Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 12.1s

Alternatives: 16

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-*r/ 98.8%

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

   3. associate-/l* 98.8%

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

   4. distribute-frac-neg 98.8%

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

   5. exp-neg 98.8%

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

   6. associate-/r/ 98.8%

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

   7. /-rgt-identity 98.8%

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

   8. associate-*l* 98.8%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	return 1.0f / ((1.0f + (1.0f / t_0)) * (s + (s * 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 / ((1.0e0 + (1.0e0 / t_0)) * (s + (s * t_0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-*r/ 98.8%

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

   3. associate-/l* 98.8%

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

   4. distribute-frac-neg 98.8%

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

   5. exp-neg 98.8%

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

   6. associate-/r/ 98.8%

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

   7. /-rgt-identity 98.8%

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

   8. associate-*l* 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

5. Applied egg-rr 62.1%

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

   1. *-un-lft-identity 62.1%

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

   2. neg-mul-1 62.1%

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

   3. times-frac 62.1%

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

   4. metadata-eval 62.1%

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

   5. metadata-eval 62.1%

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

   6. times-frac 62.1%

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

   7. neg-mul-1 62.1%

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

   8. *-un-lft-identity 62.1%

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

   9. distribute-frac-neg 62.1%

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

   10. exp-neg 62.1%

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

   11. div-inv 62.1%

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

   12. exp-prod 59.4%

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

   13. add-sqr-sqrt 59.4%

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

   14. sqrt-unprod 59.4%

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

   15. sqr-neg 59.4%

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

   16. sqrt-unprod -0.0%

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

   17. add-sqr-sqrt 85.1%

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

   18. exp-prod 94.5%

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

   19. div-inv 94.5%

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

   20. neg-mul-1 94.5%

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

   21. *-un-lft-identity 94.5%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 3: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-*r/ 98.8%

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

   3. associate-/l* 98.8%

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

   4. distribute-frac-neg 98.8%

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

   5. exp-neg 98.8%

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

   6. associate-/r/ 98.8%

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

   7. /-rgt-identity 98.8%

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

   8. associate-*l* 98.8%

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

3. Simplified 98.9%

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

   1. fma-udef 98.9%

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

5. Applied egg-rr 62.1%

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

   1. *-un-lft-identity 62.1%

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

   2. neg-mul-1 62.1%

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

   3. times-frac 62.1%

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

   4. metadata-eval 62.1%

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

   5. metadata-eval 62.1%

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

   6. times-frac 62.1%

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

   7. neg-mul-1 62.1%

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

   8. *-un-lft-identity 62.1%

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

   9. distribute-frac-neg 62.1%

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

   10. exp-neg 62.1%

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

   11. div-inv 62.1%

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

   12. exp-prod 59.4%

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

   13. add-sqr-sqrt 59.4%

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

   14. sqrt-unprod 59.4%

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

   15. sqr-neg 59.4%

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

   16. sqrt-unprod -0.0%

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

   17. add-sqr-sqrt 85.1%

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

   18. exp-prod 94.5%

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

   19. div-inv 94.5%

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

   20. neg-mul-1 94.5%

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

   21. *-un-lft-identity 94.5%

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

7. Applied egg-rr 98.9%

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

   1. rec-exp 98.9%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

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

Alternative 4: 96.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-*r/ 98.8%

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

   3. associate-*l* 98.9%

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

   4. times-frac 98.4%

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

   5. associate-*r/ 98.5%

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

   6. associate-/l* 98.4%

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

   7. distribute-frac-neg 98.4%

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

   8. exp-neg 98.4%

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

3. Simplified 98.5%

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

4. Taylor expanded in s around inf 94.5%

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

5. Final simplification 94.5%

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

Alternative 5: 94.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. associate-*l* 98.9%

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

   2. +-commutative 98.9%

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

   3. +-commutative 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around inf 92.2%

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

5. Final simplification 92.2%

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

Alternative 6: 95.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{t_0}{4 + \frac{x}{s} \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s + s \cdot t_0\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x 4.0000000126843074e-29)
     (* (/ 1.0 s) (/ t_0 (+ 4.0 (* (/ x s) 4.0))))
     (/ 1.0 (* (+ s (* s t_0)) 2.0)))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= 4.0000000126843074e-29f) {
		tmp = (1.0f / s) * (t_0 / (4.0f + ((x / s) * 4.0f)));
	} else {
		tmp = 1.0f / ((s + (s * t_0)) * 2.0f);
	}
	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 <= 4.0000000126843074e-29) then
        tmp = (1.0e0 / s) * (t_0 / (4.0e0 + ((x / s) * 4.0e0)))
    else
        tmp = 1.0e0 / ((s + (s * t_0)) * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000126843074e-29))
		tmp = Float32(Float32(Float32(1.0) / s) * Float32(t_0 / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(4.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(s + Float32(s * t_0)) * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(4.0000000126843074e-29))
		tmp = (single(1.0) / s) * (t_0 / (single(4.0) + ((x / s) * single(4.0))));
	else
		tmp = single(1.0) / ((s + (s * t_0)) * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000001e-29

   1. Initial program 98.4%

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

      1. associate-*l* 98.5%

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

      2. +-commutative 98.5%

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

      3. +-commutative 98.5%

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

   3. Simplified 98.5%

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

      1. *-un-lft-identity 98.5%

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

      2. times-frac 98.4%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in x around 0 93.5%

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

   if 4.00000001e-29 < x

   1. Initial program 99.4%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 99.4%

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

      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. Step-by-step derivation

      1. fma-udef 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in s around inf 92.7%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{e^{\frac{x}{s}}}{4 + \frac{x}{s} \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s + s \cdot e^{\frac{x}{s}}\right) \cdot 2}\\ \end{array} \]

Alternative 7: 94.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{t_0}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + t_0 \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x -9.999999796611898e-32)
     (* (/ t_0 s) 0.25)
     (/ (/ 1.0 s) (+ 2.0 (* t_0 2.0))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= -9.999999796611898e-32f) {
		tmp = (t_0 / s) * 0.25f;
	} else {
		tmp = (1.0f / s) / (2.0f + (t_0 * 2.0f));
	}
	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 <= (-9.999999796611898e-32)) then
        tmp = (t_0 / s) * 0.25e0
    else
        tmp = (1.0e0 / s) / (2.0e0 + (t_0 * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999796611898e-32))
		tmp = Float32(Float32(t_0 / s) * Float32(0.25));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(2.0) + Float32(t_0 * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(-9.999999796611898e-32))
		tmp = (t_0 / s) * single(0.25);
	else
		tmp = (single(1.0) / s) / (single(2.0) + (t_0 * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.9999998e-32

   1. Initial program 98.2%

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

      1. associate-*l* 98.2%

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

      2. +-commutative 98.2%

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

      3. +-commutative 98.2%

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

   3. Simplified 98.2%

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

      1. add-sqr-sqrt 98.1%

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

   5. Applied egg-rr 98.0%

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

      1. unpow2 98.0%

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

   7. Simplified 98.0%

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

      1. div-inv 98.0%

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

      2. unpow-prod-down 98.0%

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

      3. pow2 98.0%

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

      4. add-sqr-sqrt 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in x around 0 92.5%

       \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \color{blue}{0.25} \]

   if -9.9999998e-32 < x

   1. Initial program 99.4%

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

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

      4. times-frac 98.7%

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

      5. associate-*r/ 98.7%

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

      6. associate-/l* 98.7%

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

      7. distribute-frac-neg 98.7%

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

      8. exp-neg 98.7%

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

   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. Step-by-step derivation

      1. expm1-log1p-u 96.3%

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

      2. expm1-udef 95.1%

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

   5. Applied egg-rr 90.6%

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

      1. expm1-def 90.9%

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

      2. expm1-log1p 92.4%

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

      3. associate-/r* 92.4%

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

      4. *-lft-identity 92.4%

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

      5. associate-*l/ 91.6%

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

      6. associate-*r/ 91.6%

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

      7. *-rgt-identity 91.6%

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

      8. +-commutative 91.6%

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

      9. +-commutative 91.6%

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

      10. associate-+l+ 91.6%

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

      11. count-2 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + e^{\frac{x}{s}} \cdot 2}\\ \end{array} \]

Alternative 8: 94.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{t_0}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s + s \cdot t_0\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x -9.999999796611898e-32)
     (* (/ t_0 s) 0.25)
     (/ 1.0 (* (+ s (* s t_0)) 2.0)))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= -9.999999796611898e-32f) {
		tmp = (t_0 / s) * 0.25f;
	} else {
		tmp = 1.0f / ((s + (s * t_0)) * 2.0f);
	}
	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 <= (-9.999999796611898e-32)) then
        tmp = (t_0 / s) * 0.25e0
    else
        tmp = 1.0e0 / ((s + (s * t_0)) * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999796611898e-32))
		tmp = Float32(Float32(t_0 / s) * Float32(0.25));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(s + Float32(s * t_0)) * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(-9.999999796611898e-32))
		tmp = (t_0 / s) * single(0.25);
	else
		tmp = single(1.0) / ((s + (s * t_0)) * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.9999998e-32

   1. Initial program 98.2%

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

      1. associate-*l* 98.2%

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

      2. +-commutative 98.2%

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

      3. +-commutative 98.2%

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

   3. Simplified 98.2%

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

      1. add-sqr-sqrt 98.1%

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

   5. Applied egg-rr 98.0%

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

      1. unpow2 98.0%

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

   7. Simplified 98.0%

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

      1. div-inv 98.0%

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

      2. unpow-prod-down 98.0%

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

      3. pow2 98.0%

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

      4. add-sqr-sqrt 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in x around 0 92.5%

       \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \color{blue}{0.25} \]

   if -9.9999998e-32 < x

   1. Initial program 99.4%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 99.4%

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

      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. Step-by-step derivation

      1. fma-udef 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in s around inf 92.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s + s \cdot e^{\frac{x}{s}}\right) \cdot 2}\\ \end{array} \]

Alternative 9: 80.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999796611898e-32)
   (* (/ (exp (/ x s)) s) 0.25)
   (/ 1.0 (fma x (/ x s) (* s 4.0)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999796611898e-32f) {
		tmp = (expf((x / s)) / s) * 0.25f;
	} else {
		tmp = 1.0f / fmaf(x, (x / s), (s * 4.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999796611898e-32))
		tmp = Float32(Float32(exp(Float32(x / s)) / s) * Float32(0.25));
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(x / s), Float32(s * Float32(4.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -9.9999998e-32

   1. Initial program 98.2%

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

      1. associate-*l* 98.2%

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

      2. +-commutative 98.2%

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

      3. +-commutative 98.2%

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

   3. Simplified 98.2%

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

      1. add-sqr-sqrt 98.1%

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

   5. Applied egg-rr 98.0%

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

      1. unpow2 98.0%

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

   7. Simplified 98.0%

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

      1. div-inv 98.0%

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

      2. unpow-prod-down 98.0%

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

      3. pow2 98.0%

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

      4. add-sqr-sqrt 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in x around 0 92.5%

       \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \color{blue}{0.25} \]

   if -9.9999998e-32 < x

   1. Initial program 99.4%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 99.4%

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

      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 34.9%

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

      1. associate-+r+ 34.9%

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

      2. distribute-rgt-out 34.9%

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

      3. metadata-eval 34.9%

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

      4. associate-+r+ 34.9%

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

      5. mul-1-neg 34.9%

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

      6. unsub-neg 34.9%

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

      7. fma-def 34.9%

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

      8. unpow2 34.9%

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

      9. sqr-abs 34.9%

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

      10. *-commutative 34.9%

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

      11. unpow2 34.9%

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

      12. sqr-abs 34.9%

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

   6. Simplified 34.9%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 34.9%

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

      2. associate-/l* 35.0%

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

   8. Applied egg-rr 35.0%

      \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\color{blue}{\left(2 \cdot \frac{x}{\frac{s}{x}} + s \cdot 4\right)} - \frac{x \cdot x}{s}\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 35.0%

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

      2. mul0-rgt 35.0%

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

      3. +-commutative 35.0%

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

      4. fma-def 35.0%

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

      5. div-inv 35.0%

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

      6. clear-num 35.0%

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

      7. associate-/l* 35.6%

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

      8. div-inv 35.6%

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

      9. clear-num 35.6%

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

   10. Applied egg-rr 35.6%

       \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(\mathsf{fma}\left(2, x \cdot \frac{x}{s}, s \cdot 4\right) - x \cdot \frac{x}{s}\right) + 0}} \]
   11. Step-by-step derivation

       1. *-lft-identity 35.6%

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

       2. +-rgt-identity 35.6%

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

       3. fma-udef 35.6%

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

       4. associate-*r/ 34.7%

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

       5. unpow2 34.7%

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

       6. *-commutative 34.7%

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

       7. +-commutative 34.7%

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

       8. associate-*r/ 34.9%

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

       9. unpow2 34.9%

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

       10. associate--l+ 34.9%

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

       11. unpow2 34.9%

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

       12. associate-*r/ 34.7%

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

       13. unpow2 34.7%

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

       14. associate-*r/ 35.6%

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

       15. *-lft-identity 35.6%

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

       16. distribute-rgt-out-- 72.9%

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

       17. metadata-eval 72.9%

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

   12. Simplified 72.9%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \]

Alternative 10: 78.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0015999999595806003:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0015999999595806003)
   (* (/ (exp (/ x s)) s) 0.25)
   (/ 1.0 (* x (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0015999999595806003f) {
		tmp = (expf((x / s)) / s) * 0.25f;
	} 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 <= 0.0015999999595806003e0) then
        tmp = (exp((x / s)) / s) * 0.25e0
    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(0.0015999999595806003))
		tmp = Float32(Float32(exp(Float32(x / s)) / s) * Float32(0.25));
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0015999999595806003))
		tmp = (exp((x / s)) / s) * single(0.25);
	else
		tmp = single(1.0) / (x * (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00159999996

   1. Initial program 98.4%

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

      1. associate-*l* 98.5%

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

      2. +-commutative 98.5%

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

      3. +-commutative 98.5%

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

   3. Simplified 98.5%

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

      1. add-sqr-sqrt 98.1%

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

   5. Applied egg-rr 88.9%

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

      1. unpow2 88.9%

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

   7. Simplified 88.9%

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

      1. div-inv 88.9%

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

      2. unpow-prod-down 88.5%

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

      3. pow2 88.5%

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

      4. add-sqr-sqrt 88.8%

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

   9. Applied egg-rr 88.8%

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

   10. Taylor expanded in x around 0 80.8%

       \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \color{blue}{0.25} \]

   if 0.00159999996 < 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 3.5%

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

      1. associate-+r+ 3.5%

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

      2. distribute-rgt-out 3.5%

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

      3. metadata-eval 3.5%

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

      4. associate-+r+ 3.5%

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

      5. mul-1-neg 3.5%

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

      6. unsub-neg 3.5%

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

      7. fma-def 3.5%

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

      8. unpow2 3.5%

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

      9. sqr-abs 3.5%

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

      10. *-commutative 3.5%

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

      11. unpow2 3.5%

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

      12. sqr-abs 3.5%

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

   6. Simplified 3.5%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 3.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)\right)} \]

      2. expm1-udef 22.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1} \]

      3. mul0-rgt 22.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{0} + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1 \]

      4. associate-+r- 22.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(0 + \mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}}\right)} - 1 \]

      5. associate-/l* 22.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \color{blue}{\frac{x}{\frac{s}{x}}}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}\right)} - 1 \]

      6. associate-/l* 22.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \color{blue}{\frac{x}{\frac{s}{x}}}}\right)} - 1 \]

   8. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 3.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)\right)} \]

      2. expm1-log1p 3.5%

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

      3. /-rgt-identity 3.5%

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

      4. /-rgt-identity 3.5%

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

      5. +-lft-identity 3.5%

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

      6. fma-def 3.5%

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

      7. +-commutative 3.5%

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

      8. fma-def 3.5%

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

      9. associate-/r/ 3.5%

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

      10. associate-/r/ 3.5%

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

   10. Simplified 3.5%

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

   11. Taylor expanded in s around 0 32.3%

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

       1. sub-neg 32.3%

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

       2. mul-1-neg 32.3%

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

       3. distribute-rgt-out 77.3%

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

       4. metadata-eval 77.3%

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

       5. *-rgt-identity 77.3%

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

       6. unpow2 77.3%

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

       7. associate-*r/ 77.3%

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

   13. Simplified 77.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0015999999595806003:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 11: 64.5% accurate, 26.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{s}\\ \mathbf{if}\;x \leq -5000000000 \lor \neg \left(x \leq 10\right):\\ \;\;\;\;\frac{1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot 4 + 2 \cdot t_0\right) - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* x (/ x s))))
   (if (or (<= x -5000000000.0) (not (<= x 10.0)))
     (/ 1.0 t_0)
     (/ 1.0 (- (+ (* s 4.0) (* 2.0 t_0)) t_0)))))

C

float code(float x, float s) {
	float t_0 = x * (x / s);
	float tmp;
	if ((x <= -5000000000.0f) || !(x <= 10.0f)) {
		tmp = 1.0f / t_0;
	} else {
		tmp = 1.0f / (((s * 4.0f) + (2.0f * t_0)) - t_0);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x * (x / s)
    if ((x <= (-5000000000.0e0)) .or. (.not. (x <= 10.0e0))) then
        tmp = 1.0e0 / t_0
    else
        tmp = 1.0e0 / (((s * 4.0e0) + (2.0e0 * t_0)) - t_0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x * Float32(x / s))
	tmp = Float32(0.0)
	if ((x <= Float32(-5000000000.0)) || !(x <= Float32(10.0)))
		tmp = Float32(Float32(1.0) / t_0);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(s * Float32(4.0)) + Float32(Float32(2.0) * t_0)) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x * (x / s);
	tmp = single(0.0);
	if ((x <= single(-5000000000.0)) || ~((x <= single(10.0))))
		tmp = single(1.0) / t_0;
	else
		tmp = single(1.0) / (((s * single(4.0)) + (single(2.0) * t_0)) - t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5e9 or 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 2.1%

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

      1. associate-+r+ 2.1%

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

      2. distribute-rgt-out 2.1%

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

      3. metadata-eval 2.1%

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

      4. associate-+r+ 2.1%

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

      5. mul-1-neg 2.1%

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

      6. unsub-neg 2.1%

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

      7. fma-def 2.1%

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

      8. unpow2 2.1%

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

      9. sqr-abs 2.1%

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

      10. *-commutative 2.1%

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

      11. unpow2 2.1%

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

      12. sqr-abs 2.1%

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

   6. Simplified 2.1%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 2.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)\right)} \]

      2. expm1-udef 16.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1} \]

      3. mul0-rgt 16.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{0} + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1 \]

      4. associate-+r- 16.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(0 + \mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}}\right)} - 1 \]

      5. associate-/l* 16.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \color{blue}{\frac{x}{\frac{s}{x}}}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}\right)} - 1 \]

      6. associate-/l* 16.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \color{blue}{\frac{x}{\frac{s}{x}}}}\right)} - 1 \]

   8. Applied egg-rr 16.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 2.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)\right)} \]

      2. expm1-log1p 2.1%

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

      3. /-rgt-identity 2.1%

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

      4. /-rgt-identity 2.1%

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

      5. +-lft-identity 2.1%

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

      6. fma-def 2.1%

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

      7. +-commutative 2.1%

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

      8. fma-def 2.1%

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

      9. associate-/r/ 2.1%

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

      10. associate-/r/ 2.1%

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

   10. Simplified 2.1%

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

   11. Taylor expanded in s around 0 31.7%

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

       1. sub-neg 31.7%

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

       2. mul-1-neg 31.7%

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

       3. distribute-rgt-out 85.6%

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

       4. metadata-eval 85.6%

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

       5. *-rgt-identity 85.6%

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

       6. unpow2 85.6%

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

       7. associate-*r/ 85.6%

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

   13. Simplified 85.6%

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

   if -5e9 < x < 10

   1. Initial program 97.9%

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

      1. *-lft-identity 97.9%

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

      2. associate-*r/ 97.9%

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

      3. associate-/l* 97.8%

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

      4. distribute-frac-neg 97.8%

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

      5. exp-neg 97.8%

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

      6. associate-/r/ 97.8%

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

      7. /-rgt-identity 97.8%

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

      8. associate-*l* 97.8%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in s around inf 50.4%

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

      1. associate-+r+ 50.4%

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

      2. distribute-rgt-out 50.4%

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

      3. metadata-eval 50.4%

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

      4. associate-+r+ 50.4%

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

      5. mul-1-neg 50.4%

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

      6. unsub-neg 50.4%

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

      7. fma-def 50.4%

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

      8. unpow2 50.4%

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

      9. sqr-abs 50.4%

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

      10. *-commutative 50.4%

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

      11. unpow2 50.4%

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

      12. sqr-abs 50.4%

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

   6. Simplified 50.4%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 47.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)\right)} \]

      2. expm1-udef 64.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1} \]

      3. mul0-rgt 64.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{0} + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1 \]

      4. associate-+r- 64.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(0 + \mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}}\right)} - 1 \]

      5. associate-/l* 64.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \color{blue}{\frac{x}{\frac{s}{x}}}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}\right)} - 1 \]

      6. associate-/l* 64.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \color{blue}{\frac{x}{\frac{s}{x}}}}\right)} - 1 \]

   8. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 48.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)\right)} \]

      2. expm1-log1p 51.4%

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

      3. /-rgt-identity 51.4%

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

      4. /-rgt-identity 51.4%

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

      5. +-lft-identity 51.4%

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

      6. fma-def 51.4%

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

      7. +-commutative 51.4%

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

      8. fma-def 51.4%

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

      9. associate-/r/ 51.4%

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

      10. associate-/r/ 51.4%

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

   10. Simplified 51.4%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, 4, 2 \cdot \left(\frac{x}{s} \cdot x\right)\right) - \frac{x}{s} \cdot x}} \]
   11. Step-by-step derivation

       1. fma-udef 51.4%

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

       2. *-commutative 51.4%

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

   12. Applied egg-rr 51.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5000000000 \lor \neg \left(x \leq 10\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot 4 + 2 \cdot \left(x \cdot \frac{x}{s}\right)\right) - x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 12: 65.6% accurate, 29.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50000:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 1999999991808:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{2 \cdot \left(x \cdot x\right) - x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -50000.0)
   (/ 1.0 (* x (/ x s)))
   (if (<= x 1999999991808.0)
     (/ 1.0 (+ (* s 4.0) (/ (- (* 2.0 (* x x)) (* x x)) s)))
     (/ s (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -50000.0f) {
		tmp = 1.0f / (x * (x / s));
	} else if (x <= 1999999991808.0f) {
		tmp = 1.0f / ((s * 4.0f) + (((2.0f * (x * x)) - (x * x)) / s));
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-50000.0e0)) then
        tmp = 1.0e0 / (x * (x / s))
    else if (x <= 1999999991808.0e0) then
        tmp = 1.0e0 / ((s * 4.0e0) + (((2.0e0 * (x * x)) - (x * x)) / s))
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-50000.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	elseif (x <= Float32(1999999991808.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(Float32(Float32(Float32(2.0) * Float32(x * x)) - Float32(x * x)) / s)));
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-50000.0))
		tmp = single(1.0) / (x * (x / s));
	elseif (x <= single(1999999991808.0))
		tmp = single(1.0) / ((s * single(4.0)) + (((single(2.0) * (x * x)) - (x * x)) / s));
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5e4

   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 2.6%

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

      1. associate-+r+ 2.6%

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

      2. distribute-rgt-out 2.6%

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

      3. metadata-eval 2.6%

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

      4. associate-+r+ 2.6%

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

      5. mul-1-neg 2.6%

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

      6. unsub-neg 2.6%

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

      7. fma-def 2.6%

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

      8. unpow2 2.6%

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

      9. sqr-abs 2.6%

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

      10. *-commutative 2.6%

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

      11. unpow2 2.6%

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

      12. sqr-abs 2.6%

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

   6. Simplified 2.6%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)\right)} \]

      2. expm1-udef 21.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1} \]

      3. mul0-rgt 21.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{0} + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1 \]

      4. associate-+r- 21.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(0 + \mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}}\right)} - 1 \]

      5. associate-/l* 21.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \color{blue}{\frac{x}{\frac{s}{x}}}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}\right)} - 1 \]

      6. associate-/l* 21.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \color{blue}{\frac{x}{\frac{s}{x}}}}\right)} - 1 \]

   8. Applied egg-rr 21.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)\right)} \]

      2. expm1-log1p 2.6%

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

      3. /-rgt-identity 2.6%

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

      4. /-rgt-identity 2.6%

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

      5. +-lft-identity 2.6%

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

      6. fma-def 2.6%

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

      7. +-commutative 2.6%

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

      8. fma-def 2.6%

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

      9. associate-/r/ 2.6%

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

      10. associate-/r/ 2.6%

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

   10. Simplified 2.6%

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

   11. Taylor expanded in s around 0 27.2%

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

       1. sub-neg 27.2%

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

       2. mul-1-neg 27.2%

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

       3. distribute-rgt-out 81.5%

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

       4. metadata-eval 81.5%

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

       5. *-rgt-identity 81.5%

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

       6. unpow2 81.5%

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

       7. associate-*r/ 81.5%

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

   13. Simplified 81.5%

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

   if -5e4 < x < 1999999990000

   1. Initial program 98.0%

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

      1. *-lft-identity 98.0%

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

      2. associate-*r/ 98.0%

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

      3. associate-/l* 98.0%

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

      4. distribute-frac-neg 98.0%

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

      5. exp-neg 98.0%

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

      6. associate-/r/ 98.0%

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

      7. /-rgt-identity 98.0%

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

      8. associate-*l* 98.0%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in s around inf 46.9%

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

      1. associate-+r+ 46.9%

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

      2. distribute-rgt-out 46.9%

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

      3. metadata-eval 46.9%

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

      4. associate-+r+ 46.9%

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

      5. mul-1-neg 46.9%

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

      6. unsub-neg 46.9%

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

      7. fma-def 46.9%

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

      8. unpow2 46.9%

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

      9. sqr-abs 46.9%

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

      10. *-commutative 46.9%

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

      11. unpow2 46.9%

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

      12. sqr-abs 46.9%

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

   6. Simplified 46.9%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 46.9%

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

      2. associate-/l* 47.0%

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

   8. Applied egg-rr 47.0%

      \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\color{blue}{\left(2 \cdot \frac{x}{\frac{s}{x}} + s \cdot 4\right)} - \frac{x \cdot x}{s}\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 47.0%

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

      2. mul0-rgt 47.0%

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

      3. +-commutative 47.0%

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

      4. fma-def 47.0%

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

      5. div-inv 47.0%

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

      6. clear-num 47.0%

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

      7. associate-/l* 47.8%

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

      8. div-inv 47.8%

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

      9. clear-num 47.8%

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

   10. Applied egg-rr 47.8%

       \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(\mathsf{fma}\left(2, x \cdot \frac{x}{s}, s \cdot 4\right) - x \cdot \frac{x}{s}\right) + 0}} \]
   11. Step-by-step derivation

       1. *-lft-identity 47.8%

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

       2. +-rgt-identity 47.8%

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

       3. fma-udef 47.8%

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

       4. associate-*r/ 46.7%

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

       5. unpow2 46.7%

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

       6. *-commutative 46.7%

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

       7. +-commutative 46.7%

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

       8. associate-+r- 46.7%

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

       9. *-commutative 46.7%

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

       10. associate-*r/ 46.7%

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

       11. associate-*r/ 46.9%

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

       12. unpow2 46.9%

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

       13. div-sub 51.5%

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

       14. unpow2 51.5%

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

       15. unpow2 51.5%

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

   12. Simplified 51.5%

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

   if 1999999990000 < 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 0.6%

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

      1. associate-+r+ 0.6%

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

      2. distribute-rgt-out 0.6%

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

      3. metadata-eval 0.6%

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

      4. associate-+r+ 0.6%

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

      5. mul-1-neg 0.6%

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

      6. unsub-neg 0.6%

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

      7. fma-def 0.6%

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

      8. unpow2 0.6%

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

      9. sqr-abs 0.6%

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

      10. *-commutative 0.6%

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

      11. unpow2 0.6%

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

      12. sqr-abs 0.6%

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

   6. Simplified 0.6%

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

   7. Taylor expanded in x around inf 96.2%

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

      1. unpow2 96.2%

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

   9. Simplified 96.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50000:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 1999999991808:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{2 \cdot \left(x \cdot x\right) - x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 13: 64.5% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10} \lor \neg \left(x \leq 0.0015999999595806003\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -9.999999717180685e-10) (not (<= x 0.0015999999595806003)))
   (/ 1.0 (* x (/ x s)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -9.999999717180685e-10f) || !(x <= 0.0015999999595806003f)) {
		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 <= (-9.999999717180685e-10)) .or. (.not. (x <= 0.0015999999595806003e0))) 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(-9.999999717180685e-10)) || !(x <= Float32(0.0015999999595806003)))
		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(-9.999999717180685e-10)) || ~((x <= single(0.0015999999595806003))))
		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 < -9.99999972e-10 or 0.00159999996 < x

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-*r/ 99.8%

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

      3. associate-/l* 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. exp-neg 99.8%

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

      6. associate-/r/ 99.8%

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

      7. /-rgt-identity 99.8%

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

      8. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 3.9%

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

      1. associate-+r+ 3.9%

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

      2. distribute-rgt-out 3.9%

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

      3. metadata-eval 3.9%

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

      4. associate-+r+ 3.9%

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

      5. mul-1-neg 3.9%

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

      6. unsub-neg 3.9%

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

      7. fma-def 3.9%

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

      8. unpow2 3.9%

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

      9. sqr-abs 3.9%

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

      10. *-commutative 3.9%

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

      11. unpow2 3.9%

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

      12. sqr-abs 3.9%

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

   6. Simplified 3.9%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 3.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)\right)} \]

      2. expm1-udef 27.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1} \]

      3. mul0-rgt 27.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{0} + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}\right)} - 1 \]

      4. associate-+r- 27.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(0 + \mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}}\right)} - 1 \]

      5. associate-/l* 27.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \color{blue}{\frac{x}{\frac{s}{x}}}, s \cdot 4\right)\right) - \frac{x \cdot x}{s}}\right)} - 1 \]

      6. associate-/l* 27.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \color{blue}{\frac{x}{\frac{s}{x}}}}\right)} - 1 \]

   8. Applied egg-rr 27.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 3.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(0 + \mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)\right) - \frac{x}{\frac{s}{x}}}\right)\right)} \]

      2. expm1-log1p 3.9%

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

      3. /-rgt-identity 3.9%

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

      4. /-rgt-identity 3.9%

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

      5. +-lft-identity 3.9%

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

      6. fma-def 3.9%

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

      7. +-commutative 3.9%

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

      8. fma-def 3.9%

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

      9. associate-/r/ 3.9%

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

      10. associate-/r/ 3.9%

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

   10. Simplified 3.9%

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

   11. Taylor expanded in s around 0 27.4%

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

       1. sub-neg 27.4%

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

       2. mul-1-neg 27.4%

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

       3. distribute-rgt-out 70.4%

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

       4. metadata-eval 70.4%

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

       5. *-rgt-identity 70.4%

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

       6. unpow2 70.4%

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

       7. associate-*r/ 70.4%

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

   13. Simplified 70.4%

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

   if -9.99999972e-10 < x < 0.00159999996

   1. Initial program 97.6%

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

      1. associate-*l* 97.6%

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

      2. +-commutative 97.6%

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

      3. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in s around inf 58.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10} \lor \neg \left(x \leq 0.0015999999595806003\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 14: 63.2% accurate, 55.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0015999999595806003:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999717180685e-10)
   (/ s (* x x))
   (if (<= x 0.0015999999595806003) (/ 0.25 s) (* s (/ 1.0 (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999717180685e-10f) {
		tmp = s / (x * x);
	} else if (x <= 0.0015999999595806003f) {
		tmp = 0.25f / s;
	} else {
		tmp = s * (1.0f / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999717180685e-10)) then
        tmp = s / (x * x)
    else if (x <= 0.0015999999595806003e0) then
        tmp = 0.25e0 / s
    else
        tmp = s * (1.0e0 / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999717180685e-10))
		tmp = Float32(s / Float32(x * x));
	elseif (x <= Float32(0.0015999999595806003))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s * Float32(Float32(1.0) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999717180685e-10))
		tmp = s / (x * x);
	elseif (x <= single(0.0015999999595806003))
		tmp = single(0.25) / s;
	else
		tmp = s * (single(1.0) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9.99999972e-10

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-*r/ 99.8%

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

      3. associate-/l* 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. exp-neg 99.8%

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

      6. associate-/r/ 99.8%

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

      7. /-rgt-identity 99.8%

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

      8. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 4.2%

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

      1. associate-+r+ 4.2%

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

      2. distribute-rgt-out 4.2%

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

      3. metadata-eval 4.2%

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

      4. associate-+r+ 4.2%

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

      5. mul-1-neg 4.2%

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

      6. unsub-neg 4.2%

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

      7. fma-def 4.2%

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

      8. unpow2 4.2%

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

      9. sqr-abs 4.2%

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

      10. *-commutative 4.2%

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

      11. unpow2 4.2%

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

      12. sqr-abs 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in x around inf 62.7%

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

      1. unpow2 62.7%

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

   9. Simplified 62.7%

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

   if -9.99999972e-10 < x < 0.00159999996

   1. Initial program 97.6%

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

      1. associate-*l* 97.6%

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

      2. +-commutative 97.6%

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

      3. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in s around inf 58.5%

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

   if 0.00159999996 < 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 3.5%

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

      1. associate-+r+ 3.5%

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

      2. distribute-rgt-out 3.5%

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

      3. metadata-eval 3.5%

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

      4. associate-+r+ 3.5%

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

      5. mul-1-neg 3.5%

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

      6. unsub-neg 3.5%

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

      7. fma-def 3.5%

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

      8. unpow2 3.5%

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

      9. sqr-abs 3.5%

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

      10. *-commutative 3.5%

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

      11. unpow2 3.5%

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

      12. sqr-abs 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in x around inf 75.8%

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

      1. unpow2 75.8%

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

   9. Simplified 75.8%

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

       1. div-inv 75.8%

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

   11. Applied egg-rr 75.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0015999999595806003:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \]

Alternative 15: 63.2% accurate, 66.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10} \lor \neg \left(x \leq 0.0015999999595806003\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -9.999999717180685e-10) (not (<= x 0.0015999999595806003)))
   (/ s (* x x))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -9.999999717180685e-10f) || !(x <= 0.0015999999595806003f)) {
		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 <= (-9.999999717180685e-10)) .or. (.not. (x <= 0.0015999999595806003e0))) 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(-9.999999717180685e-10)) || !(x <= Float32(0.0015999999595806003)))
		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(-9.999999717180685e-10)) || ~((x <= single(0.0015999999595806003))))
		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 < -9.99999972e-10 or 0.00159999996 < x

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-*r/ 99.8%

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

      3. associate-/l* 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. exp-neg 99.8%

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

      6. associate-/r/ 99.8%

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

      7. /-rgt-identity 99.8%

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

      8. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 3.9%

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

      1. associate-+r+ 3.9%

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

      2. distribute-rgt-out 3.9%

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

      3. metadata-eval 3.9%

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

      4. associate-+r+ 3.9%

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

      5. mul-1-neg 3.9%

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

      6. unsub-neg 3.9%

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

      7. fma-def 3.9%

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

      8. unpow2 3.9%

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

      9. sqr-abs 3.9%

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

      10. *-commutative 3.9%

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

      11. unpow2 3.9%

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

      12. sqr-abs 3.9%

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

   6. Simplified 3.9%

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

   7. Taylor expanded in x around inf 69.0%

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

      1. unpow2 69.0%

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

   9. Simplified 69.0%

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

   if -9.99999972e-10 < x < 0.00159999996

   1. Initial program 97.6%

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

      1. associate-*l* 97.6%

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

      2. +-commutative 97.6%

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

      3. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in s around inf 58.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10} \lor \neg \left(x \leq 0.0015999999595806003\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 16: 27.2% 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 98.8%

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

   1. associate-*l* 98.9%

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

   2. +-commutative 98.9%

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

   3. +-commutative 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around inf 28.3%

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

5. Final simplification 28.3%

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

Reproduce

herbie shell --seed 2023174 
(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))))))