Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 11.9s

Alternatives: 11

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 62.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s))) * fma(s, (Float32(exp(1)) ^ Float32(x / s)), s)))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 96.9%

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

   2. exp-prod 96.9%

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

   3. add-sqr-sqrt 49.3%

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

   4. fabs-sqr 49.3%

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

   5. add-sqr-sqrt 62.3%

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

6. Applied egg-rr 63.6%

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

   1. exp-1-e 62.3%

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

8. Simplified 63.6%

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

9. Final simplification 63.6%

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

Alternative 3: 60.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s + s \cdot e^{\frac{x \cdot \log e}{s}}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 0.5 (+ s (* s (exp (/ (* x (log E)) s))))))

C

float code(float x, float s) {
	return 0.5f / (s + (s * expf(((x * logf(((float) M_E))) / s))));
}

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(s + Float32(s * exp(Float32(Float32(x * log(Float32(exp(1)))) / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s + (s * exp(((x * log(single(2.71828182845904523536))) / s))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 96.9%

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

   1. *-un-lft-identity 96.9%

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

   2. exp-prod 96.9%

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

   3. add-sqr-sqrt 49.3%

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

   4. fabs-sqr 49.3%

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

   5. add-sqr-sqrt 62.3%

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

7. Applied egg-rr 62.3%

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

   1. exp-1-e 62.3%

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

9. Simplified 62.3%

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

10. Taylor expanded in x around inf 62.3%

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

11. Final simplification 62.3%

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

Alternative 4: 60.7% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* (fma s (pow E (/ x s)) s) 2.0)))

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(fma(s, (Float32(exp(1)) ^ Float32(x / s)), s) * Float32(2.0)))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 96.9%

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

   1. *-un-lft-identity 96.9%

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

   2. exp-prod 96.9%

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

   3. add-sqr-sqrt 49.3%

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

   4. fabs-sqr 49.3%

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

   5. add-sqr-sqrt 62.3%

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

7. Applied egg-rr 62.3%

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

   1. exp-1-e 62.3%

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

9. Simplified 62.3%

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

10. Final simplification 62.3%

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

Alternative 5: 60.7% accurate, 5.5× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* 2.0 (+ s (* s (pow E (/ x s)))))))

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(s + Float32(s * (Float32(exp(1)) ^ Float32(x / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(2.0) * (s + (s * (single(2.71828182845904523536) ^ (x / s)))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 96.9%

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

   1. fma-udef 96.9%

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

   2. add-sqr-sqrt 49.3%

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

   3. fabs-sqr 49.3%

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

   4. add-sqr-sqrt 62.3%

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

7. Applied egg-rr 62.3%

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

   1. *-un-lft-identity 62.3%

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

   2. pow-exp 62.3%

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

   3. e-exp-1 62.3%

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

9. Applied egg-rr 62.3%

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

10. Final simplification 62.3%

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

Alternative 6: 60.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s + s \cdot e^{\frac{x}{s}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (+ s (* s (exp (/ x s))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 96.9%

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

   1. fma-udef 96.9%

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

   2. add-sqr-sqrt 49.3%

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

   3. fabs-sqr 49.3%

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

   4. add-sqr-sqrt 62.3%

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

7. Applied egg-rr 62.3%

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

8. Taylor expanded in x around inf 62.3%

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

9. Final simplification 62.3%

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

Alternative 7: 45.8% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999949504854e-6) (/ 0.25 s) (* (/ s x) (/ 1.0 x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999949504854e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s / x) * (1.0f / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999949504854e-6) then
        tmp = 0.25e0 / s
    else
        tmp = (s / x) * (1.0e0 / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999949504854e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s / x) * Float32(Float32(1.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999949504854e-6))
		tmp = single(0.25) / s;
	else
		tmp = (s / x) * (single(1.0) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-6

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf 36.2%

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

   if 1.99999999e-6 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in s around -inf 27.5%

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

      1. +-commutative 27.5%

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

      2. mul-1-neg 27.5%

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

      3. distribute-lft1-in 73.5%

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

      4. metadata-eval 73.5%

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

      5. associate-*r/ 73.5%

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

      6. mul-1-neg 73.5%

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

      7. remove-double-neg 73.5%

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

      8. associate-+r+ 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in s around 0 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. unpow2 71.8%

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

      3. times-frac 71.8%

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

   10. Applied egg-rr 71.8%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 56.4× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / ((s * 4.0f) + (x * (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) + (x * (x / s)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around -inf 40.6%

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

   1. +-commutative 40.6%

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

   2. mul-1-neg 40.6%

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

   3. distribute-lft1-in 65.6%

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

   4. metadata-eval 65.6%

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

   5. associate-*r/ 65.6%

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

   6. mul-1-neg 65.6%

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

   7. remove-double-neg 65.6%

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

   8. associate-+r+ 65.6%

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

7. Simplified 65.6%

   \[\leadsto \frac{1}{\color{blue}{\left(s \cdot 4 + 0\right) + \frac{x \cdot x}{s}}} \]
8. Step-by-step derivation

   1. +-commutative 65.6%

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

   2. add-sqr-sqrt 65.6%

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

   3. fma-def 65.6%

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

   4. sqrt-div 65.6%

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

   5. sqrt-prod 35.6%

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

   6. add-sqr-sqrt 65.5%

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

   7. sqrt-div 65.5%

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

   8. sqrt-prod 35.6%

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

   9. add-sqr-sqrt 65.9%

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

   10. +-rgt-identity 65.9%

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

9. Applied egg-rr 65.9%

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

    1. fma-udef 65.9%

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

    2. unpow2 65.9%

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

11. Simplified 65.9%

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

    1. unpow2 65.9%

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

    2. div-inv 65.9%

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

    3. associate-*l* 65.9%

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

    4. times-frac 65.9%

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

    5. *-un-lft-identity 65.9%

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

    6. add-sqr-sqrt 65.9%

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

13. Applied egg-rr 65.9%

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

14. Final simplification 65.9%

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

Alternative 9: 66.3% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around -inf 40.6%

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

   1. +-commutative 40.6%

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

   2. mul-1-neg 40.6%

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

   3. distribute-lft1-in 65.6%

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

   4. metadata-eval 65.6%

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

   5. associate-*r/ 65.6%

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

   6. mul-1-neg 65.6%

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

   7. remove-double-neg 65.6%

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

   8. associate-+r+ 65.6%

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

7. Simplified 65.6%

   \[\leadsto \frac{1}{\color{blue}{\left(s \cdot 4 + 0\right) + \frac{x \cdot x}{s}}} \]
8. Step-by-step derivation

   1. +-commutative 65.6%

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

   2. add-sqr-sqrt 65.6%

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

   3. fma-def 65.6%

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

   4. sqrt-div 65.6%

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

   5. sqrt-prod 35.6%

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

   6. add-sqr-sqrt 65.5%

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

   7. sqrt-div 65.5%

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

   8. sqrt-prod 35.6%

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

   9. add-sqr-sqrt 65.9%

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

   10. +-rgt-identity 65.9%

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

9. Applied egg-rr 65.9%

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

    1. fma-udef 65.9%

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

    2. unpow2 65.9%

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

11. Simplified 65.9%

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

    1. unpow2 65.9%

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

    2. frac-2neg 65.9%

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

    3. clear-num 65.9%

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

    4. frac-2neg 65.9%

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

    5. add-sqr-sqrt 35.6%

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

    6. sqrt-prod 65.5%

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

    7. unpow2 65.5%

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

    8. sqrt-div 65.5%

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

    9. frac-times 65.5%

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

    10. *-un-lft-identity 65.5%

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

    11. sqrt-div 65.5%

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

    12. unpow2 65.5%

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

    13. sqrt-prod 35.6%

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

    14. add-sqr-sqrt 65.9%

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

13. Applied egg-rr 65.9%

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

    1. associate-/l/ 65.9%

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

    2. associate-/r* 65.9%

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

    3. associate-*r/ 65.9%

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

    4. rem-square-sqrt 65.9%

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

15. Simplified 65.9%

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

16. Final simplification 65.9%

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

Alternative 10: 45.8% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999949504854e-6) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999949504854e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999949504854e-6) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999949504854e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999949504854e-6))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-6

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf 36.2%

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

   if 1.99999999e-6 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in s around -inf 27.5%

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

      1. +-commutative 27.5%

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

      2. mul-1-neg 27.5%

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

      3. distribute-lft1-in 73.5%

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

      4. metadata-eval 73.5%

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

      5. associate-*r/ 73.5%

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

      6. mul-1-neg 73.5%

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

      7. remove-double-neg 73.5%

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

      8. associate-+r+ 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in s around 0 71.8%

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

      1. unpow2 71.8%

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

   10. Applied egg-rr 71.8%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.3% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 25.5%

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

6. Final simplification 25.5%

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

Reproduce

herbie shell --seed 2024021 
(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))))))