Logistic distribution

Percentage Accurate: 99.6% → 99.7%

Time: 13.3s

Alternatives: 17

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{\left|x\right|}{s}\\ \frac{e^{t\_0}}{s \cdot e^{2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) - t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* -0.5 (/ (fabs x) s))))
   (/ (exp t_0) (* s (exp (- (* 2.0 (log1p (exp (/ (fabs x) (- s))))) t_0))))))

C

float code(float x, float s) {
	float t_0 = -0.5f * (fabsf(x) / s);
	return expf(t_0) / (s * expf(((2.0f * log1pf(expf((fabsf(x) / -s)))) - t_0)));
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-0.5) * Float32(abs(x) / s))
	return Float32(exp(t_0) / Float32(s * exp(Float32(Float32(Float32(2.0) * log1p(exp(Float32(abs(x) / Float32(-s))))) - t_0))))
end

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.5

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-/.f32 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.5

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.7%

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

10. Applied rewrites 99.7%

    \[\leadsto \color{blue}{\frac{e^{-0.5 \cdot \frac{\left|x\right|}{s}}}{s \cdot e^{2 \cdot \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right) - -0.5 \cdot \frac{\left|x\right|}{s}}}} \]

11. Final simplification 99.7%

    \[\leadsto \frac{e^{-0.5 \cdot \frac{\left|x\right|}{s}}}{s \cdot e^{2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right) - -0.5 \cdot \frac{\left|x\right|}{s}}} \]
12. Add Preprocessing

Alternative 2: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.5

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-/.f32 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(\frac{\left|x\right|}{s}\right)}}{\mathsf{fma}\left(e^{\frac{\left|x\right|}{-s}}, s, s\right) \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)} \]
8. Add Preprocessing

Alternative 3: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\left|x\right|}{s} + \mathsf{fma}\left(x \cdot x, \frac{0.5}{s \cdot s}, 2\right)}}{s}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0)
     (/ (/ (/ 1.0 (+ (/ (fabs x) s) (fma (* x x) (/ 0.5 (* s s)) 2.0))) s) 2.0)
     (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = ((1.0f / ((fabsf(x) / s) + fmaf((x * x), (0.5f / (s * s)), 2.0f))) / s) / 2.0f;
	} else {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(Float32(abs(x) / s) + fma(Float32(x * x), Float32(Float32(0.5) / Float32(s * s)), Float32(2.0)))) / s) / Float32(2.0));
	else
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   1. Initial program 99.5%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. sqr-abs N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 77.6

          \[\leadsto \frac{\frac{\frac{1}{\frac{\left|x\right|}{s} + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{s \cdot s}}, 2\right)}}{s}}{2} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left|x\right|}{s} + \mathsf{fma}\left(x \cdot x, \frac{0.5}{s \cdot s}, 2\right)}}}{s}}{2} \]

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.4%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 67.5%

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

   7. Applied rewrites 90.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)\right)} \leq 0:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\left|x\right|}{s} + \mathsf{fma}\left(x \cdot x, \frac{0.5}{s \cdot s}, 2\right)}}{s}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.5

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-/.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 5: 99.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.5

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-/.f32 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.5

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 6: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. associate-/r* N/A

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

   3. associate-/l/ N/A

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

   4. exp-neg N/A

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

   5. neg-mul-1 N/A

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

   6. unpow2 N/A

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

   7. neg-mul-1 N/A

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

   8. associate-/r* N/A

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 7: 95.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{1}{\mathsf{fma}\left(1, e^{\frac{s - 0}{s \cdot \frac{s}{\left|x\right|}}}, 1\right)}}{s}}{2} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ (/ 1.0 (fma 1.0 (exp (/ (- s 0.0) (* s (/ s (fabs x))))) 1.0)) s) 2.0))

C

float code(float x, float s) {
	return ((1.0f / fmaf(1.0f, expf(((s - 0.0f) / (s * (s / fabsf(x))))), 1.0f)) / s) / 2.0f;
}

Julia

function code(x, s)
	return Float32(Float32(Float32(Float32(1.0) / fma(Float32(1.0), exp(Float32(Float32(s - Float32(0.0)) / Float32(s * Float32(s / abs(x))))), Float32(1.0))) / s) / Float32(2.0))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 94.8%

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

7. Applied rewrites 94.9%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. neg-sub0 N/A

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

   4. div-sub N/A

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

   5. distribute-neg-frac2 N/A

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

   6. clear-num N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. frac-sub N/A

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

   10. lower-/.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-neg.f32 N/A

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

   18. lower-/.f32 94.9

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

9. Applied rewrites 94.9%

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

10. Taylor expanded in s around 0

    \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(1, e^{\frac{\color{blue}{0} - \left(\mathsf{neg}\left(s\right)\right) \cdot -1}{\left(\mathsf{neg}\left(s\right)\right) \cdot \frac{s}{\left|x\right|}}}, 1\right)}}{s}}{2} \]
11. Step-by-step derivation

12. Applied rewrites 94.9%

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

13. Final simplification 94.9%

    \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(1, e^{\frac{s - 0}{s \cdot \frac{s}{\left|x\right|}}}, 1\right)}}{s}}{2} \]
14. Add Preprocessing

Alternative 8: 95.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{1}{\mathsf{fma}\left(1, e^{\left|x\right| \cdot \frac{1}{s}}, 1\right)}}{s}}{2} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ (/ 1.0 (fma 1.0 (exp (* (fabs x) (/ 1.0 s))) 1.0)) s) 2.0))

C

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

Julia

function code(x, s)
	return Float32(Float32(Float32(Float32(1.0) / fma(Float32(1.0), exp(Float32(abs(x) * Float32(Float32(1.0) / s))), Float32(1.0))) / s) / Float32(2.0))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 94.8%

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

7. Applied rewrites 94.9%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 94.9

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

9. Applied rewrites 94.9%

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

10. Final simplification 94.9%

    \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(1, e^{\left|x\right| \cdot \frac{1}{s}}, 1\right)}}{s}}{2} \]
11. Add Preprocessing

Alternative 9: 95.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{-1 - e^{\frac{\left|x\right|}{s}}}}{s \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 94.8%

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

7. Applied rewrites 94.9%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. *-lft-identity N/A

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

   7. +-commutative N/A

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

   8. lift-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 94.9

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

9. Applied rewrites 94.9%

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

10. Final simplification 94.9%

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

Alternative 10: 95.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right) \cdot 2} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (fma s (exp (/ (fabs x) s)) s) 2.0)))

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(fma(s, exp(Float32(abs(x) / s)), s) * Float32(2.0)))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 94.8%

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

7. Applied rewrites 94.9%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/l/ N/A

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

   6. lift-fma.f32 N/A

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

   7. *-lft-identity N/A

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

   8. +-commutative N/A

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

   9. lift-+.f32 N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

9. Applied rewrites 94.8%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right) \cdot 2}} \]
10. Add Preprocessing

Alternative 11: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.5

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

5. Applied rewrites 94.5%

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

6. Final simplification 94.5%

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

Alternative 12: 87.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ t_1 := -\left|x\right|\\ t_2 := \frac{0.5}{s \cdot s}\\ \mathbf{if}\;t\_1 \leq -2.20000002915631 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{fma}\left(x \cdot x, t\_2, t\_0\right) + \mathsf{fma}\left(\frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.16666666666666666, 2\right)}}{s}}{2}\\ \mathbf{elif}\;t\_1 \leq -4.00000012549758 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{\frac{1}{t\_0 + \mathsf{fma}\left(x \cdot x, t\_2, 2\right)}}{s}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)) (t_1 (- (fabs x))) (t_2 (/ 0.5 (* s s))))
   (if (<= t_1 -2.20000002915631e-14)
     (/
      (/
       (/
        1.0
        (+
         (fma (* x x) t_2 t_0)
         (fma (/ (* (fabs x) (* x x)) (* s (* s s))) 0.16666666666666666 2.0)))
       s)
      2.0)
     (if (<= t_1 -4.00000012549758e-22)
       (/ (/ (/ 1.0 (+ t_0 (fma (* x x) t_2 2.0))) s) 2.0)
       (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s)))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	float t_1 = -fabsf(x);
	float t_2 = 0.5f / (s * s);
	float tmp;
	if (t_1 <= -2.20000002915631e-14f) {
		tmp = ((1.0f / (fmaf((x * x), t_2, t_0) + fmaf(((fabsf(x) * (x * x)) / (s * (s * s))), 0.16666666666666666f, 2.0f))) / s) / 2.0f;
	} else if (t_1 <= -4.00000012549758e-22f) {
		tmp = ((1.0f / (t_0 + fmaf((x * x), t_2, 2.0f))) / s) / 2.0f;
	} else {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	t_1 = Float32(-abs(x))
	t_2 = Float32(Float32(0.5) / Float32(s * s))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-2.20000002915631e-14))
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(fma(Float32(x * x), t_2, t_0) + fma(Float32(Float32(abs(x) * Float32(x * x)) / Float32(s * Float32(s * s))), Float32(0.16666666666666666), Float32(2.0)))) / s) / Float32(2.0));
	elseif (t_1 <= Float32(-4.00000012549758e-22))
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(t_0 + fma(Float32(x * x), t_2, Float32(2.0)))) / s) / Float32(2.0));
	else
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 (fabs.f32 x)) < -2.20000003e-14

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{2}, \frac{\frac{1}{2}}{{s}^{2}}, \frac{\left|x\right|}{s}\right)} + \left(2 + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right)}}{s}}{2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(\color{blue}{\left|x\right| \cdot \left|x\right|}, \frac{\frac{1}{2}}{{s}^{2}}, \frac{\left|x\right|}{s}\right) + \left(2 + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right)}}{s}}{2} \]

      9. sqr-abs N/A

         \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{\frac{1}{2}}{{s}^{2}}, \frac{\left|x\right|}{s}\right) + \left(2 + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right)}}{s}}{2} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{\frac{1}{2}}{{s}^{2}}, \frac{\left|x\right|}{s}\right) + \left(2 + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right)}}{s}}{2} \]

      11. lower-/.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\frac{1}{2}}{{s}^{2}}}, \frac{\left|x\right|}{s}\right) + \left(2 + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right)}}{s}}{2} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(x \cdot x, \frac{\frac{1}{2}}{\color{blue}{s \cdot s}}, \frac{\left|x\right|}{s}\right) + \left(2 + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right)}}{s}}{2} \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(x \cdot x, \frac{\frac{1}{2}}{\color{blue}{s \cdot s}}, \frac{\left|x\right|}{s}\right) + \left(2 + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right)}}{s}}{2} \]

      14. lower-/.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(x \cdot x, \frac{\frac{1}{2}}{s \cdot s}, \color{blue}{\frac{\left|x\right|}{s}}\right) + \left(2 + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right)}}{s}}{2} \]

      15. lower-fabs.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(x \cdot x, \frac{\frac{1}{2}}{s \cdot s}, \frac{\color{blue}{\left|x\right|}}{s}\right) + \left(2 + \frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right)}}{s}}{2} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{\mathsf{fma}\left(x \cdot x, \frac{\frac{1}{2}}{s \cdot s}, \frac{\left|x\right|}{s}\right) + \color{blue}{\left(\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}} + 2\right)}}}{s}}{2} \]

   10. Applied rewrites 88.4%

       \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{s \cdot s}, \frac{\left|x\right|}{s}\right) + \mathsf{fma}\left(\frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 0.16666666666666666, 2\right)}}}{s}}{2} \]

   if -2.20000003e-14 < (neg.f32 (fabs.f32 x)) < -4.00000013e-22

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-fabs.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. sqr-abs N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 90.7

          \[\leadsto \frac{\frac{\frac{1}{\frac{\left|x\right|}{s} + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{s \cdot s}}, 2\right)}}{s}}{2} \]

   10. Applied rewrites 90.7%

       \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{\left|x\right|}{s} + \mathsf{fma}\left(x \cdot x, \frac{0.5}{s \cdot s}, 2\right)}}}{s}}{2} \]

   if -4.00000013e-22 < (neg.f32 (fabs.f32 x))

   1. Initial program 97.8%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 47.3%

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

   7. Applied rewrites 78.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 81.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{1}{2 + \frac{\left|x\right| + \frac{\mathsf{fma}\left(0.16666666666666666, \left|x\right| \cdot \frac{x \cdot x}{s}, \left(x \cdot x\right) \cdot 0.5\right)}{s}}{s}}}{s}}{2} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (/
   (/
    1.0
    (+
     2.0
     (/
      (+
       (fabs x)
       (/
        (fma 0.16666666666666666 (* (fabs x) (/ (* x x) s)) (* (* x x) 0.5))
        s))
      s)))
   s)
  2.0))

C

float code(float x, float s) {
	return ((1.0f / (2.0f + ((fabsf(x) + (fmaf(0.16666666666666666f, (fabsf(x) * ((x * x) / s)), ((x * x) * 0.5f)) / s)) / s))) / s) / 2.0f;
}

Julia

function code(x, s)
	return Float32(Float32(Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(abs(x) + Float32(fma(Float32(0.16666666666666666), Float32(abs(x) * Float32(Float32(x * x) / s)), Float32(Float32(x * x) * Float32(0.5))) / s)) / s))) / s) / Float32(2.0))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 94.8%

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 79.3%

    \[\leadsto \frac{\frac{\frac{1}{\color{blue}{2 - \frac{\left(-\left|x\right|\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \left|x\right| \cdot \frac{x \cdot x}{s}, \left(x \cdot x\right) \cdot 0.5\right)}{s}}{s}}}}{s}}{2} \]

11. Final simplification 79.3%

    \[\leadsto \frac{\frac{\frac{1}{2 + \frac{\left|x\right| + \frac{\mathsf{fma}\left(0.16666666666666666, \left|x\right| \cdot \frac{x \cdot x}{s}, \left(x \cdot x\right) \cdot 0.5\right)}{s}}{s}}}{s}}{2} \]
12. Add Preprocessing

Alternative 14: 51.0% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 94.8%

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-fabs.f32 48.9

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

10. Applied rewrites 48.9%

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

Alternative 15: 75.7% accurate, 7.9× speedup

Math

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

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(Float32(Float32(x * x) / 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 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. associate-/r* N/A

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

   3. associate-/l/ N/A

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

   4. exp-neg N/A

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

   5. neg-mul-1 N/A

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

   6. unpow2 N/A

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

   7. neg-mul-1 N/A

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

   8. associate-/r* N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around -inf

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

10. Applied rewrites 73.2%

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

11. Final simplification 73.2%

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

Alternative 16: 27.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s))

C

float code(float x, float s) {
	return fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
}

Julia

function code(x, s)
	return Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s)
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 19.2%

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

7. Applied rewrites 25.1%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s} \]
8. Add Preprocessing

Alternative 17: 27.4% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 25.1

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

5. Applied rewrites 25.1%

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

Reproduce

herbie shell --seed 2024232 
(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))))))