Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 14.3s

Alternatives: 12

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ t_1 := e^{-t\_0}\\ t_2 := 1 + t\_1\\ \mathbf{if}\;\frac{t\_1}{t\_2 \cdot \left(s \cdot t\_2\right)} \leq 0.5:\\ \;\;\;\;\frac{1}{s \cdot e^{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)) (t_1 (exp (- t_0))) (t_2 (+ 1.0 t_1)))
   (if (<= (/ t_1 (* t_2 (* s t_2))) 0.5)
     (/ 1.0 (* s (exp t_0)))
     (/ 1.0 (fma x (/ x s) (* s 4.0))))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	float t_1 = expf(-t_0);
	float t_2 = 1.0f + t_1;
	float tmp;
	if ((t_1 / (t_2 * (s * t_2))) <= 0.5f) {
		tmp = 1.0f / (s * expf(t_0));
	} else {
		tmp = 1.0f / fmaf(x, (x / s), (s * 4.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	t_1 = exp(Float32(-t_0))
	t_2 = Float32(Float32(1.0) + t_1)
	tmp = Float32(0.0)
	if (Float32(t_1 / Float32(t_2 * Float32(s * t_2))) <= Float32(0.5))
		tmp = Float32(Float32(1.0) / Float32(s * exp(t_0)));
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(x / s), Float32(s * Float32(4.0))));
	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))))) < 0.5

   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. Step-by-step derivation

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-exp.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 99.9

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

   6. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{fma}\left(2, \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right), \frac{\left|x\right|}{s}\right)} \cdot s}} \]

   7. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fabs.f32 99.6

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

   9. Applied rewrites 99.6%

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

   if 0.5 < (/.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 98.8%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

   7. Applied rewrites 83.5%

      \[\leadsto \frac{1}{\color{blue}{-s \cdot \left(\frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \mathsf{fma}\left(5, \frac{x \cdot x}{s}, 0\right)\right)}{-s} + -4\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s} - -4 \cdot s}} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 85.8

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

   10. Applied rewrites 85.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(1 + e^{-\frac{\left|x\right|}{s}}\right) \cdot \left(s \cdot \left(1 + e^{-\frac{\left|x\right|}{s}}\right)\right)} \leq 0.5:\\ \;\;\;\;\frac{1}{s \cdot e^{\frac{\left|x\right|}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 1.999999987845058e-8)
     t_0
     (/ 1.0 (fma x (/ x s) (* s 4.0))))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 1.999999987845058e-8f) {
		tmp = t_0;
	} else {
		tmp = 1.0f / fmaf(x, (x / s), (s * 4.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(1.999999987845058e-8))
		tmp = t_0;
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(x / s), Float32(s * Float32(4.0))));
	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))))) < 1.99999999e-8

   1. Initial program 100.0%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around 0

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fabs.f32 99.5

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

   7. Applied rewrites 99.5%

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

   if 1.99999999e-8 < (/.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 98.7%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

   7. Applied rewrites 82.5%

      \[\leadsto \frac{1}{\color{blue}{-s \cdot \left(\frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \mathsf{fma}\left(5, \frac{x \cdot x}{s}, 0\right)\right)}{-s} + -4\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s} - -4 \cdot s}} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 84.9

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

   10. Applied rewrites 84.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(1 + e^{-\frac{\left|x\right|}{s}}\right) \cdot \left(s \cdot \left(1 + e^{-\frac{\left|x\right|}{s}}\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;e^{-\frac{\left|x\right|}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 2000:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 2000.0)
     (/ 1.0 (* s (fma x (/ x (* s s)) 4.0)))
     (/ 1.0 (fma x (/ x s) (* s 4.0))))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 2000.0f) {
		tmp = 1.0f / (s * fmaf(x, (x / (s * s)), 4.0f));
	} else {
		tmp = 1.0f / fmaf(x, (x / s), (s * 4.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(2000.0))
		tmp = Float32(Float32(1.0) / Float32(s * fma(x, Float32(x / Float32(s * s)), Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(x / s), Float32(s * Float32(4.0))));
	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))))) < 2e3

   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. Step-by-step derivation

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

   7. Applied rewrites 22.4%

      \[\leadsto \frac{1}{\color{blue}{-s \cdot \left(\frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \mathsf{fma}\left(5, \frac{x \cdot x}{s}, 0\right)\right)}{-s} + -4\right)}} \]

   8. Taylor expanded in s around inf

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

      1. distribute-rgt-in N/A

         \[\leadsto \frac{1}{\color{blue}{4 \cdot s + \left(-4 \cdot \frac{{x}^{2}}{{s}^{2}} + 5 \cdot \frac{{x}^{2}}{{s}^{2}}\right) \cdot s}} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{1}{4 \cdot s + \color{blue}{\left(\frac{{x}^{2}}{{s}^{2}} \cdot \left(-4 + 5\right)\right)} \cdot s} \]

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

         \[\leadsto \frac{1}{4 \cdot s + \color{blue}{\frac{{x}^{2} \cdot 1}{{s}^{2}}} \cdot s} \]

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{1}{4 \cdot s + \frac{\mathsf{neg}\left({x}^{2} \cdot \color{blue}{\left(-5 + 4\right)}\right)}{{s}^{2}} \cdot s} \]

      8. distribute-rgt-out N/A

         \[\leadsto \frac{1}{4 \cdot s + \frac{\mathsf{neg}\left(\color{blue}{\left(-5 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}\right)}{{s}^{2}} \cdot s} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{1}{4 \cdot s + \frac{\color{blue}{-1 \cdot \left(-5 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}}{{s}^{2}} \cdot s} \]

      10. associate-*r/ N/A

          \[\leadsto \frac{1}{4 \cdot s + \color{blue}{\left(-1 \cdot \frac{-5 \cdot {x}^{2} + 4 \cdot {x}^{2}}{{s}^{2}}\right)} \cdot s} \]

      11. distribute-rgt-in N/A

          \[\leadsto \frac{1}{\color{blue}{s \cdot \left(4 + -1 \cdot \frac{-5 \cdot {x}^{2} + 4 \cdot {x}^{2}}{{s}^{2}}\right)}} \]

      12. lower-*.f32 N/A

          \[\leadsto \frac{1}{\color{blue}{s \cdot \left(4 + -1 \cdot \frac{-5 \cdot {x}^{2} + 4 \cdot {x}^{2}}{{s}^{2}}\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{s \cdot \color{blue}{\left(-1 \cdot \frac{-5 \cdot {x}^{2} + 4 \cdot {x}^{2}}{{s}^{2}} + 4\right)}} \]

   10. Applied rewrites 84.6%

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

   if 2e3 < (/.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 98.7%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

   7. Applied rewrites 83.2%

      \[\leadsto \frac{1}{\color{blue}{-s \cdot \left(\frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \mathsf{fma}\left(5, \frac{x \cdot x}{s}, 0\right)\right)}{-s} + -4\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s} - -4 \cdot s}} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 85.9

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

   10. Applied rewrites 85.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(1 + e^{-\frac{\left|x\right|}{s}}\right) \cdot \left(s \cdot \left(1 + e^{-\frac{\left|x\right|}{s}}\right)\right)} \leq 2000:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(x, \frac{x}{s \cdot s}, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.5f) {
		tmp = 1.0f / ((x * x) / s);
	} else {
		tmp = 0.25f / 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 = exp(-(abs(x) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / (t_1 * (s * t_1))) <= 0.5e0) then
        tmp = 1.0e0 / ((x * x) / s)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp(-(abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / (t_1 * (s * t_1))) <= single(0.5))
		tmp = single(1.0) / ((x * x) / s);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = 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))))) < 0.5

   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. Step-by-step derivation

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

   7. Applied rewrites 20.2%

      \[\leadsto \frac{1}{\color{blue}{-s \cdot \left(\frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \mathsf{fma}\left(5, \frac{x \cdot x}{s}, 0\right)\right)}{-s} + -4\right)}} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-4 \cdot {x}^{2} + 5 \cdot {x}^{2}}{s}}} \]
   9. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(-4 + 5\right)}}{s}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{{x}^{2} \cdot \color{blue}{1}}{s}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2}}}{s}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]

      5. unpow2 N/A

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

      6. lower-*.f32 58.8

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

   10. Applied rewrites 58.8%

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

   if 0.5 < (/.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 98.8%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 81.5

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

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

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

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-fabs.f32 N/A

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

   12. remove-double-neg N/A

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

   13. lift-neg.f32 N/A

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

   14. remove-double-neg N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 7: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

   1. lift-fabs.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift-pow.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-*.f32 N/A

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right), -\frac{\left|x\right|}{s}\right)}}{s}} \]
7. Add Preprocessing

Alternative 8: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ \frac{t\_0}{\left(s \cdot \left(2 - \frac{\mathsf{fma}\left(x, \frac{x}{s} \cdot -0.5, \left|x\right|\right)}{s}\right)\right) \cdot \left(1 + t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))))
   (/
    t_0
    (* (* s (- 2.0 (/ (fma x (* (/ x s) -0.5) (fabs x)) s))) (+ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	return t_0 / ((s * (2.0f - (fmaf(x, ((x / s) * -0.5f), fabsf(x)) / s))) * (1.0f + t_0));
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	return Float32(t_0 / Float32(Float32(s * Float32(Float32(2.0) - Float32(fma(x, Float32(Float32(x / s) * Float32(-0.5)), abs(x)) / s))) * Float32(Float32(1.0) + t_0)))
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. associate-*r/ N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. div-sub N/A

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

   11. unsub-neg N/A

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

   12. mul-1-neg N/A

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

   13. +-commutative N/A

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

5. Applied rewrites 95.1%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-fabs.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 95.1

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

7. Applied rewrites 95.9%

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

8. Final simplification 95.9%

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

Alternative 9: 95.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.3

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

5. Applied rewrites 94.3%

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

6. Final simplification 94.3%

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

Alternative 10: 94.8% 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(-Float32(abs(x) / 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 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 94.0

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

5. Applied rewrites 94.0%

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

6. Final simplification 94.0%

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

Alternative 11: 66.2% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. lift-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

7. Applied rewrites 34.3%

   \[\leadsto \frac{1}{\color{blue}{-s \cdot \left(\frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \mathsf{fma}\left(5, \frac{x \cdot x}{s}, 0\right)\right)}{-s} + -4\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s} - -4 \cdot s}} \]
9. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 64.8

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

10. Applied rewrites 64.8%

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

Alternative 12: 27.1% 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 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 21.7

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

5. Applied rewrites 21.7%

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

Reproduce

herbie shell --seed 2024216 
(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))))))