Logistic distribution

Percentage Accurate: 99.5% → 99.4%

Time: 21.9s

Alternatives: 18

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{\left|x\right|}{s}}\\ \frac{e^{-\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}{\mathsf{fma}\left(s, {\left(e^{t_0}\right)}^{t_0}, s\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (sqrt (/ (fabs x) s))))
   (/
    (exp (- (log1p (exp (/ (fabs x) (- s))))))
    (fma s (pow (exp t_0) t_0) s))))

C

float code(float x, float s) {
	float t_0 = sqrtf((fabsf(x) / s));
	return expf(-log1pf(expf((fabsf(x) / -s)))) / fmaf(s, powf(expf(t_0), t_0), s);
}

Julia

function code(x, s)
	t_0 = sqrt(Float32(abs(x) / s))
	return Float32(exp(Float32(-log1p(exp(Float32(abs(x) / Float32(-s)))))) / fma(s, (exp(t_0) ^ t_0), s))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. exp-prod 99.8%

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

5. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-udef 99.8%

      \[\leadsto \frac{e^{-\color{blue}{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}}{\mathsf{fma}\left(s, {\left(e^{\sqrt{\frac{\left|x\right|}{s}}}\right)}^{\left(\sqrt{\frac{\left|x\right|}{s}}\right)}, s\right)} \]

7. Applied egg-rr 99.8%

   \[\leadsto \frac{\color{blue}{e^{-\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}}{\mathsf{fma}\left(s, {\left(e^{\sqrt{\frac{\left|x\right|}{s}}}\right)}^{\left(\sqrt{\frac{\left|x\right|}{s}}\right)}, s\right)} \]

8. Final simplification 99.8%

   \[\leadsto \frac{e^{-\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)}}{\mathsf{fma}\left(s, {\left(e^{\sqrt{\frac{\left|x\right|}{s}}}\right)}^{\left(\sqrt{\frac{\left|x\right|}{s}}\right)}, s\right)} \]

Alternative 2: 53.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. exp-prod 99.8%

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

5. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 99.6%

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

7. Applied egg-rr 99.6%

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

   1. expm1-def 99.7%

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

   2. expm1-log1p 99.8%

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

   3. unpow1 99.8%

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

   4. sqr-pow 56.9%

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

   5. fabs-sqr 56.9%

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

   6. sqr-pow 56.9%

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

   7. unpow1 56.9%

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

9. Simplified 56.9%

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

10. Final simplification 56.9%

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

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. exp-prod 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-udef 99.7%

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

   3. *-commutative 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 52.6%

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

   6. fabs-sqr 52.6%

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

   7. sqr-pow 63.1%

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

   8. unpow1 63.1%

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

   9. neg-mul-1 63.1%

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

   10. unpow1 63.1%

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

   11. sqr-pow 52.6%

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

   12. fabs-sqr 52.6%

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

   13. sqr-pow 99.7%

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

   14. unpow1 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (s * ((1.0f + 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 * ((1.0e0 + exp((-x / s))) * (1.0e0 + exp((x / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. exp-prod 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-udef 99.7%

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

   3. *-commutative 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 52.6%

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

   6. fabs-sqr 52.6%

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

   7. sqr-pow 63.1%

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

   8. unpow1 63.1%

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

   9. neg-mul-1 63.1%

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

   10. unpow1 63.1%

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

   11. sqr-pow 52.6%

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

   12. fabs-sqr 52.6%

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

   13. sqr-pow 99.7%

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

   14. unpow1 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in s around 0 99.7%

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

    1. distribute-frac-neg 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. exp-prod 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-udef 99.7%

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

   3. *-commutative 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 52.6%

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

   6. fabs-sqr 52.6%

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

   7. sqr-pow 63.1%

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

   8. unpow1 63.1%

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

   9. neg-mul-1 63.1%

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

   10. unpow1 63.1%

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

   11. sqr-pow 52.6%

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

   12. fabs-sqr 52.6%

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

   13. sqr-pow 99.7%

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

   14. unpow1 99.7%

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

8. Simplified 99.7%

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

   1. fma-udef 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

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

Alternative 6: 95.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.7%

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

5. Taylor expanded in s around inf 94.7%

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

6. Final simplification 94.7%

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

Alternative 7: 59.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. exp-prod 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-udef 99.7%

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

   3. *-commutative 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 52.6%

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

   6. fabs-sqr 52.6%

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

   7. sqr-pow 63.1%

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

   8. unpow1 63.1%

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

   9. neg-mul-1 63.1%

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

   10. unpow1 63.1%

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

   11. sqr-pow 52.6%

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

   12. fabs-sqr 52.6%

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

   13. sqr-pow 99.7%

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

   14. unpow1 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in x around 0 60.1%

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

10. Final simplification 60.1%

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

Alternative 8: 94.6% 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 99.6%

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

2. Taylor expanded in s around inf 94.3%

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

   1. *-commutative 94.3%

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

4. Simplified 94.3%

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

5. Final simplification 94.3%

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

Alternative 9: 85.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{s}\\ \mathbf{if}\;x \leq 0.012000000104308128:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + s \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* x (/ x s))))
   (if (<= x 0.012000000104308128)
     (/
      1.0
      (*
       (+ 1.0 (+ 1.0 (- (* 0.5 (* (/ x s) (/ x s))) (/ x s))))
       (+ s (* s (exp (/ x s))))))
     (/ 1.0 (+ (cbrt (* t_0 (* t_0 t_0))) (* s 4.0))))))

C

float code(float x, float s) {
	float t_0 = x * (x / s);
	float tmp;
	if (x <= 0.012000000104308128f) {
		tmp = 1.0f / ((1.0f + (1.0f + ((0.5f * ((x / s) * (x / s))) - (x / s)))) * (s + (s * expf((x / s)))));
	} else {
		tmp = 1.0f / (cbrtf((t_0 * (t_0 * t_0))) + (s * 4.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(x * Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(0.012000000104308128))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(0.5) * Float32(Float32(x / s) * Float32(x / s))) - Float32(x / s)))) * Float32(s + Float32(s * exp(Float32(x / s))))));
	else
		tmp = Float32(Float32(1.0) / Float32(cbrt(Float32(t_0 * Float32(t_0 * t_0))) + Float32(s * Float32(4.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0120000001

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in s around inf 70.7%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{\left|x\right|}{s} + 0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 70.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + -1 \cdot \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      2. neg-mul-1 70.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{\left(-\frac{\left|x\right|}{s}\right)}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      3. unsub-neg 70.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      4. unpow2 70.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      5. unpow2 70.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\left|x\right| \cdot \left|x\right|}{\color{blue}{s \cdot s}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      6. times-frac 79.3%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \color{blue}{\left(\frac{\left|x\right|}{s} \cdot \frac{\left|x\right|}{s}\right)} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      7. unpow1 79.3%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{1}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      8. sqr-pow 33.1%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      9. fabs-sqr 33.1%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      10. sqr-pow 95.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{1}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      11. unpow1 95.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{x}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      12. unpow1 95.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{1}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      13. sqr-pow 33.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      14. fabs-sqr 33.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      15. sqr-pow 79.3%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{1}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      16. unpow1 79.3%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{x}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   7. Simplified 94.6%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 94.6%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)\right)}\right)} \]

      2. expm1-udef 78.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(e^{\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)} - 1\right)}\right)} \]

   9. Applied egg-rr 78.4%

      \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(e^{\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)} - 1\right)}\right)} \]
   10. Step-by-step derivation

       1. expm1-def 94.6%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot e^{\frac{\left|x\right|}{s}}\right)\right)}\right)} \]

       2. expm1-log1p 94.6%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{s \cdot e^{\frac{\left|x\right|}{s}}}\right)} \]

       3. unpow1 94.6%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|\color{blue}{{x}^{1}}\right|}{s}}\right)} \]

       4. sqr-pow 33.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}}\right)} \]

       5. fabs-sqr 33.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}}\right)} \]

       6. sqr-pow 81.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\color{blue}{{x}^{1}}}{s}}\right)} \]

       7. unpow1 81.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\color{blue}{x}}{s}}\right)} \]

   11. Simplified 81.9%

       \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{s \cdot e^{\frac{x}{s}}}\right)} \]

   if 0.0120000001 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in s around -inf 24.6%

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

      1. +-commutative 24.6%

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

      2. +-commutative 24.6%

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

      3. mul-1-neg 24.6%

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

      4. distribute-lft1-in 73.9%

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

      5. metadata-eval 73.9%

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

      6. associate-*r/ 73.9%

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

      7. mul-1-neg 73.9%

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

      8. remove-double-neg 73.9%

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

      9. +-commutative 73.9%

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

      10. associate-+l+ 73.9%

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

   7. Simplified 73.9%

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

      1. add-cbrt-cube 100.0%

         \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(\frac{x}{\frac{s}{x}} \cdot \frac{x}{\frac{s}{x}}\right) \cdot \frac{x}{\frac{s}{x}}}} + \left(s \cdot 4 + 0\right)} \]

      2. div-inv 100.0%

         \[\leadsto \frac{1}{\sqrt[3]{\left(\color{blue}{\left(x \cdot \frac{1}{\frac{s}{x}}\right)} \cdot \frac{x}{\frac{s}{x}}\right) \cdot \frac{x}{\frac{s}{x}}} + \left(s \cdot 4 + 0\right)} \]

      3. clear-num 100.0%

         \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \color{blue}{\frac{x}{s}}\right) \cdot \frac{x}{\frac{s}{x}}\right) \cdot \frac{x}{\frac{s}{x}}} + \left(s \cdot 4 + 0\right)} \]

      4. div-inv 100.0%

         \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{s}{x}}\right)}\right) \cdot \frac{x}{\frac{s}{x}}} + \left(s \cdot 4 + 0\right)} \]

      5. clear-num 100.0%

         \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \color{blue}{\frac{x}{s}}\right)\right) \cdot \frac{x}{\frac{s}{x}}} + \left(s \cdot 4 + 0\right)} \]

      6. div-inv 100.0%

         \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \frac{x}{s}\right)\right) \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{s}{x}}\right)}} + \left(s \cdot 4 + 0\right)} \]

      7. clear-num 100.0%

         \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \frac{x}{s}\right)\right) \cdot \left(x \cdot \color{blue}{\frac{x}{s}}\right)} + \left(s \cdot 4 + 0\right)} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \frac{x}{s}\right)\right) \cdot \left(x \cdot \frac{x}{s}\right)}} + \left(s \cdot 4 + 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.012000000104308128:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\left(x \cdot \frac{x}{s}\right) \cdot \left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \frac{x}{s}\right)\right)} + s \cdot 4}\\ \end{array} \]

Alternative 10: 79.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{s}\\ \frac{1}{\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + s \cdot 4} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* x (/ x s)))) (/ 1.0 (+ (cbrt (* t_0 (* t_0 t_0))) (* s 4.0)))))

C

float code(float x, float s) {
	float t_0 = x * (x / s);
	return 1.0f / (cbrtf((t_0 * (t_0 * t_0))) + (s * 4.0f));
}

Julia

function code(x, s)
	t_0 = Float32(x * Float32(x / s))
	return Float32(Float32(1.0) / Float32(cbrt(Float32(t_0 * Float32(t_0 * t_0))) + Float32(s * Float32(4.0))))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.7%

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

5. Taylor expanded in s around -inf 40.2%

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

   1. +-commutative 40.2%

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

   2. +-commutative 40.2%

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

   3. mul-1-neg 40.2%

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

   4. distribute-lft1-in 64.8%

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

   5. metadata-eval 64.8%

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

   6. associate-*r/ 64.8%

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

   7. mul-1-neg 64.8%

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

   8. remove-double-neg 64.8%

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

   9. +-commutative 64.8%

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

   10. associate-+l+ 64.8%

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

7. Simplified 65.6%

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

   1. add-cbrt-cube 80.3%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(\frac{x}{\frac{s}{x}} \cdot \frac{x}{\frac{s}{x}}\right) \cdot \frac{x}{\frac{s}{x}}}} + \left(s \cdot 4 + 0\right)} \]

   2. div-inv 80.3%

      \[\leadsto \frac{1}{\sqrt[3]{\left(\color{blue}{\left(x \cdot \frac{1}{\frac{s}{x}}\right)} \cdot \frac{x}{\frac{s}{x}}\right) \cdot \frac{x}{\frac{s}{x}}} + \left(s \cdot 4 + 0\right)} \]

   3. clear-num 80.3%

      \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \color{blue}{\frac{x}{s}}\right) \cdot \frac{x}{\frac{s}{x}}\right) \cdot \frac{x}{\frac{s}{x}}} + \left(s \cdot 4 + 0\right)} \]

   4. div-inv 80.3%

      \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{s}{x}}\right)}\right) \cdot \frac{x}{\frac{s}{x}}} + \left(s \cdot 4 + 0\right)} \]

   5. clear-num 80.3%

      \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \color{blue}{\frac{x}{s}}\right)\right) \cdot \frac{x}{\frac{s}{x}}} + \left(s \cdot 4 + 0\right)} \]

   6. div-inv 80.3%

      \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \frac{x}{s}\right)\right) \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{s}{x}}\right)}} + \left(s \cdot 4 + 0\right)} \]

   7. clear-num 80.3%

      \[\leadsto \frac{1}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \frac{x}{s}\right)\right) \cdot \left(x \cdot \color{blue}{\frac{x}{s}}\right)} + \left(s \cdot 4 + 0\right)} \]

9. Applied egg-rr 80.3%

   \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \frac{x}{s}\right)\right) \cdot \left(x \cdot \frac{x}{s}\right)}} + \left(s \cdot 4 + 0\right)} \]

10. Final simplification 80.3%

    \[\leadsto \frac{1}{\sqrt[3]{\left(x \cdot \frac{x}{s}\right) \cdot \left(\left(x \cdot \frac{x}{s}\right) \cdot \left(x \cdot \frac{x}{s}\right)\right)} + s \cdot 4} \]

Alternative 11: 78.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{s}{x \cdot x}\\ \mathbf{if}\;x \leq 0.003000000026077032:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(x + s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ s (* x x))))
   (if (<= x 0.003000000026077032)
     (/
      1.0
      (*
       (+ 1.0 (+ 1.0 (- (* 0.5 (* (/ x s) (/ x s))) (/ x s))))
       (+ s (+ x s))))
     (cbrt (* t_0 (* t_0 t_0))))))

C

float code(float x, float s) {
	float t_0 = s / (x * x);
	float tmp;
	if (x <= 0.003000000026077032f) {
		tmp = 1.0f / ((1.0f + (1.0f + ((0.5f * ((x / s) * (x / s))) - (x / s)))) * (s + (x + s)));
	} else {
		tmp = cbrtf((t_0 * (t_0 * t_0)));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(s / Float32(x * x))
	tmp = Float32(0.0)
	if (x <= Float32(0.003000000026077032))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(0.5) * Float32(Float32(x / s) * Float32(x / s))) - Float32(x / s)))) * Float32(s + Float32(x + s))));
	else
		tmp = cbrt(Float32(t_0 * Float32(t_0 * t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00300000003

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in s around inf 70.4%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{\left|x\right|}{s} + 0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 70.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + -1 \cdot \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      2. neg-mul-1 70.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{\left(-\frac{\left|x\right|}{s}\right)}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      3. unsub-neg 70.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      4. unpow2 70.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      5. unpow2 70.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\left|x\right| \cdot \left|x\right|}{\color{blue}{s \cdot s}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      6. times-frac 79.1%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \color{blue}{\left(\frac{\left|x\right|}{s} \cdot \frac{\left|x\right|}{s}\right)} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      7. unpow1 79.1%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{1}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      8. sqr-pow 32.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      9. fabs-sqr 32.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      10. sqr-pow 95.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{1}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      11. unpow1 95.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{x}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      12. unpow1 95.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{1}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      13. sqr-pow 32.4%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      14. fabs-sqr 32.4%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      15. sqr-pow 79.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{1}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      16. unpow1 79.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{x}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   7. Simplified 94.6%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   8. Taylor expanded in s around inf 71.7%

      \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(s + \left|x\right|\right)}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 71.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(\left|x\right| + s\right)}\right)} \]

      2. unpow1 71.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\left|\color{blue}{{x}^{1}}\right| + s\right)\right)} \]

      3. sqr-pow 22.1%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + s\right)\right)} \]

      4. fabs-sqr 22.1%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + s\right)\right)} \]

      5. sqr-pow 72.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\color{blue}{{x}^{1}} + s\right)\right)} \]

      6. unpow1 72.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\color{blue}{x} + s\right)\right)} \]

   10. Simplified 72.4%

       \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(x + s\right)}\right)} \]

   if 0.00300000003 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in s around -inf 25.4%

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

      1. +-commutative 25.4%

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

      2. +-commutative 25.4%

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

      3. mul-1-neg 25.4%

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

      4. distribute-lft1-in 73.3%

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

      5. metadata-eval 73.3%

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

      6. associate-*r/ 73.3%

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

      7. mul-1-neg 73.3%

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

      8. remove-double-neg 73.3%

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

      9. +-commutative 73.3%

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

      10. associate-+l+ 73.3%

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

   7. Simplified 73.3%

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

   8. Taylor expanded in x around inf 70.5%

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

      1. unpow2 70.5%

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

   10. Simplified 70.5%

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

       1. add-cbrt-cube 96.2%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{s}{x \cdot x} \cdot \frac{s}{x \cdot x}\right) \cdot \frac{s}{x \cdot x}}} \]

   12. Applied egg-rr 96.2%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{s}{x \cdot x} \cdot \frac{s}{x \cdot x}\right) \cdot \frac{s}{x \cdot x}}} \]
   13. Step-by-step derivation

       1. associate-*l* 96.2%

          \[\leadsto \sqrt[3]{\color{blue}{\frac{s}{x \cdot x} \cdot \left(\frac{s}{x \cdot x} \cdot \frac{s}{x \cdot x}\right)}} \]

   14. Simplified 96.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.003000000026077032:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(x + s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{s}{x \cdot x} \cdot \left(\frac{s}{x \cdot x} \cdot \frac{s}{x \cdot x}\right)}\\ \end{array} \]

Alternative 12: 77.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2000000:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(x + s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2000000.0)
   (/
    1.0
    (* (+ 1.0 (+ 1.0 (- (* 0.5 (* (/ x s) (/ x s))) (/ x s)))) (+ s (+ x s))))
   (/ s (cbrt (* (* x x) (* (* x x) (* x x)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2000000.0f) {
		tmp = 1.0f / ((1.0f + (1.0f + ((0.5f * ((x / s) * (x / s))) - (x / s)))) * (s + (x + s)));
	} else {
		tmp = s / cbrtf(((x * x) * ((x * x) * (x * x))));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2000000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(0.5) * Float32(Float32(x / s) * Float32(x / s))) - Float32(x / s)))) * Float32(s + Float32(x + s))));
	else
		tmp = Float32(s / cbrt(Float32(Float32(x * x) * Float32(Float32(x * x) * Float32(x * x)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 2e6

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in s around inf 71.8%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{\left|x\right|}{s} + 0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 71.8%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + -1 \cdot \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      2. neg-mul-1 71.8%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{\left(-\frac{\left|x\right|}{s}\right)}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      3. unsub-neg 71.8%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      4. unpow2 71.8%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      5. unpow2 71.8%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\left|x\right| \cdot \left|x\right|}{\color{blue}{s \cdot s}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      6. times-frac 79.9%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \color{blue}{\left(\frac{\left|x\right|}{s} \cdot \frac{\left|x\right|}{s}\right)} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      7. unpow1 79.9%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{1}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      8. sqr-pow 36.3%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      9. fabs-sqr 36.3%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      10. sqr-pow 94.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{1}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      11. unpow1 94.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{x}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      12. unpow1 94.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{1}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      13. sqr-pow 36.3%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      14. fabs-sqr 36.3%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      15. sqr-pow 79.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{1}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      16. unpow1 79.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{x}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   7. Simplified 94.4%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   8. Taylor expanded in s around inf 70.8%

      \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(s + \left|x\right|\right)}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 70.8%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(\left|x\right| + s\right)}\right)} \]

      2. unpow1 70.8%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\left|\color{blue}{{x}^{1}}\right| + s\right)\right)} \]

      3. sqr-pow 24.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + s\right)\right)} \]

      4. fabs-sqr 24.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + s\right)\right)} \]

      5. sqr-pow 71.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\color{blue}{{x}^{1}} + s\right)\right)} \]

      6. unpow1 71.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\color{blue}{x} + s\right)\right)} \]

   10. Simplified 71.4%

       \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(x + s\right)}\right)} \]

   if 2e6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in s around -inf 22.6%

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

      1. +-commutative 22.6%

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

      2. +-commutative 22.6%

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

      3. mul-1-neg 22.6%

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

      4. distribute-lft1-in 81.2%

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

      5. metadata-eval 81.2%

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

      6. associate-*r/ 81.2%

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

      7. mul-1-neg 81.2%

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

      8. remove-double-neg 81.2%

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

      9. +-commutative 81.2%

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

      10. associate-+l+ 81.2%

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

   7. Simplified 81.2%

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

   8. Taylor expanded in x around inf 80.4%

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

      1. unpow2 80.4%

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

   10. Simplified 80.4%

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

       1. add-cbrt-cube 100.0%

          \[\leadsto \frac{s}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}}} \]

   12. Applied egg-rr 100.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2000000:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(x + s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}\\ \end{array} \]

Alternative 13: 75.5% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 50000000188416:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(x + s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s} + s \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 50000000188416.0)
   (/
    1.0
    (* (+ 1.0 (+ 1.0 (- (* 0.5 (* (/ x s) (/ x s))) (/ x s)))) (+ s (+ x s))))
   (/ 1.0 (+ (* x (/ x s)) (* s 4.0)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 50000000188416.0f) {
		tmp = 1.0f / ((1.0f + (1.0f + ((0.5f * ((x / s) * (x / s))) - (x / s)))) * (s + (x + s)));
	} else {
		tmp = 1.0f / ((x * (x / s)) + (s * 4.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 50000000188416.0e0) then
        tmp = 1.0e0 / ((1.0e0 + (1.0e0 + ((0.5e0 * ((x / s) * (x / s))) - (x / s)))) * (s + (x + s)))
    else
        tmp = 1.0e0 / ((x * (x / s)) + (s * 4.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(50000000188416.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(0.5) * Float32(Float32(x / s) * Float32(x / s))) - Float32(x / s)))) * Float32(s + Float32(x + s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x / s)) + Float32(s * Float32(4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(50000000188416.0))
		tmp = single(1.0) / ((single(1.0) + (single(1.0) + ((single(0.5) * ((x / s) * (x / s))) - (x / s)))) * (s + (x + s)));
	else
		tmp = single(1.0) / ((x * (x / s)) + (s * single(4.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000002e13

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in s around inf 72.7%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{\left|x\right|}{s} + 0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 72.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + -1 \cdot \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      2. neg-mul-1 72.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{\left(-\frac{\left|x\right|}{s}\right)}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      3. unsub-neg 72.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      4. unpow2 72.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      5. unpow2 72.7%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\left|x\right| \cdot \left|x\right|}{\color{blue}{s \cdot s}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      6. times-frac 80.2%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \color{blue}{\left(\frac{\left|x\right|}{s} \cdot \frac{\left|x\right|}{s}\right)} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      7. unpow1 80.2%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{1}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      8. sqr-pow 40.2%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      9. fabs-sqr 40.2%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      10. sqr-pow 93.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{1}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      11. unpow1 93.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{x}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      12. unpow1 93.9%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{1}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      13. sqr-pow 40.2%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      14. fabs-sqr 40.2%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      15. sqr-pow 80.2%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{1}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      16. unpow1 80.2%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{x}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   7. Simplified 93.5%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   8. Taylor expanded in s around inf 71.4%

      \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(s + \left|x\right|\right)}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 71.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(\left|x\right| + s\right)}\right)} \]

      2. unpow1 71.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\left|\color{blue}{{x}^{1}}\right| + s\right)\right)} \]

      3. sqr-pow 28.9%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + s\right)\right)} \]

      4. fabs-sqr 28.9%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + s\right)\right)} \]

      5. sqr-pow 72.0%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\color{blue}{{x}^{1}} + s\right)\right)} \]

      6. unpow1 72.0%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(\color{blue}{x} + s\right)\right)} \]

   10. Simplified 72.0%

       \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{\left(x + s\right)}\right)} \]

   if 5.00000002e13 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in s around -inf 15.0%

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

      1. +-commutative 15.0%

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

      2. +-commutative 15.0%

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

      3. mul-1-neg 15.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*r/ 100.0%

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

      7. mul-1-neg 100.0%

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

      8. remove-double-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. associate-+l+ 100.0%

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

   7. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{s} \cdot x} + \left(s \cdot 4 + 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50000000188416:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \left(x + s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s} + s \cdot 4}\\ \end{array} \]

Alternative 14: 72.5% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10000000000:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s} + s \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 10000000000.0)
   (/ 1.0 (* (+ 1.0 (+ 1.0 (- (* 0.5 (* (/ x s) (/ x s))) (/ x s)))) (+ s s)))
   (/ 1.0 (+ (* x (/ x s)) (* s 4.0)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 10000000000.0f) {
		tmp = 1.0f / ((1.0f + (1.0f + ((0.5f * ((x / s) * (x / s))) - (x / s)))) * (s + s));
	} else {
		tmp = 1.0f / ((x * (x / s)) + (s * 4.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 10000000000.0e0) then
        tmp = 1.0e0 / ((1.0e0 + (1.0e0 + ((0.5e0 * ((x / s) * (x / s))) - (x / s)))) * (s + s))
    else
        tmp = 1.0e0 / ((x * (x / s)) + (s * 4.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(10000000000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(0.5) * Float32(Float32(x / s) * Float32(x / s))) - Float32(x / s)))) * Float32(s + s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x / s)) + Float32(s * Float32(4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(10000000000.0))
		tmp = single(1.0) / ((single(1.0) + (single(1.0) + ((single(0.5) * ((x / s) * (x / s))) - (x / s)))) * (s + s));
	else
		tmp = single(1.0) / ((x * (x / s)) + (s * single(4.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1e10

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in s around inf 72.4%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(-1 \cdot \frac{\left|x\right|}{s} + 0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 72.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + -1 \cdot \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      2. neg-mul-1 72.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{\left(-\frac{\left|x\right|}{s}\right)}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      3. unsub-neg 72.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \color{blue}{\left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)}\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      4. unpow2 72.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      5. unpow2 72.4%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \frac{\left|x\right| \cdot \left|x\right|}{\color{blue}{s \cdot s}} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      6. times-frac 80.1%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \color{blue}{\left(\frac{\left|x\right|}{s} \cdot \frac{\left|x\right|}{s}\right)} - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      7. unpow1 80.1%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{1}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      8. sqr-pow 39.0%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      9. fabs-sqr 39.0%

         \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      10. sqr-pow 94.2%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{{x}^{1}}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      11. unpow1 94.2%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{\color{blue}{x}}{s} \cdot \frac{\left|x\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      12. unpow1 94.2%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{1}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      13. sqr-pow 39.0%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      14. fabs-sqr 39.0%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      15. sqr-pow 80.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{{x}^{1}}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

      16. unpow1 80.1%

          \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{\color{blue}{x}}{s}\right) - \frac{\left|x\right|}{s}\right)\right)\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   7. Simplified 93.8%

      \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

   8. Taylor expanded in s around inf 67.1%

      \[\leadsto \frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + \color{blue}{s}\right)} \]

   if 1e10 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in s around -inf 20.4%

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

      1. +-commutative 20.4%

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

      2. +-commutative 20.4%

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

      3. mul-1-neg 20.4%

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

      4. distribute-lft1-in 94.3%

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

      5. metadata-eval 94.3%

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

      6. associate-*r/ 94.3%

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

      7. mul-1-neg 94.3%

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

      8. remove-double-neg 94.3%

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

      9. +-commutative 94.3%

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

      10. associate-+l+ 94.3%

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

   7. Simplified 94.3%

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

      1. associate-/r/ 94.3%

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

   9. Applied egg-rr 94.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{s} \cdot x} + \left(s \cdot 4 + 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10000000000:\\ \;\;\;\;\frac{1}{\left(1 + \left(1 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)\right)\right) \cdot \left(s + s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s} + s \cdot 4}\\ \end{array} \]

Alternative 15: 65.2% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.7%

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

5. Taylor expanded in s around -inf 40.2%

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

   1. +-commutative 40.2%

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

   2. +-commutative 40.2%

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

   3. mul-1-neg 40.2%

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

   4. distribute-lft1-in 64.8%

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

   5. metadata-eval 64.8%

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

   6. associate-*r/ 64.8%

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

   7. mul-1-neg 64.8%

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

   8. remove-double-neg 64.8%

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

   9. +-commutative 64.8%

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

   10. associate-+l+ 64.8%

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

7. Simplified 65.6%

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

   1. associate-/r/ 65.6%

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

9. Applied egg-rr 65.6%

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

10. Final simplification 65.6%

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

Alternative 16: 43.6% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.006000000052154064:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.006000000052154064) (/ 0.25 s) (* s (/ 1.0 (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.006000000052154064f) {
		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 <= 0.006000000052154064e0) 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(0.006000000052154064))
		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(0.006000000052154064))
		tmp = single(0.25) / s;
	else
		tmp = s * (single(1.0) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00600000005

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in s around inf 34.2%

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

   if 0.00600000005 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in s around -inf 24.4%

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

      1. +-commutative 24.4%

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

      2. +-commutative 24.4%

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

      3. mul-1-neg 24.4%

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

      4. distribute-lft1-in 72.9%

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

      5. metadata-eval 72.9%

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

      6. associate-*r/ 72.9%

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

      7. mul-1-neg 72.9%

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

      8. remove-double-neg 72.9%

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

      9. +-commutative 72.9%

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

      10. associate-+l+ 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in x around inf 71.3%

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

      1. unpow2 71.3%

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

   10. Simplified 71.3%

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

       1. div-inv 71.3%

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

   12. Applied egg-rr 71.3%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.006000000052154064:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \]

Alternative 17: 43.6% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.006000000052154064:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.006000000052154064) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.006000000052154064f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.006000000052154064e0) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.006000000052154064))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.006000000052154064))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00600000005

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in s around inf 34.2%

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

   if 0.00600000005 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in s around -inf 24.4%

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

      1. +-commutative 24.4%

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

      2. +-commutative 24.4%

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

      3. mul-1-neg 24.4%

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

      4. distribute-lft1-in 72.9%

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

      5. metadata-eval 72.9%

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

      6. associate-*r/ 72.9%

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

      7. mul-1-neg 72.9%

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

      8. remove-double-neg 72.9%

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

      9. +-commutative 72.9%

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

      10. associate-+l+ 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in x around inf 71.3%

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

      1. unpow2 71.3%

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

   10. Simplified 71.3%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.006000000052154064:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 18: 25.7% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around inf 26.1%

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

5. Final simplification 26.1%

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

Reproduce

herbie shell --seed 2023278 
(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))))))