Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 10.4s

Alternatives: 11

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around 0 99.3%

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

   1. associate-+r+ 99.3%

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

   2. +-commutative 99.3%

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

   3. mul-1-neg 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 2: 96.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.2%

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

   2. mul-1-neg 99.2%

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

   3. distribute-frac-neg 99.2%

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

   4. +-commutative 99.2%

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

   5. mul-1-neg 99.2%

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

   6. distribute-frac-neg 99.2%

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

4. Simplified 99.2%

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

   1. unpow2 99.2%

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

   2. distribute-lft-in 99.3%

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

6. Applied egg-rr 95.5%

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

   1. *-rgt-identity 95.5%

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

   2. distribute-lft-in 95.5%

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

   3. unpow2 95.5%

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

8. Simplified 95.5%

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

9. Final simplification 95.5%

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

Alternative 3: 96.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ (exp (/ (fabs x) s)) 3.0)))

C

float code(float x, float s) {
	return (1.0f / s) / (expf((fabsf(x) / s)) + 3.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / (exp((abs(x) / s)) + 3.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(exp(Float32(abs(x) / s)) + Float32(3.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / s) / (exp((abs(x) / s)) + single(3.0));
end

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around inf 95.2%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]

5. Final simplification 95.2%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3} \]

Alternative 4: 94.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.2%

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

   2. mul-1-neg 99.2%

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

   3. distribute-frac-neg 99.2%

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

   4. +-commutative 99.2%

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

   5. mul-1-neg 99.2%

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

   6. distribute-frac-neg 99.2%

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

4. Simplified 99.2%

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

5. Taylor expanded in s around inf 92.9%

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

6. Final simplification 92.9%

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

Alternative 5: 85.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;{\left(x \cdot -262144\right)}^{-262144}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0020000000949949026)
   (pow (* x -262144.0) -262144.0)
   (/ 1.0 (* s (+ 2.0 (* 2.0 (exp (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0020000000949949026f) {
		tmp = powf((x * -262144.0f), -262144.0f);
	} else {
		tmp = 1.0f / (s * (2.0f + (2.0f * expf((x / s)))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.0020000000949949026e0)) then
        tmp = (x * (-262144.0e0)) ** (-262144.0e0)
    else
        tmp = 1.0e0 / (s * (2.0e0 + (2.0e0 * exp((x / s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0020000000949949026))
		tmp = Float32(x * Float32(-262144.0)) ^ Float32(-262144.0);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * exp(Float32(x / s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0020000000949949026))
		tmp = (x * single(-262144.0)) ^ single(-262144.0);
	else
		tmp = single(1.0) / (s * (single(2.0) + (single(2.0) * exp((x / s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.00200000009

   1. Initial program 100.0%

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

   2. Taylor expanded in s around inf 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(-262144 \cdot x\right)}^{-262144}} \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {\color{blue}{\left(x \cdot -262144\right)}}^{-262144} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(x \cdot -262144\right)}^{-262144}} \]

   if -0.00200000009 < x

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in s around 0 99.1%

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

      1. associate-+r+ 99.1%

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

      2. +-commutative 99.1%

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

      3. mul-1-neg 99.1%

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

   6. Simplified 99.1%

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

      1. add-sqr-sqrt 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. distribute-lft-in 99.1%

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

   10. Applied egg-rr 83.9%

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

       1. distribute-lft-out 83.9%

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

       2. associate-+r+ 83.8%

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

       3. count-2 83.8%

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

   12. Simplified 83.8%

       \[\leadsto \frac{1}{\color{blue}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;{\left(x \cdot -262144\right)}^{-262144}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \]

Alternative 6: 85.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;{\left(x \cdot -262144\right)}^{-262144}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot e^{\frac{x}{s}}}}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0020000000949949026)
   (pow (* x -262144.0) -262144.0)
   (/ (/ 1.0 (* s (exp (/ x s)))) 4.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0020000000949949026f) {
		tmp = powf((x * -262144.0f), -262144.0f);
	} else {
		tmp = (1.0f / (s * expf((x / s)))) / 4.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.0020000000949949026e0)) then
        tmp = (x * (-262144.0e0)) ** (-262144.0e0)
    else
        tmp = (1.0e0 / (s * exp((x / s)))) / 4.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0020000000949949026))
		tmp = Float32(x * Float32(-262144.0)) ^ Float32(-262144.0);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(s * exp(Float32(x / s)))) / Float32(4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0020000000949949026))
		tmp = (x * single(-262144.0)) ^ single(-262144.0);
	else
		tmp = (single(1.0) / (s * exp((x / s)))) / single(4.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.00200000009

   1. Initial program 100.0%

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

   2. Taylor expanded in s around inf 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(-262144 \cdot x\right)}^{-262144}} \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {\color{blue}{\left(x \cdot -262144\right)}}^{-262144} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(x \cdot -262144\right)}^{-262144}} \]

   if -0.00200000009 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 98.9%

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

      1. associate-/r* 98.9%

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

      2. mul-1-neg 98.9%

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

      3. distribute-frac-neg 98.9%

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

      4. +-commutative 98.9%

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

      5. mul-1-neg 98.9%

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

      6. distribute-frac-neg 98.9%

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

   4. Simplified 98.9%

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

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

   6. Applied egg-rr 87.5%

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

      1. unpow-1 87.5%

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

   8. Simplified 87.5%

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

   9. Taylor expanded in s around inf 83.2%

      \[\leadsto \frac{\frac{1}{s \cdot e^{\frac{x}{s}}}}{\color{blue}{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;{\left(x \cdot -262144\right)}^{-262144}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot e^{\frac{x}{s}}}}{4}\\ \end{array} \]

Alternative 7: 85.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x \cdot -262144\right)}^{-262144}\\ \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 0.0010000000474974513:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (pow (* x -262144.0) -262144.0)))
   (if (<= x -0.20000000298023224)
     t_0
     (if (<= x 4.0000000781659255e-25)
       (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
       (if (<= x 0.0010000000474974513)
         (/ (/ 1.0 s) (+ 4.0 (/ (* x x) (* s s))))
         t_0)))))

C

float code(float x, float s) {
	float t_0 = powf((x * -262144.0f), -262144.0f);
	float tmp;
	if (x <= -0.20000000298023224f) {
		tmp = t_0;
	} else if (x <= 4.0000000781659255e-25f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else if (x <= 0.0010000000474974513f) {
		tmp = (1.0f / s) / (4.0f + ((x * x) / (s * s)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (x * (-262144.0e0)) ** (-262144.0e0)
    if (x <= (-0.20000000298023224e0)) then
        tmp = t_0
    else if (x <= 4.0000000781659255e-25) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else if (x <= 0.0010000000474974513e0) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) / (s * s)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x * Float32(-262144.0)) ^ Float32(-262144.0)
	tmp = Float32(0.0)
	if (x <= Float32(-0.20000000298023224))
		tmp = t_0;
	elseif (x <= Float32(4.0000000781659255e-25))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	elseif (x <= Float32(0.0010000000474974513))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = (x * single(-262144.0)) ^ single(-262144.0);
	tmp = single(0.0);
	if (x <= single(-0.20000000298023224))
		tmp = t_0;
	elseif (x <= single(4.0000000781659255e-25))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	elseif (x <= single(0.0010000000474974513))
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) / (s * s)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.200000003 or 0.00100000005 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in s around inf 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(-262144 \cdot x\right)}^{-262144}} \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {\color{blue}{\left(x \cdot -262144\right)}}^{-262144} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(x \cdot -262144\right)}^{-262144}} \]

   if -0.200000003 < x < 4.00000008e-25

   1. Initial program 98.9%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in s around 0 99.2%

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

      1. associate-/r* 99.1%

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

      2. associate-+r+ 99.1%

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

      3. +-commutative 99.1%

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

      4. mul-1-neg 99.1%

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

      5. distribute-frac-neg 99.1%

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

      6. associate-+l+ 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in s around inf 50.0%

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

      1. associate-+r+ 50.0%

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

      2. distribute-lft1-in 50.0%

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

      3. metadata-eval 50.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 50.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. +-rgt-identity 50.0%

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

      6. +-commutative 50.0%

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

      7. unpow2 50.0%

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

      8. sqr-abs 50.0%

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

      9. unpow2 50.0%

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

      10. times-frac 70.5%

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

   9. Simplified 70.5%

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

   if 4.00000008e-25 < x < 0.00100000005

   1. Initial program 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in s around 0 96.7%

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

      1. associate-/r* 96.7%

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

      2. associate-+r+ 96.8%

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

      3. +-commutative 96.8%

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

      4. mul-1-neg 96.8%

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

      5. distribute-frac-neg 96.8%

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

      6. associate-+l+ 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in s around inf 77.7%

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

      1. associate-+r+ 77.7%

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

      2. distribute-lft1-in 77.7%

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

      3. metadata-eval 77.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 77.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. +-rgt-identity 77.7%

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

      6. +-commutative 77.7%

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

      7. unpow2 77.7%

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

      8. sqr-abs 77.7%

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

      9. unpow2 77.7%

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

   9. Simplified 77.7%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{4 + \frac{x \cdot x}{s \cdot s}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;{\left(x \cdot -262144\right)}^{-262144}\\ \mathbf{elif}\;x \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 0.0010000000474974513:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot -262144\right)}^{-262144}\\ \end{array} \]

Alternative 8: 79.2% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000781659255e-25)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (/ (/ 1.0 s) (+ 4.0 (/ (* x x) (* s s))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000781659255e-25f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) / (s * s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.0000000781659255e-25) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) / (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000781659255e-25))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000781659255e-25))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000008e-25

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in s around 0 99.6%

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

      1. associate-/r* 99.5%

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

      2. associate-+r+ 99.5%

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

      3. +-commutative 99.5%

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

      4. mul-1-neg 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. associate-+l+ 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in s around inf 46.4%

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

      1. associate-+r+ 46.4%

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

      2. distribute-lft1-in 46.4%

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

      3. metadata-eval 46.4%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 68.4%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. +-rgt-identity 68.4%

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

      6. +-commutative 68.4%

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

      7. unpow2 68.4%

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

      8. sqr-abs 68.4%

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

      9. unpow2 68.4%

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

      10. times-frac 79.0%

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

   9. Simplified 79.0%

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

   if 4.00000008e-25 < x

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in s around 0 99.1%

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

      1. associate-/r* 99.1%

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

      2. associate-+r+ 99.1%

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

      3. +-commutative 99.1%

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

      4. mul-1-neg 99.1%

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

      5. distribute-frac-neg 99.1%

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

      6. associate-+l+ 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in s around inf 53.7%

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

      1. associate-+r+ 53.7%

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

      2. distribute-lft1-in 53.7%

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

      3. metadata-eval 53.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 84.1%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. +-rgt-identity 84.1%

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

      6. +-commutative 84.1%

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

      7. unpow2 84.1%

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

      8. sqr-abs 84.1%

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

      9. unpow2 84.1%

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

   9. Simplified 84.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{4 + \frac{x \cdot x}{s \cdot s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 9: 76.4% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around 0 99.3%

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

   1. associate-/r* 99.3%

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

   2. associate-+r+ 99.3%

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

   3. +-commutative 99.3%

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

   4. mul-1-neg 99.3%

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

   5. distribute-frac-neg 99.3%

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

   6. associate-+l+ 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in s around inf 49.7%

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

   1. associate-+r+ 49.7%

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

   2. distribute-lft1-in 49.7%

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

   3. metadata-eval 49.7%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

   4. mul0-lft 75.5%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

   5. +-rgt-identity 75.5%

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

   6. +-commutative 75.5%

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

   7. unpow2 75.5%

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

   8. sqr-abs 75.5%

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

   9. unpow2 75.5%

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

   10. times-frac 78.0%

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

9. Simplified 78.0%

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

10. Final simplification 78.0%

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

Alternative 10: 5.3% accurate, 206.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 262144 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* x 262144.0))

C

float code(float x, float s) {
	return x * 262144.0f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = x * 262144.0e0
end function

Julia

function code(x, s)
	return Float32(x * Float32(262144.0))
end

MATLAB

function tmp = code(x, s)
	tmp = x * single(262144.0);
end

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in s around inf 93.9%

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

4. Applied egg-rr 4.9%

   \[\leadsto \color{blue}{-262144 \cdot x + \mathsf{fma}\left(-262144 \cdot x, -3, -262144 \cdot x\right)} \]
5. Step-by-step derivation

   1. *-commutative 4.9%

      \[\leadsto \color{blue}{x \cdot -262144} + \mathsf{fma}\left(-262144 \cdot x, -3, -262144 \cdot x\right) \]

   2. fma-udef 4.9%

      \[\leadsto x \cdot -262144 + \color{blue}{\left(\left(-262144 \cdot x\right) \cdot -3 + -262144 \cdot x\right)} \]

   3. *-commutative 4.9%

      \[\leadsto x \cdot -262144 + \left(\color{blue}{-3 \cdot \left(-262144 \cdot x\right)} + -262144 \cdot x\right) \]

   4. associate-*r* 4.9%

      \[\leadsto x \cdot -262144 + \left(\color{blue}{\left(-3 \cdot -262144\right) \cdot x} + -262144 \cdot x\right) \]

   5. distribute-rgt-out 4.9%

      \[\leadsto x \cdot -262144 + \color{blue}{x \cdot \left(-3 \cdot -262144 + -262144\right)} \]

   6. distribute-lft-out 5.2%

      \[\leadsto \color{blue}{x \cdot \left(-262144 + \left(-3 \cdot -262144 + -262144\right)\right)} \]

   7. metadata-eval 5.2%

      \[\leadsto x \cdot \left(-262144 + \left(\color{blue}{786432} + -262144\right)\right) \]

   8. metadata-eval 5.2%

      \[\leadsto x \cdot \left(-262144 + \color{blue}{524288}\right) \]

   9. metadata-eval 5.2%

      \[\leadsto x \cdot \color{blue}{262144} \]

6. Simplified 5.2%

   \[\leadsto \color{blue}{x \cdot 262144} \]

7. Final simplification 5.2%

   \[\leadsto x \cdot 262144 \]

Alternative 11: 26.2% 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 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around inf 25.6%

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

5. Final simplification 25.6%

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

Reproduce

herbie shell --seed 2023189 
(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))))))