Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 14.8s

Alternatives: 12

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: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return expf(-logf((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 = exp(-log((2.0e0 + (2.0e0 * cosh((abs(x) / s)))))) / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

   1. inv-pow N/A

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

   2. pow-to-exp N/A

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

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

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

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

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

6. Applied egg-rr 99.4%

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

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

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. +-commutative N/A

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

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

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

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

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

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

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

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

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

   12. fabs-lowering-fabs.f32 99.4%

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

8. Applied egg-rr 99.4%

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

Alternative 2: 99.5% 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 99.3%

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

3. Simplified 99.4%

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

   1. associate-+r+ N/A

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

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

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

   3. distribute-frac-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-frac-neg N/A

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

   6. cosh-undef N/A

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

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

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

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

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

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

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

   10. fabs-lowering-fabs.f32 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 99.5% 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 99.3%

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

3. Simplified 99.4%

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

   1. associate-/l/ N/A

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

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

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

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

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

   4. associate-+r+ N/A

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

   5. +-commutative N/A

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

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

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

   7. distribute-frac-neg N/A

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

   8. +-commutative N/A

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

   9. distribute-frac-neg N/A

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

   10. cosh-undef N/A

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

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

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

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

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

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

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

   14. fabs-lowering-fabs.f32 99.4%

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

6. Applied egg-rr 99.4%

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

Alternative 4: 85.2% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\frac{s \cdot s}{x}}\\ t_1 := \frac{x}{\frac{s}{x}}\\ \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{\frac{s}{\frac{x}{s}}} - -4}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{256}{\left(\frac{t\_0}{\frac{\frac{s \cdot \left(s \cdot s\right)}{x \cdot x}}{t\_1}} + 64\right) \cdot \left(16 - \frac{-4 + t\_0}{\frac{s}{t\_1}}\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (/ (* s s) x))) (t_1 (/ x (/ s x))))
   (if (<= x 1.9999999996399175e-23)
     (/ (/ 1.0 (- (/ x (/ s (/ x s))) -4.0)) s)
     (/
      (/
       256.0
       (*
        (+ (/ t_0 (/ (/ (* s (* s s)) (* x x)) t_1)) 64.0)
        (- 16.0 (/ (+ -4.0 t_0) (/ s t_1)))))
      s))))

C

float code(float x, float s) {
	float t_0 = x / ((s * s) / x);
	float t_1 = x / (s / x);
	float tmp;
	if (x <= 1.9999999996399175e-23f) {
		tmp = (1.0f / ((x / (s / (x / s))) - -4.0f)) / s;
	} else {
		tmp = (256.0f / (((t_0 / (((s * (s * s)) / (x * x)) / t_1)) + 64.0f) * (16.0f - ((-4.0f + t_0) / (s / t_1))))) / 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) :: t_1
    real(4) :: tmp
    t_0 = x / ((s * s) / x)
    t_1 = x / (s / x)
    if (x <= 1.9999999996399175e-23) then
        tmp = (1.0e0 / ((x / (s / (x / s))) - (-4.0e0))) / s
    else
        tmp = (256.0e0 / (((t_0 / (((s * (s * s)) / (x * x)) / t_1)) + 64.0e0) * (16.0e0 - (((-4.0e0) + t_0) / (s / t_1))))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(Float32(s * s) / x))
	t_1 = Float32(x / Float32(s / x))
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(x / Float32(s / Float32(x / s))) - Float32(-4.0))) / s);
	else
		tmp = Float32(Float32(Float32(256.0) / Float32(Float32(Float32(t_0 / Float32(Float32(Float32(s * Float32(s * s)) / Float32(x * x)) / t_1)) + Float32(64.0)) * Float32(Float32(16.0) - Float32(Float32(Float32(-4.0) + t_0) / Float32(s / t_1))))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / ((s * s) / x);
	t_1 = x / (s / x);
	tmp = single(0.0);
	if (x <= single(1.9999999996399175e-23))
		tmp = (single(1.0) / ((x / (s / (x / s))) - single(-4.0))) / s;
	else
		tmp = (single(256.0) / (((t_0 / (((s * (s * s)) / (x * x)) / t_1)) + single(64.0)) * (single(16.0) - ((single(-4.0) + t_0) / (s / t_1))))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2e-23

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 77.3%

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

   7. Simplified 77.3%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. /-lowering-/.f32 77.0%

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

   9. Applied egg-rr 77.0%

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

       1. associate-/l* N/A

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

       2. frac-times N/A

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

       3. associate-*l/ N/A

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

       4. associate-/r/ N/A

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

       5. metadata-eval N/A

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

       6. sub-neg N/A

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

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

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

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

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

       9. associate-/l* N/A

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

       10. clear-num N/A

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

       11. un-div-inv N/A

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

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

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

       13. /-lowering-/.f32 80.6%

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

   11. Applied egg-rr 80.6%

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

   if 2e-23 < x

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 83.0%

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

   7. Simplified 83.0%

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

   9. Applied egg-rr 13.5%

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{\frac{x}{\frac{s \cdot s}{x}}}{\left(s \cdot \left(\frac{s}{x \cdot x} \cdot s\right)\right) \cdot \frac{s}{x \cdot x}} + 64}{16 + \frac{\frac{x}{\frac{s \cdot s}{x}} + -4}{\frac{s \cdot s}{x \cdot x}}}}}}{s} \]

   11. Applied egg-rr 10.6%

       \[\leadsto \frac{\color{blue}{\frac{256 - \frac{\frac{\frac{x}{\frac{s \cdot s}{x}} + -4}{\frac{s}{\frac{x}{\frac{s}{x}}}}}{\frac{s \cdot s}{\left(\frac{x}{\frac{s \cdot s}{x}} + -4\right) \cdot \left(x \cdot x\right)}}}{\left(\frac{\frac{x}{\frac{s \cdot s}{x}}}{\frac{\frac{s \cdot \left(s \cdot s\right)}{x \cdot x}}{\frac{x}{\frac{s}{x}}}} + 64\right) \cdot \left(16 - \frac{\frac{x}{\frac{s \cdot s}{x}} + -4}{\frac{s}{\frac{x}{\frac{s}{x}}}}\right)}}}{s} \]

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{256}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, s\right), x\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(x, \mathsf{/.f32}\left(s, x\right)\right)\right)\right), 64\right), \mathsf{\_.f32}\left(16, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, s\right), x\right)\right), -4\right), \mathsf{/.f32}\left(s, \mathsf{/.f32}\left(x, \mathsf{/.f32}\left(s, x\right)\right)\right)\right)\right)\right)\right), s\right) \]
   13. Step-by-step derivation

   14. Simplified 92.7%

       \[\leadsto \frac{\frac{\color{blue}{256}}{\left(\frac{\frac{x}{\frac{s \cdot s}{x}}}{\frac{\frac{s \cdot \left(s \cdot s\right)}{x \cdot x}}{\frac{x}{\frac{s}{x}}}} + 64\right) \cdot \left(16 - \frac{\frac{x}{\frac{s \cdot s}{x}} + -4}{\frac{s}{\frac{x}{\frac{s}{x}}}}\right)}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{\frac{s}{\frac{x}{s}}} - -4}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{256}{\left(\frac{\frac{x}{\frac{s \cdot s}{x}}}{\frac{\frac{s \cdot \left(s \cdot s\right)}{x \cdot x}}{\frac{x}{\frac{s}{x}}}} + 64\right) \cdot \left(16 - \frac{-4 + \frac{x}{\frac{s \cdot s}{x}}}{\frac{s}{\frac{x}{\frac{s}{x}}}}\right)}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 29.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-4}{\frac{\frac{\frac{x}{\frac{s \cdot s}{x}}}{\frac{s}{x}}}{\frac{s}{x}} + -16}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ -4.0 (+ (/ (/ (/ x (/ (* s s) x)) (/ s x)) (/ s x)) -16.0)) s))

C

float code(float x, float s) {
	return (-4.0f / ((((x / ((s * s) / x)) / (s / x)) / (s / x)) + -16.0f)) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = ((-4.0e0) / ((((x / ((s * s) / x)) / (s / x)) / (s / x)) + (-16.0e0))) / s
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(-4.0) / Float32(Float32(Float32(Float32(x / Float32(Float32(s * s) / x)) / Float32(s / x)) / Float32(s / x)) + Float32(-16.0))) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = (single(-4.0) / ((((x / ((s * s) / x)) / (s / x)) / (s / x)) + single(-16.0))) / s;
end

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

   6. +-lft-identity N/A

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

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

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

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

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f32 79.6%

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

7. Simplified 79.6%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

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

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

   4. sub-neg N/A

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

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

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

   6. associate-/l* N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

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

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

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

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

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

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

9. Applied egg-rr 26.4%

   \[\leadsto \frac{\color{blue}{\frac{\frac{x}{\frac{s \cdot s}{x}} + -4}{\frac{\frac{\frac{x}{\frac{s \cdot s}{x}}}{\frac{s}{x}}}{\frac{s}{x}} + -16}}}{s} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{-4}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, s\right), x\right)\right), \mathsf{/.f32}\left(s, x\right)\right), \mathsf{/.f32}\left(s, x\right)\right), -16\right)\right), s\right) \]
11. Step-by-step derivation

12. Simplified 87.6%

    \[\leadsto \frac{\frac{\color{blue}{-4}}{\frac{\frac{\frac{x}{\frac{s \cdot s}{x}}}{\frac{s}{x}}}{\frac{s}{x}} + -16}}{s} \]
13. Add Preprocessing

Alternative 6: 50.6% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{x \cdot x}{s \cdot s}}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.000000031374395e-22)
   (/ 0.25 s)
   (/ (/ 1.0 (/ (* x x) (* s s))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.000000031374395e-22f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / ((x * x) / (s * s))) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.000000031374395e-22) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / ((x * x) / (s * s))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(x * x) / Float32(s * s))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.000000031374395e-22))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / ((x * x) / (s * s))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000003e-22

   1. Initial program 99.1%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 34.6%

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

   5. Simplified 34.6%

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

   if 1.00000003e-22 < x

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 73.5%

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

   10. Simplified 73.5%

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

Alternative 7: 49.5% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot s}{x \cdot x}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.000000031374395e-22) (/ 0.25 s) (/ (/ (* s s) (* x x)) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.000000031374395e-22f) {
		tmp = 0.25f / s;
	} else {
		tmp = ((s * s) / (x * x)) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.000000031374395e-22) then
        tmp = 0.25e0 / s
    else
        tmp = ((s * s) / (x * x)) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(s * s) / Float32(x * x)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.000000031374395e-22))
		tmp = single(0.25) / s;
	else
		tmp = ((s * s) / (x * x)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000003e-22

   1. Initial program 99.1%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 34.6%

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

   5. Simplified 34.6%

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

   if 1.00000003e-22 < x

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 83.0%

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

   7. Simplified 83.0%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f32 79.3%

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

   9. Applied egg-rr 79.3%

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

   10. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f32 71.0%

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

   12. Simplified 71.0%

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

Alternative 8: 79.9% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

   6. +-lft-identity N/A

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

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

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

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

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f32 79.6%

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

7. Simplified 79.6%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

   6. associate-/l* N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

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

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

   11. *-lowering-*.f32 83.4%

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

9. Applied egg-rr 83.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{x}{\frac{s \cdot s}{x}} + 4}} \]
10. Add Preprocessing

Alternative 9: 80.1% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

   6. +-lft-identity N/A

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

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

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

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

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f32 79.6%

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

7. Simplified 79.6%

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

   1. associate-/l/ N/A

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

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

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

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

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

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

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

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

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

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

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

   10. *-lowering-*.f32 83.3%

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

9. Applied egg-rr 83.3%

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

Alternative 10: 45.9% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999873689376e-6) (/ 0.25 s) (/ 1.0 (/ x (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999873689376e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x / (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999873689376e-6) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x / (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999873689376e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999873689376e-6))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x / (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999987e-6

   1. Initial program 99.0%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 35.4%

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

   5. Simplified 35.4%

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

   if 4.99999987e-6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   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 66.4%

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

   10. Simplified 66.4%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-*l/ N/A

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

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

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

       4. *-commutative N/A

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

       5. clear-num N/A

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

       6. un-div-inv N/A

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

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

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

       8. /-lowering-/.f32 67.0%

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

   12. Applied egg-rr 67.0%

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

Alternative 11: 45.3% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999873689376e-6) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999873689376e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999873689376e-6) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999873689376e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999873689376e-6))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999987e-6

   1. Initial program 99.0%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 35.4%

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

   5. Simplified 35.4%

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

   if 4.99999987e-6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   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 66.4%

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

   10. Simplified 66.4%

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

Alternative 12: 26.8% 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 99.3%

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

3. Taylor expanded in s around inf

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

   1. /-lowering-/.f32 26.4%

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

5. Simplified 26.4%

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

Reproduce

herbie shell --seed 2024158 
(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))))))