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

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

       19. metadata-eval 90.2%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, s\right)\right), s\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{3}{2}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 90.2%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\frac{\frac{1}{s \cdot s}}{s} \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot x\right)}^{1.5}\right)\right)}} \]
   13. Step-by-step derivation

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

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

       2. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

       7. pow2 N/A

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

       8. pow-pow N/A

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

       9. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{*.f32}\left(\frac{-1}{6}, \left({x}^{3}\right)\right)\right)\right)\right) \]

       10. cube-mult N/A

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

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

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

       12. *-lowering-*.f32 90.5%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 90.5%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\frac{1}{s \cdot \left(s \cdot s\right)} \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \frac{x}{s \cdot s} + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{1}{s \cdot \left(s \cdot s\right)} \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \frac{x}{s \cdot s} + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999858590343e-10)
   (/ 1.0 (* s (+ (* x (/ x (* s s))) 4.0)))
   (/ 1.0 (* s (/ (* -0.16666666666666666 (* x (* x x))) (* s (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999858590343e-10f) {
		tmp = 1.0f / (s * ((x * (x / (s * s))) + 4.0f));
	} else {
		tmp = 1.0f / (s * ((-0.16666666666666666f * (x * (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 <= 4.999999858590343e-10) then
        tmp = 1.0e0 / (s * ((x * (x / (s * s))) + 4.0e0))
    else
        tmp = 1.0e0 / (s * (((-0.16666666666666666e0) * (x * (x * x))) / (s * (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999858590343e-10))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(x * Float32(x / Float32(s * s))) + Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(-0.16666666666666666) * Float32(x * Float32(x * x))) / Float32(s * Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999858590343e-10))
		tmp = single(1.0) / (s * ((x * (x / (s * s))) + single(4.0)));
	else
		tmp = single(1.0) / (s * ((single(-0.16666666666666666) * (x * (x * x))) / (s * (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999986e-10

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 76.5%

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

   7. Simplified 76.5%

      \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\frac{x \cdot x}{s \cdot s} + 4\right)}} \]
   8. 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 83.0%

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

   9. Applied egg-rr 83.0%

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

   if 4.99999986e-10 < 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%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   7. Simplified 14.4%

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

   8. Taylor expanded in s around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left({s}^{3}\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(s \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f32 90.2%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   10. Simplified 90.2%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\frac{\left|x\right| \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left(\frac{1}{\left(s \cdot s\right) \cdot s}\right), \left(\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       5. associate-/r* N/A

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, s\right)\right), s\right), \left(\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. sqr-abs N/A

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

       12. unpow3 N/A

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

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

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

       14. sqr-pow N/A

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

       15. pow-prod-down N/A

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

       16. sqr-abs N/A

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

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

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

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

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

       19. metadata-eval 91.4%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, s\right)\right), s\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{3}{2}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 91.4%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\frac{\frac{1}{s \cdot s}}{s} \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot x\right)}^{1.5}\right)\right)}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

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

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

       3. *-commutative N/A

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

       4. associate-/l/ N/A

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

       5. un-div-inv N/A

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

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

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

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

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

       8. pow2 N/A

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

       9. pow-pow N/A

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

       10. metadata-eval N/A

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

       11. cube-mult N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(s \cdot \left(s \cdot s\right)\right)\right), s\right)\right) \]

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

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot x\right)\right)\right), \left(s \cdot \left(s \cdot s\right)\right)\right), s\right)\right) \]

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

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(s \cdot \left(s \cdot s\right)\right)\right), s\right)\right) \]

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

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \left(s \cdot s\right)\right)\right), s\right)\right) \]

       15. *-lowering-*.f32 90.2%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right)\right), s\right)\right) \]

   14. Applied egg-rr 90.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)} \cdot s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \frac{x}{s \cdot s} + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s} + s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.99999991225835e-14)
   (/ 1.0 (+ (/ (* x x) s) (* s 4.0)))
   (/ (* (* s s) 6.0) (* x (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.99999991225835e-14f) {
		tmp = 1.0f / (((x * x) / s) + (s * 4.0f));
	} else {
		tmp = ((s * s) * 6.0f) / (x * (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.99999991225835e-14) then
        tmp = 1.0e0 / (((x * x) / s) + (s * 4.0e0))
    else
        tmp = ((s * s) * 6.0e0) / (x * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.99999991225835e-14))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * x) / s) + Float32(s * Float32(4.0))));
	else
		tmp = Float32(Float32(Float32(s * s) * Float32(6.0)) / Float32(x * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.99999991225835e-14))
		tmp = single(1.0) / (((x * x) / s) + (s * single(4.0)));
	else
		tmp = ((s * s) * single(6.0)) / (x * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999991e-14

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 76.3%

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

   7. Simplified 76.3%

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

      1. distribute-rgt-in N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. pow2 N/A

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

      5. pow-flip N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f32 62.1%

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

   9. Applied egg-rr 62.1%

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

   if 4.99999991e-14 < 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%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   7. Simplified 17.2%

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

   8. Taylor expanded in s around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left({s}^{3}\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(s \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f32 88.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   10. Simplified 88.5%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\frac{\left|x\right| \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left(\frac{1}{\left(s \cdot s\right) \cdot s}\right), \left(\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       5. associate-/r* N/A

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, s\right)\right), s\right), \left(\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. sqr-abs N/A

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

       12. unpow3 N/A

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

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

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

       14. sqr-pow N/A

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

       15. pow-prod-down N/A

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

       16. sqr-abs N/A

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

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

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

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

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

       19. metadata-eval 90.8%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, s\right)\right), s\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{3}{2}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 90.8%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\frac{\frac{1}{s \cdot s}}{s} \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot x\right)}^{1.5}\right)\right)}} \]

   13. Taylor expanded in x around -inf

       \[\leadsto \color{blue}{6 \cdot \frac{{s}^{2}}{{x}^{3}}} \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{6 \cdot {s}^{2}}{\color{blue}{{x}^{3}}} \]

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

          \[\leadsto \mathsf{/.f32}\left(\left(6 \cdot {s}^{2}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\left({s}^{2} \cdot 6\right), \left({\color{blue}{x}}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({s}^{2}\right), 6\right), \left({\color{blue}{x}}^{3}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot s\right), 6\right), \left({x}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), 6\right), \left({x}^{3}\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), 6\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), 6\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), 6\right), \mathsf{*.f32}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), 6\right), \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       11. *-lowering-*.f32 83.3%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), 6\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right) \]

   15. Simplified 83.3%

       \[\leadsto \color{blue}{\frac{\left(s \cdot s\right) \cdot 6}{x \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s} + s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.99999991225835e-14)
   (/ 1.0 (+ (/ (* x x) s) (* s 4.0)))
   (/ (* s s) (* -0.16666666666666666 (* x (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.99999991225835e-14f) {
		tmp = 1.0f / (((x * x) / s) + (s * 4.0f));
	} else {
		tmp = (s * s) / (-0.16666666666666666f * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.99999991225835e-14) then
        tmp = 1.0e0 / (((x * x) / s) + (s * 4.0e0))
    else
        tmp = (s * s) / ((-0.16666666666666666e0) * (x * (x * x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.99999991225835e-14))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * x) / s) + Float32(s * Float32(4.0))));
	else
		tmp = Float32(Float32(s * s) / Float32(Float32(-0.16666666666666666) * Float32(x * Float32(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.99999991225835e-14))
		tmp = single(1.0) / (((x * x) / s) + (s * single(4.0)));
	else
		tmp = (s * s) / (single(-0.16666666666666666) * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999991e-14

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 76.3%

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

   7. Simplified 76.3%

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

      1. distribute-rgt-in N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. pow2 N/A

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

      5. pow-flip N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f32 62.1%

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

   9. Applied egg-rr 62.1%

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

   if 4.99999991e-14 < 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%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   7. Simplified 17.2%

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

   8. Taylor expanded in s around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left({s}^{3}\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(s \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f32 88.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   10. Simplified 88.5%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\frac{\left|x\right| \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left(\frac{1}{\left(s \cdot s\right) \cdot s}\right), \left(\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       5. associate-/r* N/A

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, s\right)\right), s\right), \left(\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. sqr-abs N/A

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

       12. unpow3 N/A

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

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

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

       14. sqr-pow N/A

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

       15. pow-prod-down N/A

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

       16. sqr-abs N/A

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

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

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

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

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

       19. metadata-eval 90.8%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, s\right)\right), s\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{3}{2}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 90.8%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\frac{\frac{1}{s \cdot s}}{s} \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot x\right)}^{1.5}\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{\frac{1}{s \cdot s}}{s} \cdot \left(\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}\right)}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{s}}{\frac{\frac{1}{s \cdot s}}{s}}}{\color{blue}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}}} \]

       3. clear-num N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{\frac{1}{s \cdot s}}{s}}{\frac{1}{s}}}}{\color{blue}{\frac{-1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{1}{s \cdot s}}{s} \cdot \frac{1}{\frac{1}{s}}}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       5. remove-double-div N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{1}{s \cdot s}}{s} \cdot s}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{s \cdot \frac{\frac{1}{s \cdot s}}{s}}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       7. associate-/l/ N/A

          \[\leadsto \frac{\frac{1}{s \cdot \frac{1}{s \cdot \left(s \cdot s\right)}}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       8. un-div-inv N/A

          \[\leadsto \frac{\frac{1}{\frac{s}{s \cdot \left(s \cdot s\right)}}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       9. unpow1 N/A

          \[\leadsto \frac{\frac{1}{\frac{{s}^{1}}{s \cdot \left(s \cdot s\right)}}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       10. cube-unmult N/A

           \[\leadsto \frac{\frac{1}{\frac{{s}^{1}}{{s}^{3}}}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       11. pow-div N/A

           \[\leadsto \frac{\frac{1}{{s}^{\left(1 - 3\right)}}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{{s}^{-2}}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       13. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{{s}^{\left(2 \cdot -1\right)}}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       14. pow-flip N/A

           \[\leadsto \frac{{s}^{\left(\mathsf{neg}\left(2 \cdot -1\right)\right)}}{\color{blue}{\frac{-1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       15. metadata-eval N/A

           \[\leadsto \frac{{s}^{\left(\mathsf{neg}\left(-2\right)\right)}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       16. metadata-eval N/A

           \[\leadsto \frac{{s}^{2}}{\frac{-1}{6} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

       17. pow2 N/A

           \[\leadsto \frac{s \cdot s}{\color{blue}{\frac{-1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{3}{2}}} \]

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

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

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

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

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

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

   14. Applied egg-rr 83.0%

       \[\leadsto \color{blue}{\frac{s \cdot s}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.0% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x}{s}\\ \mathbf{if}\;x \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{t\_0 + s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \frac{t\_0}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (* x x) s)))
   (if (<= x 4.99999991225835e-14)
     (/ 1.0 (+ t_0 (* s 4.0)))
     (/ 1.0 (* s (/ t_0 s))))))

C

float code(float x, float s) {
	float t_0 = (x * x) / s;
	float tmp;
	if (x <= 4.99999991225835e-14f) {
		tmp = 1.0f / (t_0 + (s * 4.0f));
	} else {
		tmp = 1.0f / (s * (t_0 / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (x * x) / s
    if (x <= 4.99999991225835e-14) then
        tmp = 1.0e0 / (t_0 + (s * 4.0e0))
    else
        tmp = 1.0e0 / (s * (t_0 / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(x * x) / s)
	tmp = Float32(0.0)
	if (x <= Float32(4.99999991225835e-14))
		tmp = Float32(Float32(1.0) / Float32(t_0 + Float32(s * Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(t_0 / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = (x * x) / s;
	tmp = single(0.0);
	if (x <= single(4.99999991225835e-14))
		tmp = single(1.0) / (t_0 + (s * single(4.0)));
	else
		tmp = single(1.0) / (s * (t_0 / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999991e-14

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 76.3%

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

   7. Simplified 76.3%

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

      1. distribute-rgt-in N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. pow2 N/A

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

      5. pow-flip N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f32 62.1%

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

   9. Applied egg-rr 62.1%

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

   if 4.99999991e-14 < 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%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 79.8%

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

   10. Simplified 79.8%

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

Alternative 11: 49.6% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \frac{\frac{x \cdot x}{s}}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.99999992980668e-14) (/ 0.25 s) (/ 1.0 (* s (/ (/ (* x x) s) s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.99999992980668e-14f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (s * (((x * x) / s) / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 3.99999992980668e-14) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (s * (((x * x) / s) / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.99999992980668e-14))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(x * x) / s) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.99999992980668e-14))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (s * (((x * x) / s) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.99999993e-14

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 29.5%

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

   7. Simplified 29.5%

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

   if 3.99999993e-14 < 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%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 79.8%

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

   10. Simplified 79.8%

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

Alternative 12: 81.0% accurate, 47.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(x \cdot \frac{x}{s \cdot s} + 4\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ (* x (/ x (* s s))) 4.0))))

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(x * Float32(x / Float32(s * s))) + Float32(4.0))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf

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

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

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. sum3-define N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

   5. distribute-lft1-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

   7. mul0-lft N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

   8. sum3-undefine N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

   9. +-lft-identity N/A

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

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

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

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

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

   12. unpow2 N/A

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

   13. sqr-abs N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 78.5%

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

7. Simplified 78.5%

   \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\frac{x \cdot x}{s \cdot s} + 4\right)}} \]
8. 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%

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

9. Applied egg-rr 83.1%

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

10. Final simplification 83.1%

    \[\leadsto \frac{1}{s \cdot \left(x \cdot \frac{x}{s \cdot s} + 4\right)} \]
11. Add Preprocessing

Alternative 13: 76.4% 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 99.8%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf

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

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

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. sum3-define N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

   5. distribute-lft1-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

   7. mul0-lft N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

   8. sum3-undefine N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

   9. +-lft-identity N/A

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

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

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

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

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

   12. unpow2 N/A

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

   13. sqr-abs N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 78.5%

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

7. Simplified 78.5%

   \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\frac{x \cdot x}{s \cdot s} + 4\right)}} \]
8. 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 77.5%

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

9. Applied egg-rr 77.5%

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

10. Final simplification 77.5%

    \[\leadsto \frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)} \]
11. Add Preprocessing

Alternative 14: 45.6% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0006500000017695129:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0006500000017695129) (/ 0.25 s) (/ 1.0 (/ (* x x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0006500000017695129f) {
		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 <= 0.0006500000017695129e0) 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(0.0006500000017695129))
		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(0.0006500000017695129))
		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 < 6.50000002e-4

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 28.7%

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

   7. Simplified 28.7%

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

   if 6.50000002e-4 < 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%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in s around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{{x}^{2}}{s}\right)}\right) \]
   9. 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 73.0%

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

   10. Simplified 73.0%

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

Alternative 15: 45.6% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0006500000017695129:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0006500000017695129) (/ 0.25 s) (/ 1.0 (/ x (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0006500000017695129f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x / (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.0006500000017695129e0) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x / (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0006500000017695129))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0006500000017695129))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x / (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 6.50000002e-4

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 28.7%

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

   7. Simplified 28.7%

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

   if 6.50000002e-4 < 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%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in s around 0

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   9. 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 71.1%

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

   10. Simplified 71.1%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

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

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

       3. clear-num N/A

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

       4. associate-/r* N/A

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

       5. clear-num N/A

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

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

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

       7. /-lowering-/.f32 73.0%

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

   12. Applied egg-rr 73.0%

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

Alternative 16: 45.0% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0006500000017695129:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0006500000017695129) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0006500000017695129f) {
		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 <= 0.0006500000017695129e0) 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(0.0006500000017695129))
		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(0.0006500000017695129))
		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 < 6.50000002e-4

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 28.7%

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

   7. Simplified 28.7%

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

   if 6.50000002e-4 < 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%

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

   5. Taylor expanded in s around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-abs N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in s around 0

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   9. 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 71.1%

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

   10. Simplified 71.1%

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

Alternative 17: 27.2% accurate, 206.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf

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

   1. /-lowering-/.f32 22.2%

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

7. Simplified 22.2%

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

Reproduce

herbie shell --seed 2024170 
(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))))))