Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 15.8s

Alternatives: 13

Speedup: 5.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. /-rgt-identity 99.6%

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

   2. associate-/l/ 99.6%

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

   3. *-lft-identity 99.6%

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

   4. +-commutative 99.6%

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

   5. distribute-rgt-in 99.7%

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

   6. *-lft-identity 99.7%

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

   7. +-commutative 99.7%

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

   8. distribute-rgt-in 99.7%

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

   9. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in s around 0 99.7%

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

   1. associate-/r* 99.8%

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

   2. mul-1-neg 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

6. Simplified 99.8%

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

   1. add-log-exp 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. /-rgt-identity 99.6%

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

   2. associate-/l/ 99.6%

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

   3. *-lft-identity 99.6%

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

   4. +-commutative 99.6%

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

   5. distribute-rgt-in 99.7%

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

   6. *-lft-identity 99.7%

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

   7. +-commutative 99.7%

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

   8. distribute-rgt-in 99.7%

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

   9. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in s around 0 99.7%

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

   1. associate-/r* 99.8%

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

   2. mul-1-neg 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 88.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999682655225e-21)
   (/ (/ 1.0 s) (+ (/ (* x x) (* s s)) 4.0))
   (/ (/ 1.0 s) (+ (exp (/ x s)) 3.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999682655225e-21f) {
		tmp = (1.0f / s) / (((x * x) / (s * s)) + 4.0f);
	} else {
		tmp = (1.0f / s) / (expf((x / s)) + 3.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999682655225e-21)) then
        tmp = (1.0e0 / s) / (((x * x) / (s * s)) + 4.0e0)
    else
        tmp = (1.0e0 / s) / (exp((x / s)) + 3.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999682655225e-21))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(x * x) / Float32(s * s)) + Float32(4.0)));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(exp(Float32(x / s)) + Float32(3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999682655225e-21))
		tmp = (single(1.0) / s) / (((x * x) / (s * s)) + single(4.0));
	else
		tmp = (single(1.0) / s) / (exp((x / s)) + single(3.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999968e-21

   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 98.9%

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

   4. Taylor expanded in s around inf 50.1%

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

      1. associate-+r+ 50.1%

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

      2. distribute-lft1-in 50.1%

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

      3. metadata-eval 50.1%

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

      4. mul0-lft 85.1%

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

      5. associate-+r+ 85.1%

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

      6. unpow2 85.1%

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

      7. sqr-abs 85.1%

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

      8. unpow2 85.1%

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

      9. metadata-eval 85.1%

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

   6. Simplified 85.1%

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

   if -9.99999968e-21 < x

   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 99.1%

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

      1. add-exp-log 99.0%

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

      2. associate-+r+ 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in s around inf 93.1%

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

      1. add-exp-log 93.1%

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

      2. associate-+l+ 93.1%

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

      3. add-sqr-sqrt 74.7%

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

      4. fabs-sqr 74.7%

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

      5. add-sqr-sqrt 90.4%

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

      6. metadata-eval 90.4%

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

   8. Applied egg-rr 90.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]

Alternative 4: 88.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{e^{\frac{x}{s}} + 3}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999682655225e-21)
   (/ (/ 1.0 s) (+ (/ (* x x) (* s s)) 4.0))
   (/ (/ 1.0 (+ (exp (/ x s)) 3.0)) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999682655225e-21f) {
		tmp = (1.0f / s) / (((x * x) / (s * s)) + 4.0f);
	} else {
		tmp = (1.0f / (expf((x / s)) + 3.0f)) / 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 <= (-9.999999682655225e-21)) then
        tmp = (1.0e0 / s) / (((x * x) / (s * s)) + 4.0e0)
    else
        tmp = (1.0e0 / (exp((x / s)) + 3.0e0)) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999682655225e-21))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(x * x) / Float32(s * s)) + Float32(4.0)));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(exp(Float32(x / s)) + Float32(3.0))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999682655225e-21))
		tmp = (single(1.0) / s) / (((x * x) / (s * s)) + single(4.0));
	else
		tmp = (single(1.0) / (exp((x / s)) + single(3.0))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999968e-21

   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 98.9%

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

   4. Taylor expanded in s around inf 50.1%

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

      1. associate-+r+ 50.1%

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

      2. distribute-lft1-in 50.1%

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

      3. metadata-eval 50.1%

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

      4. mul0-lft 85.1%

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

      5. associate-+r+ 85.1%

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

      6. unpow2 85.1%

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

      7. sqr-abs 85.1%

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

      8. unpow2 85.1%

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

      9. metadata-eval 85.1%

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

   6. Simplified 85.1%

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

   if -9.99999968e-21 < x

   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 99.1%

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

   4. Taylor expanded in s around inf 93.1%

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

      1. div-inv 93.1%

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

      2. add-sqr-sqrt 74.7%

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

      3. fabs-sqr 74.7%

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

      4. add-sqr-sqrt 90.4%

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

      5. metadata-eval 90.4%

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

   6. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{1}{e^{\frac{x}{s}} + 3}} \]
   7. Step-by-step derivation

      1. associate-*l/ 91.1%

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

      2. *-un-lft-identity 91.1%

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

   8. Applied egg-rr 91.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{e^{\frac{x}{s}} + 3}}{s}\\ \end{array} \]

Alternative 5: 99.6% accurate, 5.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (s * (2.0f + (2.0f * coshf((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 + (2.0e0 * cosh((x / s)))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * cosh(Float32(x / s))))))
end

MATLAB

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

Derivation

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 99.0%

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

4. Taylor expanded in s around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. +-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. mul-1-neg 99.7%

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

   5. distribute-frac-neg 99.7%

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

6. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\left(2 + e^{\frac{\left|x\right|}{s}}\right) + e^{\frac{-\left|x\right|}{s}}\right)\right)\right)}} \]

   2. expm1-udef 77.7%

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

8. Applied egg-rr 77.7%

   \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(2 + 2 \cdot \cosh \left(\frac{x}{s}\right)\right)\right)} - 1}} \]
9. Step-by-step derivation

   1. expm1-def 99.7%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(2 + 2 \cdot \cosh \left(\frac{x}{s}\right)\right)\right)\right)}} \]

   2. expm1-log1p 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

    \[\leadsto \frac{1}{s \cdot \left(2 + 2 \cdot \cosh \left(\frac{x}{s}\right)\right)} \]

Alternative 6: 78.9% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{1}{s} \cdot \left(x \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.0000000272452012e-27)
   (/ (/ 1.0 s) (+ 4.0 (* (/ 1.0 s) (* x (/ x s)))))
   (/ (/ 1.0 s) (+ (/ (* x x) (* s s)) 4.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.0000000272452012e-27f) {
		tmp = (1.0f / s) / (4.0f + ((1.0f / s) * (x * (x / s))));
	} else {
		tmp = (1.0f / s) / (((x * x) / (s * 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 <= 1.0000000272452012e-27) then
        tmp = (1.0e0 / s) / (4.0e0 + ((1.0e0 / s) * (x * (x / s))))
    else
        tmp = (1.0e0 / s) / (((x * x) / (s * s)) + 4.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.0000000272452012e-27))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(Float32(1.0) / s) * Float32(x * Float32(x / s)))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(x * x) / Float32(s * s)) + Float32(4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.0000000272452012e-27))
		tmp = (single(1.0) / s) / (single(4.0) + ((single(1.0) / s) * (x * (x / s))));
	else
		tmp = (single(1.0) / s) / (((x * x) / (s * s)) + single(4.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000003e-27

   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 99.2%

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

   4. Taylor expanded in s around inf 48.8%

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

      1. associate-+r+ 48.8%

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

      2. distribute-lft1-in 48.8%

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

      3. metadata-eval 48.8%

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

      4. mul0-lft 71.8%

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

      5. associate-+r+ 71.8%

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

      6. unpow2 71.8%

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

      7. sqr-abs 71.8%

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

      8. unpow2 71.8%

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

      9. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. clear-num 71.8%

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

      2. inv-pow 71.8%

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

   8. Applied egg-rr 71.8%

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

      1. unpow-1 71.8%

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

      2. unpow2 71.8%

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

      3. associate-/l* 78.2%

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

      4. associate-/r/ 78.2%

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

      5. unpow2 78.2%

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

      6. associate-*r/ 79.2%

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

   10. Simplified 79.2%

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

   if 1.00000003e-27 < x

   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 98.8%

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

   4. Taylor expanded in s around inf 55.3%

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

      1. associate-+r+ 55.3%

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

      2. distribute-lft1-in 55.3%

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

      3. metadata-eval 55.3%

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

      4. mul0-lft 77.4%

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

      5. associate-+r+ 77.4%

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

      6. unpow2 77.4%

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

      7. sqr-abs 77.4%

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

      8. unpow2 77.4%

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

      9. metadata-eval 77.4%

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

   6. Simplified 77.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{1}{s} \cdot \left(x \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \end{array} \]

Alternative 7: 78.9% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{1}{\frac{s}{x} \cdot \frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.0000000272452012e-27)
   (/ (/ 1.0 s) (+ 4.0 (/ 1.0 (* (/ s x) (/ s x)))))
   (/ (/ 1.0 s) (+ (/ (* x x) (* s s)) 4.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.0000000272452012e-27f) {
		tmp = (1.0f / s) / (4.0f + (1.0f / ((s / x) * (s / x))));
	} else {
		tmp = (1.0f / s) / (((x * x) / (s * 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 <= 1.0000000272452012e-27) then
        tmp = (1.0e0 / s) / (4.0e0 + (1.0e0 / ((s / x) * (s / x))))
    else
        tmp = (1.0e0 / s) / (((x * x) / (s * s)) + 4.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.0000000272452012e-27))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(1.0) / Float32(Float32(s / x) * Float32(s / x)))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(x * x) / Float32(s * s)) + Float32(4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.0000000272452012e-27))
		tmp = (single(1.0) / s) / (single(4.0) + (single(1.0) / ((s / x) * (s / x))));
	else
		tmp = (single(1.0) / s) / (((x * x) / (s * s)) + single(4.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000003e-27

   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 99.2%

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

   4. Taylor expanded in s around inf 48.8%

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

      1. associate-+r+ 48.8%

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

      2. distribute-lft1-in 48.8%

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

      3. metadata-eval 48.8%

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

      4. mul0-lft 71.8%

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

      5. associate-+r+ 71.8%

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

      6. unpow2 71.8%

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

      7. sqr-abs 71.8%

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

      8. unpow2 71.8%

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

      9. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. clear-num 71.8%

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

      2. inv-pow 71.8%

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

   8. Applied egg-rr 71.8%

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

      1. unpow-1 71.8%

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

      2. times-frac 79.2%

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

   10. Simplified 79.2%

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

   if 1.00000003e-27 < x

   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 98.8%

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

   4. Taylor expanded in s around inf 55.3%

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

      1. associate-+r+ 55.3%

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

      2. distribute-lft1-in 55.3%

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

      3. metadata-eval 55.3%

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

      4. mul0-lft 77.4%

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

      5. associate-+r+ 77.4%

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

      6. unpow2 77.4%

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

      7. sqr-abs 77.4%

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

      8. unpow2 77.4%

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

      9. metadata-eval 77.4%

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

   6. Simplified 77.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{1}{\frac{s}{x} \cdot \frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \end{array} \]

Alternative 8: 78.9% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.0000000272452012e-27)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (/ (/ 1.0 s) (+ (/ (* x x) (* s s)) 4.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.0000000272452012e-27f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else {
		tmp = (1.0f / s) / (((x * x) / (s * 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 <= 1.0000000272452012e-27) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else
        tmp = (1.0e0 / s) / (((x * x) / (s * s)) + 4.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.0000000272452012e-27))
		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(Float32(x * x) / Float32(s * s)) + Float32(4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.0000000272452012e-27))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	else
		tmp = (single(1.0) / s) / (((x * x) / (s * s)) + single(4.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000003e-27

   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 99.2%

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

   4. Taylor expanded in s around inf 48.8%

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

      1. associate-+r+ 48.8%

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

      2. distribute-lft1-in 48.8%

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

      3. metadata-eval 48.8%

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

      4. mul0-lft 71.8%

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

      5. associate-+r+ 71.8%

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

      6. unpow2 71.8%

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

      7. sqr-abs 71.8%

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

      8. unpow2 71.8%

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

      9. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. times-frac 79.2%

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

   8. Applied egg-rr 79.2%

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

   if 1.00000003e-27 < x

   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 98.8%

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

   4. Taylor expanded in s around inf 55.3%

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

      1. associate-+r+ 55.3%

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

      2. distribute-lft1-in 55.3%

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

      3. metadata-eval 55.3%

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

      4. mul0-lft 77.4%

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

      5. associate-+r+ 77.4%

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

      6. unpow2 77.4%

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

      7. sqr-abs 77.4%

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

      8. unpow2 77.4%

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

      9. metadata-eval 77.4%

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

   6. Simplified 77.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s} + 4}\\ \end{array} \]

Alternative 9: 76.3% 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 99.6%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 51.4%

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

   1. associate-+r+ 51.4%

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

   2. distribute-lft1-in 51.4%

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

   3. metadata-eval 51.4%

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

   4. mul0-lft 74.1%

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

   5. associate-+r+ 74.1%

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

   6. unpow2 74.1%

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

   7. sqr-abs 74.1%

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

   8. unpow2 74.1%

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

   9. metadata-eval 74.1%

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

6. Simplified 74.1%

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

   1. times-frac 74.8%

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

8. Applied egg-rr 74.8%

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

9. Final simplification 74.8%

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

Alternative 10: 63.1% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -2.0000000233721948e-7) (not (<= x 1.9999999920083944e-12)))
   (/ 1.0 (* x (/ x s)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -2.0000000233721948e-7f) || !(x <= 1.9999999920083944e-12f)) {
		tmp = 1.0f / (x * (x / s));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-2.0000000233721948e-7)) .or. (.not. (x <= 1.9999999920083944e-12))) then
        tmp = 1.0e0 / (x * (x / s))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-2.0000000233721948e-7)) || !(x <= Float32(1.9999999920083944e-12)))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-2.0000000233721948e-7)) || ~((x <= single(1.9999999920083944e-12))))
		tmp = single(1.0) / (x * (x / s));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000002e-7 or 1.99999999e-12 < 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 98.6%

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

   4. Taylor expanded in s around inf 41.1%

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

      1. associate-+r+ 41.1%

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

      2. distribute-lft1-in 41.1%

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

      3. metadata-eval 41.1%

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

      4. mul0-lft 78.7%

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

      5. associate-+r+ 78.7%

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

      6. unpow2 78.7%

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

      7. sqr-abs 78.7%

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

      8. unpow2 78.7%

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

      9. metadata-eval 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in s around 0 61.2%

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

      1. unpow2 61.2%

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

   9. Simplified 61.2%

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

       1. clear-num 64.2%

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

       2. inv-pow 64.2%

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

   11. Applied egg-rr 64.2%

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

       1. unpow-1 64.2%

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

       2. associate-*r/ 64.2%

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

   13. Simplified 64.2%

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

   if -2.00000002e-7 < x < 1.99999999e-12

   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 99.6%

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

   4. Taylor expanded in s around inf 65.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 11: 63.1% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -2.0000000233721948e-7) (not (<= x 1.9999999920083944e-12)))
   (/ 1.0 (/ x (/ s x)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -2.0000000233721948e-7f) || !(x <= 1.9999999920083944e-12f)) {
		tmp = 1.0f / (x / (s / x));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-2.0000000233721948e-7)) .or. (.not. (x <= 1.9999999920083944e-12))) then
        tmp = 1.0e0 / (x / (s / x))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-2.0000000233721948e-7)) || !(x <= Float32(1.9999999920083944e-12)))
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-2.0000000233721948e-7)) || ~((x <= single(1.9999999920083944e-12))))
		tmp = single(1.0) / (x / (s / x));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000002e-7 or 1.99999999e-12 < 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 98.6%

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

   4. Taylor expanded in s around inf 41.1%

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

      1. associate-+r+ 41.1%

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

      2. distribute-lft1-in 41.1%

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

      3. metadata-eval 41.1%

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

      4. mul0-lft 78.7%

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

      5. associate-+r+ 78.7%

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

      6. unpow2 78.7%

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

      7. sqr-abs 78.7%

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

      8. unpow2 78.7%

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

      9. metadata-eval 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in s around 0 61.2%

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

      1. unpow2 61.2%

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

   9. Simplified 61.2%

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

       1. clear-num 64.2%

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

       2. inv-pow 64.2%

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

   11. Applied egg-rr 64.2%

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

       1. unpow-1 64.2%

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

       2. associate-/l* 64.2%

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

   13. Simplified 64.2%

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

   if -2.00000002e-7 < x < 1.99999999e-12

   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 99.6%

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

   4. Taylor expanded in s around inf 65.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 12: 61.9% accurate, 66.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -2.0000000233721948e-7) (not (<= x 1.9999999920083944e-12)))
   (/ s (* x x))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -2.0000000233721948e-7f) || !(x <= 1.9999999920083944e-12f)) {
		tmp = s / (x * x);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-2.0000000233721948e-7)) .or. (.not. (x <= 1.9999999920083944e-12))) then
        tmp = s / (x * x)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-2.0000000233721948e-7)) || !(x <= Float32(1.9999999920083944e-12)))
		tmp = Float32(s / Float32(x * x));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-2.0000000233721948e-7)) || ~((x <= single(1.9999999920083944e-12))))
		tmp = s / (x * x);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000002e-7 or 1.99999999e-12 < 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 98.6%

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

   4. Taylor expanded in s around inf 41.1%

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

      1. associate-+r+ 41.1%

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

      2. distribute-lft1-in 41.1%

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

      3. metadata-eval 41.1%

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

      4. mul0-lft 78.7%

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

      5. associate-+r+ 78.7%

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

      6. unpow2 78.7%

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

      7. sqr-abs 78.7%

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

      8. unpow2 78.7%

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

      9. metadata-eval 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in s around 0 61.2%

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

      1. unpow2 61.2%

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

   9. Simplified 61.2%

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

   if -2.00000002e-7 < x < 1.99999999e-12

   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 99.6%

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

   4. Taylor expanded in s around inf 65.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13: 26.8% 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 99.6%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 28.8%

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

5. Final simplification 28.8%

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

Reproduce

herbie shell --seed 2023196 
(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))))))