Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 10.8s

Alternatives: 13

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 99.2%

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

5. Applied egg-rr 97.3%

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

   1. expm1-def 97.3%

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

   2. expm1-log1p 97.3%

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

   3. +-commutative 97.3%

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

7. Simplified 97.3%

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

   1. div-inv 97.3%

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

   2. exp-prod 88.4%

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

   3. add-sqr-sqrt 88.4%

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

   4. sqrt-unprod 88.4%

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

   5. sqr-neg 88.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 63.6%

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

   8. exp-prod 64.6%

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

   9. div-inv 64.6%

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

   10. distribute-frac-neg 64.6%

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

   11. exp-neg 64.6%

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

   12. div-inv 64.6%

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

   13. exp-prod 63.6%

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

9. Applied egg-rr 99.2%

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

10. Taylor expanded in s around 0 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 99.2%

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

5. Applied egg-rr 97.3%

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

   1. expm1-def 97.3%

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

   2. expm1-log1p 97.3%

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

   3. +-commutative 97.3%

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

7. Simplified 97.3%

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

   1. div-inv 97.3%

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

   2. exp-prod 88.4%

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

   3. add-sqr-sqrt 88.4%

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

   4. sqrt-unprod 88.4%

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

   5. sqr-neg 88.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 63.6%

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

   8. exp-prod 64.6%

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

   9. div-inv 64.6%

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

   10. distribute-frac-neg 64.6%

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

   11. exp-neg 64.6%

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

   12. div-inv 64.6%

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

   13. exp-prod 63.6%

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

9. Applied egg-rr 99.2%

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

    1. rec-exp 99.2%

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

    2. distribute-neg-frac 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 3: 96.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 99.2%

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

5. Applied egg-rr 97.3%

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

   1. expm1-def 97.3%

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

   2. expm1-log1p 97.3%

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

   3. +-commutative 97.3%

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

7. Simplified 97.3%

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

8. Taylor expanded in x around 0 96.5%

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

9. Final simplification 96.5%

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

Alternative 4: 96.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.999999821029131 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{s \cdot \left(1 + \left(2 + \frac{1}{t_0}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x -5.999999821029131e-35)
     (/ 1.0 (* s (+ 1.0 (+ 2.0 (/ 1.0 t_0)))))
     (/ 1.0 (* s (+ t_0 3.0))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= -5.999999821029131e-35f) {
		tmp = 1.0f / (s * (1.0f + (2.0f + (1.0f / t_0))));
	} else {
		tmp = 1.0f / (s * (t_0 + 3.0f));
	}
	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 = exp((x / s))
    if (x <= (-5.999999821029131e-35)) then
        tmp = 1.0e0 / (s * (1.0e0 + (2.0e0 + (1.0e0 / t_0))))
    else
        tmp = 1.0e0 / (s * (t_0 + 3.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(-5.999999821029131e-35))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(1.0) + Float32(Float32(2.0) + Float32(Float32(1.0) / t_0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(t_0 + Float32(3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(-5.999999821029131e-35))
		tmp = single(1.0) / (s * (single(1.0) + (single(2.0) + (single(1.0) / t_0))));
	else
		tmp = single(1.0) / (s * (t_0 + single(3.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.99999982e-35

   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 99.3%

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

   5. Applied egg-rr 99.2%

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

      1. expm1-def 99.2%

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

      2. expm1-log1p 99.3%

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

      3. +-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. div-inv 99.3%

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

      2. exp-prod 78.0%

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

      3. add-sqr-sqrt 78.0%

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

      4. sqrt-unprod 78.0%

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

      5. sqr-neg 78.0%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 22.6%

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

      8. exp-prod 22.4%

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

      9. div-inv 22.4%

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

      10. distribute-frac-neg 22.4%

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

      11. exp-neg 22.4%

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

      12. div-inv 22.4%

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

      13. exp-prod 22.6%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in s around 0 99.3%

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

   11. Taylor expanded in x around 0 95.0%

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

   if -5.99999982e-35 < x

   1. Initial program 99.7%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 95.8%

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

      1. expm1-def 95.8%

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

      2. expm1-log1p 95.8%

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

      3. +-commutative 95.8%

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

   7. Simplified 95.8%

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

      1. div-inv 95.8%

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

      2. exp-prod 96.9%

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

      3. add-sqr-sqrt 96.9%

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

      4. sqrt-unprod 96.9%

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

      5. sqr-neg 96.9%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 96.9%

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

      8. exp-prod 99.1%

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

      9. div-inv 99.1%

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

      10. distribute-frac-neg 99.1%

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

      11. exp-neg 99.1%

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

      12. div-inv 99.1%

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

      13. exp-prod 96.9%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in s around 0 99.8%

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

   11. Taylor expanded in x around 0 97.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.999999821029131 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{s \cdot \left(1 + \left(2 + \frac{1}{e^{\frac{x}{s}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 5: 96.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.999999821029131 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{1}{t_0} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x -5.999999821029131e-35)
     (/ (/ 1.0 s) (+ (/ 1.0 t_0) 3.0))
     (/ 1.0 (* s (+ t_0 3.0))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= -5.999999821029131e-35f) {
		tmp = (1.0f / s) / ((1.0f / t_0) + 3.0f);
	} else {
		tmp = 1.0f / (s * (t_0 + 3.0f));
	}
	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 = exp((x / s))
    if (x <= (-5.999999821029131e-35)) then
        tmp = (1.0e0 / s) / ((1.0e0 / t_0) + 3.0e0)
    else
        tmp = 1.0e0 / (s * (t_0 + 3.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(-5.999999821029131e-35))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(1.0) / t_0) + Float32(3.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(t_0 + Float32(3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(-5.999999821029131e-35))
		tmp = (single(1.0) / s) / ((single(1.0) / t_0) + single(3.0));
	else
		tmp = single(1.0) / (s * (t_0 + single(3.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.99999982e-35

   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 99.3%

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

   5. Applied egg-rr 99.2%

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

      1. expm1-def 99.2%

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

      2. expm1-log1p 99.3%

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

      3. +-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. div-inv 99.3%

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

      2. exp-prod 78.0%

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

      3. add-sqr-sqrt 78.0%

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

      4. sqrt-unprod 78.0%

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

      5. sqr-neg 78.0%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 22.6%

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

      8. exp-prod 22.4%

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

      9. div-inv 22.4%

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

      10. distribute-frac-neg 22.4%

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

      11. exp-neg 22.4%

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

      12. div-inv 22.4%

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

      13. exp-prod 22.6%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in x around 0 95.0%

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

   if -5.99999982e-35 < x

   1. Initial program 99.7%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 95.8%

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

      1. expm1-def 95.8%

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

      2. expm1-log1p 95.8%

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

      3. +-commutative 95.8%

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

   7. Simplified 95.8%

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

      1. div-inv 95.8%

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

      2. exp-prod 96.9%

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

      3. add-sqr-sqrt 96.9%

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

      4. sqrt-unprod 96.9%

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

      5. sqr-neg 96.9%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 96.9%

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

      8. exp-prod 99.1%

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

      9. div-inv 99.1%

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

      10. distribute-frac-neg 99.1%

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

      11. exp-neg 99.1%

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

      12. div-inv 99.1%

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

      13. exp-prod 96.9%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in s around 0 99.8%

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

   11. Taylor expanded in x around 0 97.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.999999821029131 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{1}{e^{\frac{x}{s}}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 6: 95.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.0000000031710769 \cdot 10^{-30}:\\ \;\;\;\;\frac{t_0}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x -1.0000000031710769e-30)
     (/ t_0 (* s 4.0))
     (/ 1.0 (* s (+ t_0 3.0))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= -1.0000000031710769e-30f) {
		tmp = t_0 / (s * 4.0f);
	} else {
		tmp = 1.0f / (s * (t_0 + 3.0f));
	}
	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 = exp((x / s))
    if (x <= (-1.0000000031710769e-30)) then
        tmp = t_0 / (s * 4.0e0)
    else
        tmp = 1.0e0 / (s * (t_0 + 3.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000031710769e-30))
		tmp = Float32(t_0 / Float32(s * Float32(4.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(t_0 + Float32(3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(-1.0000000031710769e-30))
		tmp = t_0 / (s * single(4.0));
	else
		tmp = single(1.0) / (s * (t_0 + single(3.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1e-30

   1. Initial program 99.1%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. expm1-def 97.9%

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

      2. expm1-log1p 99.1%

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

      3. +-commutative 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 90.9%

      \[\leadsto \frac{e^{\frac{x}{s}}}{s \cdot \color{blue}{4}} \]

   if -1e-30 < 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 99.1%

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

   5. Applied egg-rr 95.9%

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

      1. expm1-def 95.9%

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

      2. expm1-log1p 95.9%

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

      3. +-commutative 95.9%

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

   7. Simplified 95.9%

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

      1. div-inv 95.9%

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

      2. exp-prod 96.5%

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

      3. add-sqr-sqrt 96.5%

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

      4. sqrt-unprod 96.5%

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

      5. sqr-neg 96.5%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 96.5%

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

      8. exp-prod 98.5%

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

      9. div-inv 98.5%

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

      10. distribute-frac-neg 98.5%

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

      11. exp-neg 98.5%

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

      12. div-inv 98.5%

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

      13. exp-prod 96.5%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in s around 0 99.8%

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

   11. Taylor expanded in x around 0 97.2%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000031710769 \cdot 10^{-30}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 7: 80.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000031710769 \cdot 10^{-30}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000031710769e-30)
   (/ (exp (/ x s)) (* s 4.0))
   (/ 1.0 (+ (* s 4.0) (* x (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000031710769e-30f) {
		tmp = expf((x / s)) / (s * 4.0f);
	} else {
		tmp = 1.0f / ((s * 4.0f) + (x * (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 <= (-1.0000000031710769e-30)) then
        tmp = exp((x / s)) / (s * 4.0e0)
    else
        tmp = 1.0e0 / ((s * 4.0e0) + (x * (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000031710769e-30))
		tmp = Float32(exp(Float32(x / s)) / Float32(s * Float32(4.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000031710769e-30))
		tmp = exp((x / s)) / (s * single(4.0));
	else
		tmp = single(1.0) / ((s * single(4.0)) + (x * (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1e-30

   1. Initial program 99.1%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. expm1-def 97.9%

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

      2. expm1-log1p 99.1%

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

      3. +-commutative 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 90.9%

      \[\leadsto \frac{e^{\frac{x}{s}}}{s \cdot \color{blue}{4}} \]

   if -1e-30 < x

   1. Initial program 99.6%

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

      1. *-lft-identity 99.6%

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

      2. associate-*r/ 99.6%

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

      3. associate-/l* 99.7%

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

      4. distribute-frac-neg 99.7%

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

      5. exp-neg 99.6%

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

      6. associate-/r/ 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 29.6%

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

   6. Simplified 29.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around 0 72.1%

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

      1. fma-def 72.1%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(4, s, \frac{{x}^{2}}{s}\right)}} \]

      2. unpow2 72.1%

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

   9. Simplified 72.1%

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

       1. fma-udef 72.1%

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

       2. *-commutative 72.1%

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

       3. associate-/l* 72.5%

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

       4. div-inv 72.5%

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

       5. clear-num 72.5%

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

   11. Applied egg-rr 72.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000031710769 \cdot 10^{-30}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 8: 63.5% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026 \lor \neg \left(x \leq 4.000000053405728 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -0.0020000000949949026) (not (<= x 4.000000053405728e-10)))
   (/ 1.0 (* x (/ x s)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -0.0020000000949949026f) || !(x <= 4.000000053405728e-10f)) {
		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 <= (-0.0020000000949949026e0)) .or. (.not. (x <= 4.000000053405728e-10))) 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(-0.0020000000949949026)) || !(x <= Float32(4.000000053405728e-10)))
		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(-0.0020000000949949026)) || ~((x <= single(4.000000053405728e-10))))
		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 < -0.00200000009 or 4.00000005e-10 < x

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. exp-neg 99.9%

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

      6. associate-/r/ 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 2.6%

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

   6. Simplified 2.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 2.6%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{{x}^{2}}{s}} - \frac{x \cdot x}{s}} \]
   8. Step-by-step derivation

      1. associate-*r/ 2.6%

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

      2. unpow2 2.6%

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

      3. associate-*r* 2.6%

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

   9. Simplified 2.6%

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

   10. Taylor expanded in x around 0 79.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   11. Step-by-step derivation

       1. unpow2 79.3%

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

       2. associate-*r/ 79.3%

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

   12. Simplified 79.3%

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

   if -0.00200000009 < x < 4.00000005e-10

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf 53.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026 \lor \neg \left(x \leq 4.000000053405728 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 9: 63.5% accurate, 55.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.000000053405728 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0020000000949949026)
   (/ 1.0 (* x (/ x s)))
   (if (<= x 4.000000053405728e-10) (/ 0.25 s) (/ 1.0 (/ x (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0020000000949949026f) {
		tmp = 1.0f / (x * (x / s));
	} else if (x <= 4.000000053405728e-10f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x / (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.0020000000949949026e0)) then
        tmp = 1.0e0 / (x * (x / s))
    else if (x <= 4.000000053405728e-10) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x / (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0020000000949949026))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	elseif (x <= Float32(4.000000053405728e-10))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0020000000949949026))
		tmp = single(1.0) / (x * (x / s));
	elseif (x <= single(4.000000053405728e-10))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x / (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00200000009

   1. Initial program 99.7%

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

      1. *-lft-identity 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-/l* 99.7%

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

      4. distribute-frac-neg 99.7%

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

      5. exp-neg 99.7%

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

      6. associate-/r/ 99.7%

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

      7. /-rgt-identity 99.7%

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

      8. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 2.8%

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

   6. Simplified 2.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 2.8%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{{x}^{2}}{s}} - \frac{x \cdot x}{s}} \]
   8. Step-by-step derivation

      1. associate-*r/ 2.8%

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

      2. unpow2 2.8%

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

      3. associate-*r* 2.8%

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

   9. Simplified 2.8%

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

   10. Taylor expanded in x around 0 77.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   11. Step-by-step derivation

       1. unpow2 77.4%

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

       2. associate-*r/ 77.4%

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

   12. Simplified 77.4%

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

   if -0.00200000009 < x < 4.00000005e-10

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf 53.4%

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

   if 4.00000005e-10 < 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. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 2.4%

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

   6. Simplified 2.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 80.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow2 80.9%

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

      2. associate-/l* 80.9%

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

   9. Simplified 80.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.000000053405728 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \]

Alternative 10: 63.5% accurate, 55.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.000000053405728 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0020000000949949026)
   (/ 1.0 (* x (/ x s)))
   (if (<= x 4.000000053405728e-10) (/ 0.25 s) (/ 1.0 (/ (* x x) s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0020000000949949026f) {
		tmp = 1.0f / (x * (x / s));
	} else if (x <= 4.000000053405728e-10f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / ((x * 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 = 1.0e0 / (x * (x / s))
    else if (x <= 4.000000053405728e-10) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((x * x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0020000000949949026))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	elseif (x <= Float32(4.000000053405728e-10))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0020000000949949026))
		tmp = single(1.0) / (x * (x / s));
	elseif (x <= single(4.000000053405728e-10))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((x * x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00200000009

   1. Initial program 99.7%

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

      1. *-lft-identity 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-/l* 99.7%

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

      4. distribute-frac-neg 99.7%

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

      5. exp-neg 99.7%

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

      6. associate-/r/ 99.7%

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

      7. /-rgt-identity 99.7%

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

      8. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 2.8%

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

   6. Simplified 2.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 2.8%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{{x}^{2}}{s}} - \frac{x \cdot x}{s}} \]
   8. Step-by-step derivation

      1. associate-*r/ 2.8%

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

      2. unpow2 2.8%

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

      3. associate-*r* 2.8%

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

   9. Simplified 2.8%

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

   10. Taylor expanded in x around 0 77.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   11. Step-by-step derivation

       1. unpow2 77.4%

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

       2. associate-*r/ 77.4%

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

   12. Simplified 77.4%

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

   if -0.00200000009 < x < 4.00000005e-10

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf 53.4%

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

   if 4.00000005e-10 < 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. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 2.4%

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

   6. Simplified 2.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 80.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow2 80.9%

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

   9. Simplified 80.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.000000053405728 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \]

Alternative 11: 65.4% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. *-lft-identity 99.4%

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

   2. associate-*r/ 99.4%

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

   3. associate-/l* 99.4%

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

   4. distribute-frac-neg 99.4%

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

   5. exp-neg 99.3%

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

   6. associate-/r/ 99.4%

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

   7. /-rgt-identity 99.4%

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

   8. associate-*l* 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 26.0%

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

6. Simplified 26.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

7. Taylor expanded in x around 0 68.6%

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

   1. fma-def 68.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(4, s, \frac{{x}^{2}}{s}\right)}} \]

   2. unpow2 68.6%

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

9. Simplified 68.6%

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

    1. fma-udef 68.6%

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

    2. *-commutative 68.6%

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

    3. associate-/l* 69.0%

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

    4. div-inv 69.0%

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

    5. clear-num 69.0%

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

11. Applied egg-rr 69.0%

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

12. Final simplification 69.0%

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

Alternative 12: 62.3% accurate, 66.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026 \lor \neg \left(x \leq 4.000000053405728 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -0.0020000000949949026) (not (<= x 4.000000053405728e-10)))
   (/ s (* x x))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -0.0020000000949949026f) || !(x <= 4.000000053405728e-10f)) {
		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 <= (-0.0020000000949949026e0)) .or. (.not. (x <= 4.000000053405728e-10))) 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(-0.0020000000949949026)) || !(x <= Float32(4.000000053405728e-10)))
		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(-0.0020000000949949026)) || ~((x <= single(4.000000053405728e-10))))
		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 < -0.00200000009 or 4.00000005e-10 < x

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. exp-neg 99.9%

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

      6. associate-/r/ 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 2.6%

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

   6. Simplified 2.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 77.5%

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

      1. unpow2 77.5%

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

      2. associate-/r* 77.1%

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

   9. Simplified 77.1%

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

   10. Taylor expanded in s around 0 77.5%

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

       1. unpow2 77.5%

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

   12. Simplified 77.5%

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

   if -0.00200000009 < x < 4.00000005e-10

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf 53.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026 \lor \neg \left(x \leq 4.000000053405728 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13: 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 99.4%

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

2. Taylor expanded in s around inf 26.3%

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

3. Final simplification 26.3%

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

Reproduce

herbie shell --seed 2023188 
(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))))))