Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 16.9s

Alternatives: 14

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\left(1 + e^{\frac{-x_m}{s}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x_m}{s}}\right)\right), s\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  1.0
  (* (+ 1.0 (exp (/ (- x_m) s))) (fma s (expm1 (log1p (exp (/ x_m s)))) s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / ((1.0f + expf((-x_m / s))) * fmaf(s, expm1f(log1pf(expf((x_m / s)))), s));
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + exp(Float32(Float32(-x_m) / s))) * fma(s, expm1(log1p(exp(Float32(x_m / s)))), s)))
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. add-sqr-sqrt 53.0%

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

   4. fabs-sqr 53.0%

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

   5. add-sqr-sqrt 62.6%

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

6. Applied egg-rr 62.6%

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

   1. +-commutative 62.6%

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

   2. fma-def 62.6%

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

8. Simplified 62.6%

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

   1. expm1-log1p-u 62.6%

      \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}, s\right)} \]

10. Applied egg-rr 62.6%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}, s\right)} \]
11. Step-by-step derivation

    1. distribute-frac-neg 62.6%

       \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{-\frac{\left|x\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    2. rec-exp 62.7%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    3. add-sqr-sqrt 53.0%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    4. fabs-sqr 53.0%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    5. add-sqr-sqrt 99.9%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{x}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

12. Applied egg-rr 99.9%

    \[\leadsto \frac{1}{\left(1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]
13. Step-by-step derivation

    1. rec-exp 99.8%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{e^{-\frac{x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    2. distribute-neg-frac 99.8%

       \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{\frac{-x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

14. Simplified 99.8%

    \[\leadsto \frac{1}{\left(1 + \color{blue}{e^{\frac{-x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

15. Final simplification 99.8%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{-x}{s}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]
16. Add Preprocessing

Alternative 2: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\left(1 + e^{\frac{-x_m}{s}}\right) \cdot \mathsf{fma}\left(s, 1 + \mathsf{expm1}\left(\frac{x_m}{s}\right), s\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ (- x_m) s))) (fma s (+ 1.0 (expm1 (/ x_m s))) s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / ((1.0f + expf((-x_m / s))) * fmaf(s, (1.0f + expm1f((x_m / s))), s));
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + exp(Float32(Float32(-x_m) / s))) * fma(s, Float32(Float32(1.0) + expm1(Float32(x_m / s))), s)))
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. add-sqr-sqrt 53.0%

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

   4. fabs-sqr 53.0%

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

   5. add-sqr-sqrt 62.6%

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

6. Applied egg-rr 62.6%

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

   1. +-commutative 62.6%

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

   2. fma-def 62.6%

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

8. Simplified 62.6%

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

   1. expm1-log1p-u 62.6%

      \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}, s\right)} \]

   2. expm1-udef 62.7%

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

   3. log1p-udef 62.6%

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

   4. add-exp-log 62.6%

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

10. Applied egg-rr 62.6%

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

    1. associate--l+ 62.6%

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

    2. expm1-def 62.6%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, 1 + \color{blue}{\mathsf{expm1}\left(\frac{x}{s}\right)}, s\right)} \]

12. Simplified 62.6%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, \color{blue}{1 + \mathsf{expm1}\left(\frac{x}{s}\right)}, s\right)} \]
13. Step-by-step derivation

    1. distribute-frac-neg 62.6%

       \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{-\frac{\left|x\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    2. rec-exp 62.7%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    3. add-sqr-sqrt 53.0%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    4. fabs-sqr 53.0%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    5. add-sqr-sqrt 99.9%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{x}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

14. Applied egg-rr 99.9%

    \[\leadsto \frac{1}{\left(1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot \mathsf{fma}\left(s, 1 + \mathsf{expm1}\left(\frac{x}{s}\right), s\right)} \]
15. Step-by-step derivation

    1. rec-exp 99.8%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{e^{-\frac{x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    2. distribute-neg-frac 99.8%

       \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{\frac{-x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

16. Simplified 99.8%

    \[\leadsto \frac{1}{\left(1 + \color{blue}{e^{\frac{-x}{s}}}\right) \cdot \mathsf{fma}\left(s, 1 + \mathsf{expm1}\left(\frac{x}{s}\right), s\right)} \]

17. Final simplification 99.8%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{-x}{s}}\right) \cdot \mathsf{fma}\left(s, 1 + \mathsf{expm1}\left(\frac{x}{s}\right), s\right)} \]
18. Add Preprocessing

Alternative 3: 99.7% accurate, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ (- x_m) s))) (fma s (exp (/ x_m s)) s))))

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. add-sqr-sqrt 53.0%

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

   4. fabs-sqr 53.0%

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

   5. add-sqr-sqrt 62.6%

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

6. Applied egg-rr 62.6%

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

   1. +-commutative 62.6%

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

   2. fma-def 62.6%

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

8. Simplified 62.6%

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

   1. distribute-frac-neg 62.6%

      \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{-\frac{\left|x\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

   2. rec-exp 62.7%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

   3. add-sqr-sqrt 53.0%

      \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

   4. fabs-sqr 53.0%

      \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

   5. add-sqr-sqrt 99.9%

      \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{x}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

10. Applied egg-rr 99.9%

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

    1. rec-exp 99.8%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{e^{-\frac{x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    2. distribute-neg-frac 99.8%

       \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{\frac{-x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 2.9× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (* s (* (+ 1.0 (exp (/ (- x_m) s))) (+ 1.0 (exp (/ x_m s)))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / (s * ((1.0f + expf((-x_m / s))) * (1.0f + expf((x_m / s)))));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. add-sqr-sqrt 53.0%

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

   4. fabs-sqr 53.0%

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

   5. add-sqr-sqrt 62.6%

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

6. Applied egg-rr 62.6%

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

   1. +-commutative 62.6%

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

   2. fma-def 62.6%

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

8. Simplified 62.6%

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

9. Taylor expanded in s around 0 62.6%

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

    1. associate-*r/ 62.6%

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

    2. mul-1-neg 62.6%

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

11. Simplified 62.6%

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

    1. distribute-frac-neg 62.6%

       \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{-\frac{\left|x\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    2. rec-exp 62.7%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    3. add-sqr-sqrt 53.0%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    4. fabs-sqr 53.0%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    5. add-sqr-sqrt 99.9%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{x}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

13. Applied egg-rr 99.8%

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

    1. rec-exp 99.8%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{e^{-\frac{x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    2. distribute-neg-frac 99.8%

       \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{\frac{-x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

15. Simplified 99.8%

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

16. Final simplification 99.8%

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

Alternative 5: 99.6% accurate, 2.9× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ (- x_m) s))) (* s (+ 1.0 (exp (/ x_m s)))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / ((1.0f + expf((-x_m / s))) * (s * (1.0f + expf((x_m / s)))));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. add-sqr-sqrt 53.0%

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

   4. fabs-sqr 53.0%

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

   5. add-sqr-sqrt 62.6%

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

6. Applied egg-rr 62.6%

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

   1. +-commutative 62.6%

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

   2. fma-def 62.6%

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

8. Simplified 62.6%

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

   1. expm1-log1p-u 62.6%

      \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}, s\right)} \]

10. Applied egg-rr 62.6%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)}, s\right)} \]
11. Step-by-step derivation

    1. distribute-frac-neg 62.6%

       \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{-\frac{\left|x\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    2. rec-exp 62.7%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    3. add-sqr-sqrt 53.0%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    4. fabs-sqr 53.0%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    5. add-sqr-sqrt 99.9%

       \[\leadsto \frac{1}{\left(1 + \frac{1}{e^{\frac{\color{blue}{x}}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

12. Applied egg-rr 99.9%

    \[\leadsto \frac{1}{\left(1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]
13. Step-by-step derivation

    1. rec-exp 99.8%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{e^{-\frac{x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

    2. distribute-neg-frac 99.8%

       \[\leadsto \frac{1}{\left(1 + e^{\color{blue}{\frac{-x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

14. Simplified 99.8%

    \[\leadsto \frac{1}{\left(1 + \color{blue}{e^{\frac{-x}{s}}}\right) \cdot \mathsf{fma}\left(s, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right), s\right)} \]

15. Taylor expanded in s around 0 99.8%

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

16. Final simplification 99.8%

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

Alternative 6: 95.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.5}{s + s \cdot e^{\frac{x_m}{s}}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.5 (+ s (* s (exp (/ x_m s))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.5f / (s + (s * expf((x_m / s))));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.5e0 / (s + (s * exp((x_m / s))))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.5) / Float32(s + Float32(s * exp(Float32(x_m / s)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.5) / (s + (s * exp((x_m / s))));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. add-sqr-sqrt 53.0%

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

   4. fabs-sqr 53.0%

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

   5. add-sqr-sqrt 62.6%

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

6. Applied egg-rr 62.6%

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

   1. +-commutative 62.6%

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

   2. fma-def 62.6%

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

8. Simplified 62.6%

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

9. Taylor expanded in s around inf 59.9%

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

10. Taylor expanded in x around inf 59.9%

    \[\leadsto \color{blue}{\frac{0.5}{s + s \cdot e^{\frac{x}{s}}}} \]

11. Final simplification 59.9%

    \[\leadsto \frac{0.5}{s + s \cdot e^{\frac{x}{s}}} \]
12. Add Preprocessing

Alternative 7: 50.4% accurate, 56.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.5}{s + s \cdot \left(1 + \frac{x_m}{s}\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.5 (+ s (* s (+ 1.0 (/ x_m s))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.5f / (s + (s * (1.0f + (x_m / s))));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.5e0 / (s + (s * (1.0e0 + (x_m / s))))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.5) / Float32(s + Float32(s * Float32(Float32(1.0) + Float32(x_m / s)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.5) / (s + (s * (single(1.0) + (x_m / s))));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. add-sqr-sqrt 53.0%

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

   4. fabs-sqr 53.0%

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

   5. add-sqr-sqrt 62.6%

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

6. Applied egg-rr 62.6%

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

   1. +-commutative 62.6%

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

   2. fma-def 62.6%

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

8. Simplified 62.6%

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

9. Taylor expanded in s around inf 59.9%

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

10. Taylor expanded in x around inf 59.9%

    \[\leadsto \color{blue}{\frac{0.5}{s + s \cdot e^{\frac{x}{s}}}} \]

11. Taylor expanded in x around 0 46.4%

    \[\leadsto \frac{0.5}{s + s \cdot \color{blue}{\left(1 + \frac{x}{s}\right)}} \]

12. Final simplification 46.4%

    \[\leadsto \frac{0.5}{s + s \cdot \left(1 + \frac{x}{s}\right)} \]
13. Add Preprocessing

Alternative 8: 50.9% accurate, 56.4× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 1.0 (* s (+ 4.0 (* (/ x_m s) 4.0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. distribute-lft-in 26.0%

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

   3. *-commutative 26.0%

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

   4. *-un-lft-identity 26.0%

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

   5. add-sqr-sqrt 16.7%

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

   6. fabs-sqr 16.7%

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

   7. add-sqr-sqrt 24.8%

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

   8. add-sqr-sqrt 16.6%

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

   9. fabs-sqr 16.6%

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

   10. add-sqr-sqrt 26.3%

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

6. Applied egg-rr 59.5%

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

   1. *-commutative 59.5%

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

   2. distribute-rgt1-in 59.5%

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

   3. +-commutative 59.5%

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

   4. fma-def 59.5%

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

   5. distribute-rgt-out 59.5%

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

   6. associate-*l* 59.5%

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

   7. distribute-lft-out 59.5%

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

   8. *-lft-identity 59.5%

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

   9. distribute-rgt-in 59.5%

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

   10. +-commutative 59.5%

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

   11. unpow2 59.5%

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

8. Simplified 59.5%

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

9. Taylor expanded in x around 0 46.7%

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

    1. *-commutative 46.7%

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

11. Simplified 46.7%

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

12. Final simplification 46.7%

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

Alternative 9: 65.1% accurate, 56.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{s \cdot 4 + \frac{x_m \cdot x_m}{s}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 1.0 (+ (* s 4.0) (/ (* x_m x_m) s))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in s around -inf 35.5%

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

   1. +-commutative 35.5%

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

   2. mul-1-neg 35.5%

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

   3. distribute-lft1-in 62.0%

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

   4. metadata-eval 62.0%

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

   5. associate-*r/ 62.0%

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

   6. mul-1-neg 62.0%

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

   7. remove-double-neg 62.0%

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

   8. associate-+r+ 62.0%

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

7. Simplified 62.0%

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

8. Final simplification 62.0%

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

Alternative 10: 30.1% accurate, 77.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 4.999999987376214e-7) (/ 0.25 s) (/ 0.25 x_m)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 4.999999987376214e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.25f / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 4.999999987376214e-7) then
        tmp = 0.25e0 / s
    else
        tmp = 0.25e0 / x_m
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(4.999999987376214e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.25) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(4.999999987376214e-7))
		tmp = single(0.25) / s;
	else
		tmp = single(0.25) / x_m;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999e-7

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-in 36.0%

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

      3. *-commutative 36.0%

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

      4. *-un-lft-identity 36.0%

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

      5. add-sqr-sqrt 23.1%

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

      6. fabs-sqr 23.1%

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

      7. add-sqr-sqrt 34.3%

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

      8. add-sqr-sqrt 23.0%

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

      9. fabs-sqr 23.0%

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

      10. add-sqr-sqrt 36.4%

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

   6. Applied egg-rr 43.9%

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

      1. *-commutative 43.9%

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

      2. distribute-rgt1-in 43.9%

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

      3. +-commutative 43.9%

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

      4. fma-def 43.9%

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

      5. distribute-rgt-out 43.9%

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

      6. associate-*l* 43.9%

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

      7. distribute-lft-out 43.9%

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

      8. *-lft-identity 43.9%

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

      9. distribute-rgt-in 43.9%

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

      10. +-commutative 43.9%

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

      11. unpow2 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in s around inf 34.8%

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

   if 4.99999999e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in -0.0%

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

      3. *-commutative -0.0%

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

      4. *-un-lft-identity -0.0%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt -0.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-rgt1-in 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. associate-*l* 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. *-lft-identity 100.0%

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

      9. distribute-rgt-in 100.0%

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

      10. +-commutative 100.0%

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

      11. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in s around inf 11.1%

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

       1. distribute-lft-out 11.1%

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

   11. Simplified 11.1%

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

   12. Taylor expanded in s around 0 11.1%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.9% accurate, 88.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.25 \cdot \frac{1}{x_m + s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (* 0.25 (/ 1.0 (+ x_m s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.25f * (1.0f / (x_m + s));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.25e0 * (1.0e0 / (x_m + s))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(x_m + s)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.25) * (single(1.0) / (x_m + s));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. distribute-lft-in 26.0%

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

   3. *-commutative 26.0%

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

   4. *-un-lft-identity 26.0%

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

   5. add-sqr-sqrt 16.7%

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

   6. fabs-sqr 16.7%

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

   7. add-sqr-sqrt 24.8%

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

   8. add-sqr-sqrt 16.6%

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

   9. fabs-sqr 16.6%

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

   10. add-sqr-sqrt 26.3%

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

6. Applied egg-rr 59.5%

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

   1. *-commutative 59.5%

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

   2. distribute-rgt1-in 59.5%

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

   3. +-commutative 59.5%

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

   4. fma-def 59.5%

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

   5. distribute-rgt-out 59.5%

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

   6. associate-*l* 59.5%

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

   7. distribute-lft-out 59.5%

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

   8. *-lft-identity 59.5%

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

   9. distribute-rgt-in 59.5%

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

   10. +-commutative 59.5%

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

   11. unpow2 59.5%

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

8. Simplified 59.5%

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

9. Taylor expanded in s around inf 28.3%

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

    1. distribute-lft-out 28.3%

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

11. Simplified 28.3%

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

    1. associate-/r* 28.3%

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

    2. div-inv 28.3%

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

    3. metadata-eval 28.3%

       \[\leadsto \color{blue}{0.25} \cdot \frac{1}{s + x} \]

    4. +-commutative 28.3%

       \[\leadsto 0.25 \cdot \frac{1}{\color{blue}{x + s}} \]

13. Applied egg-rr 28.3%

    \[\leadsto \color{blue}{0.25 \cdot \frac{1}{x + s}} \]

14. Final simplification 28.3%

    \[\leadsto 0.25 \cdot \frac{1}{x + s} \]
15. Add Preprocessing

Alternative 12: 29.0% accurate, 88.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.5}{x_m + s \cdot 2} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.5 (+ x_m (* s 2.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.5f / (x_m + (s * 2.0f));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.5e0 / (x_m + (s * 2.0e0))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.5) / Float32(x_m + Float32(s * Float32(2.0))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.5) / (x_m + (s * single(2.0)));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. add-sqr-sqrt 53.0%

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

   4. fabs-sqr 53.0%

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

   5. add-sqr-sqrt 62.6%

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

6. Applied egg-rr 62.6%

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

   1. +-commutative 62.6%

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

   2. fma-def 62.6%

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

8. Simplified 62.6%

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

9. Taylor expanded in s around inf 59.9%

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

10. Taylor expanded in x around inf 59.9%

    \[\leadsto \color{blue}{\frac{0.5}{s + s \cdot e^{\frac{x}{s}}}} \]

11. Taylor expanded in s around inf 28.6%

    \[\leadsto \frac{0.5}{\color{blue}{x + 2 \cdot s}} \]

12. Final simplification 28.6%

    \[\leadsto \frac{0.5}{x + s \cdot 2} \]
13. Add Preprocessing

Alternative 13: 28.9% accurate, 124.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.25}{x_m + s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 (+ x_m s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.25f / (x_m + s);
}

Fortran

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

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.25) / Float32(x_m + s))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. distribute-lft-in 26.0%

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

   3. *-commutative 26.0%

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

   4. *-un-lft-identity 26.0%

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

   5. add-sqr-sqrt 16.7%

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

   6. fabs-sqr 16.7%

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

   7. add-sqr-sqrt 24.8%

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

   8. add-sqr-sqrt 16.6%

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

   9. fabs-sqr 16.6%

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

   10. add-sqr-sqrt 26.3%

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

6. Applied egg-rr 59.5%

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

   1. *-commutative 59.5%

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

   2. distribute-rgt1-in 59.5%

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

   3. +-commutative 59.5%

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

   4. fma-def 59.5%

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

   5. distribute-rgt-out 59.5%

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

   6. associate-*l* 59.5%

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

   7. distribute-lft-out 59.5%

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

   8. *-lft-identity 59.5%

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

   9. distribute-rgt-in 59.5%

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

   10. +-commutative 59.5%

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

   11. unpow2 59.5%

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

8. Simplified 59.5%

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

9. Taylor expanded in s around inf 28.3%

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

    1. distribute-lft-out 28.3%

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

11. Simplified 28.3%

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

    1. expm1-log1p-u 26.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{4 \cdot \left(s + x\right)}\right)\right)} \]

    2. expm1-udef 60.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{4 \cdot \left(s + x\right)}\right)} - 1} \]

    3. associate-/r* 60.7%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{4}}{s + x}}\right)} - 1 \]

    4. metadata-eval 60.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.25}}{s + x}\right)} - 1 \]

    5. +-commutative 60.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{x + s}}\right)} - 1 \]

13. Applied egg-rr 60.7%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{x + s}\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 26.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{x + s}\right)\right)} \]

    2. expm1-log1p 28.3%

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

    3. +-commutative 28.3%

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

15. Simplified 28.3%

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

16. Final simplification 28.3%

    \[\leadsto \frac{0.25}{x + s} \]
17. Add Preprocessing

Alternative 14: 26.8% accurate, 206.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.25}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. distribute-lft-in 26.0%

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

   3. *-commutative 26.0%

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

   4. *-un-lft-identity 26.0%

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

   5. add-sqr-sqrt 16.7%

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

   6. fabs-sqr 16.7%

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

   7. add-sqr-sqrt 24.8%

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

   8. add-sqr-sqrt 16.6%

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

   9. fabs-sqr 16.6%

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

   10. add-sqr-sqrt 26.3%

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

6. Applied egg-rr 59.5%

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

   1. *-commutative 59.5%

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

   2. distribute-rgt1-in 59.5%

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

   3. +-commutative 59.5%

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

   4. fma-def 59.5%

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

   5. distribute-rgt-out 59.5%

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

   6. associate-*l* 59.5%

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

   7. distribute-lft-out 59.5%

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

   8. *-lft-identity 59.5%

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

   9. distribute-rgt-in 59.5%

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

   10. +-commutative 59.5%

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

   11. unpow2 59.5%

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

8. Simplified 59.5%

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

9. Taylor expanded in s around inf 26.4%

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

10. Final simplification 26.4%

    \[\leadsto \frac{0.25}{s} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))))))