Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.5s

Alternatives: 11

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. associate-+r+ N/A

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. associate-+r+ N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. neg-mul-1 N/A

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

   5. lower-exp.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-fabs.f32 99.8

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.5% 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(abs(x) / Float32(-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
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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 99.7

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ \frac{t\_0}{\left(s \cdot \left(2 + \frac{\frac{x \cdot \left(x \cdot \mathsf{fma}\left(\frac{\left|x\right|}{s}, -0.16666666666666666, 0.5\right)\right)}{s} - \left|x\right|}{s}\right)\right) \cdot \left(t\_0 + 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))))
   (/
    t_0
    (*
     (*
      s
      (+
       2.0
       (/
        (-
         (/ (* x (* x (fma (/ (fabs x) s) -0.16666666666666666 0.5))) s)
         (fabs x))
        s)))
     (+ t_0 1.0)))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	return t_0 / ((s * (2.0f + ((((x * (x * fmaf((fabsf(x) / s), -0.16666666666666666f, 0.5f))) / s) - fabsf(x)) / s))) * (t_0 + 1.0f));
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	return Float32(t_0 / Float32(Float32(s * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(x * Float32(x * fma(Float32(abs(x) / s), Float32(-0.16666666666666666), Float32(0.5)))) / s) - abs(x)) / s))) * Float32(t_0 + Float32(1.0))))
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

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in s around -inf

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

8. Applied rewrites 95.4%

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

9. Final simplification 95.4%

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

Alternative 5: 96.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ \frac{t\_0}{\mathsf{fma}\left(t\_0, s \cdot 3, s\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s))))) (/ t_0 (fma t_0 (* s 3.0) s))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	return t_0 / fmaf(t_0, (s * 3.0f), s);
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	return Float32(t_0 / fma(t_0, Float32(s * Float32(3.0)), 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
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

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. associate-+r+ N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\mathsf{fma}\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}, \color{blue}{3 \cdot s}, s\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\mathsf{fma}\left(e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}, \color{blue}{s \cdot 3}, s\right)} \]

   2. lower-*.f32 95.4

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\mathsf{fma}\left(e^{-\frac{\left|x\right|}{s}}, \color{blue}{s \cdot 3}, s\right)} \]

9. Applied rewrites 95.4%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\mathsf{fma}\left(e^{-\frac{\left|x\right|}{s}}, \color{blue}{s \cdot 3}, s\right)} \]

10. Final simplification 95.4%

    \[\leadsto \frac{e^{\frac{\left|x\right|}{-s}}}{\mathsf{fma}\left(e^{\frac{\left|x\right|}{-s}}, s \cdot 3, s\right)} \]
11. Add Preprocessing

Alternative 6: 95.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 94.0%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 94.0

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lift-/.f32 N/A

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

   11. lift-neg.f32 94.0

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

7. Applied rewrites 94.0%

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

8. Final simplification 94.0%

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

Alternative 7: 94.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(4.0)) * exp((abs(x) / 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. 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

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 93.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. div-inv N/A

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

   10. lift-exp.f32 N/A

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

   11. rec-exp N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-/.f32 N/A

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

9. Applied rewrites 93.7%

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

Alternative 8: 94.7% 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 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 93.6

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

5. Applied rewrites 93.6%

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

6. Final simplification 93.6%

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

Alternative 9: 27.2% accurate, 8.3× speedup

Math

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

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(x * Float32(Float32(-0.0625) / s)), Float32(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. 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. lift-*.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 23.3%

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

9. Applied rewrites 23.4%

   \[\leadsto \frac{\mathsf{fma}\left(x, \frac{x \cdot -0.0625}{s \cdot s}, 0.25\right)}{s} \]
10. Step-by-step derivation

11. Applied rewrites 27.7%

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

12. Final simplification 27.7%

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

Alternative 10: 27.2% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(fma(x, Float32(Float32(x / s) * Float32(Float32(-0.0625) / s)), Float32(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. 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. lift-*.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 23.3%

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

9. Applied rewrites 27.7%

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

Alternative 11: 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 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 27.7

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

5. Applied rewrites 27.7%

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

Reproduce

herbie shell --seed 2024222 
(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))))))