Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 10.2s

Alternatives: 10

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. *-lft-identity 99.4%

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

   2. associate-*r/ 99.4%

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

   3. associate-/l* 99.4%

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

   4. distribute-frac-neg 99.4%

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

   5. exp-neg 99.4%

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

   6. associate-/r/ 99.4%

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

   7. /-rgt-identity 99.4%

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

   8. associate-*l* 99.4%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 3: 97.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 98.7%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 94.2%

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

   3. frac-times 88.8%

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

   4. sqr-neg 88.8%

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

   5. sqr-neg 88.8%

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

   6. frac-times 94.2%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 98.7%

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

   9. distribute-frac-neg 98.7%

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

   10. exp-neg 98.7%

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

   11. div-inv 98.7%

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

   12. exp-prod 96.2%

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

   13. add-sqr-sqrt 96.2%

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

   14. sqrt-unprod 96.2%

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

   15. sqr-neg 96.2%

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

   16. sqrt-unprod -0.0%

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

   17. add-sqr-sqrt 95.2%

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

5. Applied egg-rr 96.3%

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

   1. rec-exp 96.3%

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

   2. distribute-neg-frac 96.3%

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

7. Simplified 96.3%

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

8. Final simplification 96.3%

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

Alternative 4: 60.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 95.5%

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

   1. expm1-log1p-u 93.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3}\right)\right)} \]

   2. expm1-udef 93.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3}\right)} - 1} \]

   3. div-inv 93.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\color{blue}{\left|x\right| \cdot \frac{1}{s}}} + 3}\right)} - 1 \]

   4. exp-prod 80.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\color{blue}{{\left(e^{\left|x\right|}\right)}^{\left(\frac{1}{s}\right)}} + 3}\right)} - 1 \]

   5. add-sqr-sqrt 38.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{{\left(e^{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)}^{\left(\frac{1}{s}\right)} + 3}\right)} - 1 \]

   6. fabs-sqr 38.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{{\left(e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{\left(\frac{1}{s}\right)} + 3}\right)} - 1 \]

   7. add-sqr-sqrt 52.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{{\left(e^{\color{blue}{x}}\right)}^{\left(\frac{1}{s}\right)} + 3}\right)} - 1 \]

   8. exp-prod 61.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\color{blue}{e^{x \cdot \frac{1}{s}}} + 3}\right)} - 1 \]

   9. div-inv 61.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\color{blue}{\frac{x}{s}}} + 3}\right)} - 1 \]

6. Applied egg-rr 61.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 61.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\right)\right)} \]

   2. expm1-log1p 63.1%

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

   3. associate-/l/ 63.5%

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

   4. *-commutative 63.5%

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

   5. +-commutative 63.5%

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

8. Simplified 63.5%

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

9. Final simplification 63.5%

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

Alternative 5: 72.6% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999996399175e-23)
   (/ 1.0 (+ (* s 4.0) (/ x (/ s x))))
   (/ 1.0 (* s (+ 4.0 (+ (/ x s) (* 0.5 (/ (* x x) (* s s)))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999996399175e-23f) {
		tmp = 1.0f / ((s * 4.0f) + (x / (s / x)));
	} else {
		tmp = 1.0f / (s * (4.0f + ((x / s) + (0.5f * ((x * x) / (s * s))))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999996399175e-23) then
        tmp = 1.0e0 / ((s * 4.0e0) + (x / (s / x)))
    else
        tmp = 1.0e0 / (s * (4.0e0 + ((x / s) + (0.5e0 * ((x * x) / (s * s))))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(x / Float32(s / x))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(x / s) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999996399175e-23))
		tmp = single(1.0) / ((s * single(4.0)) + (x / (s / x)));
	else
		tmp = single(1.0) / (s * (single(4.0) + ((x / s) + (single(0.5) * ((x * x) / (s * s))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2e-23

   1. Initial program 99.2%

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

      1. *-lft-identity 99.2%

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

      2. associate-*r/ 99.2%

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

      3. associate-/l* 99.3%

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

      4. distribute-frac-neg 99.3%

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

      5. exp-neg 99.3%

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

      6. associate-/r/ 99.2%

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

      7. /-rgt-identity 99.2%

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

      8. associate-*l* 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around -inf 48.5%

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

      1. associate-+r+ 48.6%

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

      2. +-commutative 48.6%

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

      3. *-commutative 48.6%

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

      4. fma-def 48.6%

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

      5. mul-1-neg 48.6%

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

      6. distribute-rgt1-in 72.2%

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

      7. metadata-eval 72.2%

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

      8. associate-*r/ 72.2%

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

      9. mul-1-neg 72.2%

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

      10. remove-double-neg 72.2%

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

      11. unpow2 72.2%

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

      12. sqr-abs 72.2%

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

      13. distribute-rgt-out 72.2%

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

   6. Simplified 72.2%

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

      1. fma-udef 72.2%

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

      2. associate-/l* 72.7%

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

   8. Applied egg-rr 72.7%

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

   if 2e-23 < x

   1. Initial program 99.6%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in s around inf 96.5%

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

      1. expm1-log1p-u 95.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3}\right)\right)} \]

      2. expm1-udef 95.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3}\right)} - 1} \]

      3. div-inv 95.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\color{blue}{\left|x\right| \cdot \frac{1}{s}}} + 3}\right)} - 1 \]

      4. exp-prod 75.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\color{blue}{{\left(e^{\left|x\right|}\right)}^{\left(\frac{1}{s}\right)}} + 3}\right)} - 1 \]

      5. add-sqr-sqrt 75.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{{\left(e^{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)}^{\left(\frac{1}{s}\right)} + 3}\right)} - 1 \]

      6. fabs-sqr 75.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{{\left(e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{\left(\frac{1}{s}\right)} + 3}\right)} - 1 \]

      7. add-sqr-sqrt 75.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{{\left(e^{\color{blue}{x}}\right)}^{\left(\frac{1}{s}\right)} + 3}\right)} - 1 \]

      8. exp-prod 95.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\color{blue}{e^{x \cdot \frac{1}{s}}} + 3}\right)} - 1 \]

      9. div-inv 95.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\color{blue}{\frac{x}{s}}} + 3}\right)} - 1 \]

   6. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 95.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\right)\right)} \]

      2. expm1-log1p 96.5%

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

      3. associate-/l/ 97.5%

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

      4. *-commutative 97.5%

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

      5. +-commutative 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in x around 0 79.0%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   10. Step-by-step derivation

       1. unpow2 79.0%

          \[\leadsto \frac{1}{s \cdot \left(4 + \left(\frac{x}{s} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)\right)} \]

       2. unpow2 79.0%

          \[\leadsto \frac{1}{s \cdot \left(4 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)\right)} \]

   11. Simplified 79.0%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]

Alternative 6: 65.6% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. *-lft-identity 99.4%

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

   2. associate-*r/ 99.4%

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

   3. associate-/l* 99.4%

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

   4. distribute-frac-neg 99.4%

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

   5. exp-neg 99.4%

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

   6. associate-/r/ 99.4%

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

   7. /-rgt-identity 99.4%

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

   8. associate-*l* 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around -inf 42.4%

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

   1. associate-+r+ 42.4%

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

   2. +-commutative 42.4%

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

   3. *-commutative 42.4%

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

   4. fma-def 42.4%

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

   5. mul-1-neg 42.4%

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

   6. distribute-rgt1-in 67.0%

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

   7. metadata-eval 67.0%

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

   8. associate-*r/ 67.0%

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

   9. mul-1-neg 67.0%

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

   10. remove-double-neg 67.0%

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

   11. unpow2 67.0%

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

   12. sqr-abs 67.0%

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

   13. distribute-rgt-out 67.4%

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

6. Simplified 67.4%

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

   1. fma-udef 67.4%

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

   2. associate-/l* 67.7%

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

8. Applied egg-rr 67.7%

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

9. Final simplification 67.7%

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

Alternative 7: 45.4% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 7.999999797903001e-5) (/ 0.25 s) (/ 1.0 (* x (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 7.999999797903001e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x * (x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 7.999999797903001e-5) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x * (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(7.999999797903001e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(7.999999797903001e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x * (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 7.9999998e-5

   1. Initial program 99.2%

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

   2. Taylor expanded in s around inf 39.0%

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

   if 7.9999998e-5 < x

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around -inf 30.2%

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

      1. associate-+r+ 30.2%

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

      2. +-commutative 30.2%

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

      3. *-commutative 30.2%

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

      4. fma-def 30.2%

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

      5. mul-1-neg 30.2%

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

      6. distribute-rgt1-in 74.5%

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

      7. metadata-eval 74.5%

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

      8. associate-*r/ 74.5%

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

      9. mul-1-neg 74.5%

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

      10. remove-double-neg 74.5%

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

      11. unpow2 74.5%

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

      12. sqr-abs 74.5%

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

      13. distribute-rgt-out 76.1%

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

   6. Simplified 76.1%

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

   7. Taylor expanded in s around 0 72.6%

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

      1. unpow2 72.6%

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

   9. Simplified 72.6%

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

       1. associate-/r* 72.6%

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

       2. div-inv 72.6%

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

   11. Applied egg-rr 72.6%

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

       1. clear-num 72.6%

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

       2. frac-times 76.1%

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

       3. metadata-eval 76.1%

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

   13. Applied egg-rr 76.1%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 8: 45.4% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 7.999999797903001e-5) (/ 0.25 s) (/ 1.0 (/ (* x x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 7.999999797903001e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / ((x * x) / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 7.999999797903001e-5) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((x * x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(7.999999797903001e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(7.999999797903001e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((x * x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 7.9999998e-5

   1. Initial program 99.2%

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

   2. Taylor expanded in s around inf 39.0%

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

   if 7.9999998e-5 < x

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in x around inf 76.1%

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

      1. unpow2 76.1%

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

   9. Simplified 76.1%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \]

Alternative 9: 44.8% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 7.999999797903001e-5) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 7.999999797903001e-5f) {
		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 <= 7.999999797903001e-5) 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(7.999999797903001e-5))
		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(7.999999797903001e-5))
		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 < 7.9999998e-5

   1. Initial program 99.2%

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

   2. Taylor expanded in s around inf 39.0%

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

   if 7.9999998e-5 < x

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around -inf 30.2%

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

      1. associate-+r+ 30.2%

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

      2. +-commutative 30.2%

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

      3. *-commutative 30.2%

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

      4. fma-def 30.2%

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

      5. mul-1-neg 30.2%

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

      6. distribute-rgt1-in 74.5%

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

      7. metadata-eval 74.5%

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

      8. associate-*r/ 74.5%

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

      9. mul-1-neg 74.5%

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

      10. remove-double-neg 74.5%

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

      11. unpow2 74.5%

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

      12. sqr-abs 74.5%

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

      13. distribute-rgt-out 76.1%

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

   6. Simplified 76.1%

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

   7. Taylor expanded in s around 0 72.6%

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

      1. unpow2 72.6%

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

   9. Simplified 72.6%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 10: 27.4% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in s around inf 30.8%

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

3. Final simplification 30.8%

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

Reproduce

herbie shell --seed 2023244 
(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))))))