Logistic distribution

Percentage Accurate: 99.6% → 99.4%

Time: 9.4s

Alternatives: 9

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / s) / (exp((-x / s)) + (single(2.0) + 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 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. neg-mul-1 99.5%

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

   3. times-frac 99.5%

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

   4. metadata-eval 99.5%

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

   5. metadata-eval 99.5%

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

   6. times-frac 99.5%

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

   7. neg-mul-1 99.5%

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

   8. *-un-lft-identity 99.5%

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

   9. distribute-frac-neg 99.5%

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

   10. rec-exp 99.5%

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

   11. div-inv 99.5%

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

   12. exp-prod 97.1%

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

   13. add-sqr-sqrt 97.1%

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

   14. sqrt-unprod 97.1%

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

   15. sqr-neg 97.1%

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

   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 96.8%

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

   18. exp-prod 96.0%

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

   19. div-inv 96.0%

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

5. Applied egg-rr 97.9%

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

   1. rec-exp 97.9%

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

   2. distribute-neg-frac 97.9%

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

7. Simplified 97.9%

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

8. Taylor expanded in s around 0 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+r+ 97.8%

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

   3. +-commutative 97.8%

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

   4. +-commutative 97.8%

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

   5. associate-+r+ 97.8%

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

   6. +-commutative 97.8%

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

   7. associate-*r/ 97.8%

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

   8. mul-1-neg 97.8%

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

10. Simplified 97.8%

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

11. Taylor expanded in s around 0 97.8%

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

    1. associate-/r* 97.9%

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

    2. mul-1-neg 97.9%

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

    3. *-lft-identity 97.9%

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

    4. *-lft-identity 97.9%

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

    5. unpow1 97.9%

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

    6. sqr-pow 47.9%

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

    7. fabs-sqr 47.9%

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

    8. sqr-pow 99.5%

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

    9. unpow1 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (exp((-x / s)) + (single(2.0) + 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 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. neg-mul-1 99.5%

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

   3. times-frac 99.5%

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

   4. metadata-eval 99.5%

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

   5. metadata-eval 99.5%

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

   6. times-frac 99.5%

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

   7. neg-mul-1 99.5%

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

   8. *-un-lft-identity 99.5%

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

   9. distribute-frac-neg 99.5%

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

   10. rec-exp 99.5%

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

   11. div-inv 99.5%

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

   12. exp-prod 97.1%

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

   13. add-sqr-sqrt 97.1%

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

   14. sqrt-unprod 97.1%

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

   15. sqr-neg 97.1%

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

   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 96.8%

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

   18. exp-prod 96.0%

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

   19. div-inv 96.0%

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

5. Applied egg-rr 97.9%

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

   1. rec-exp 97.9%

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

   2. distribute-neg-frac 97.9%

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

7. Simplified 97.9%

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

8. Taylor expanded in s around 0 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+r+ 97.8%

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

   3. +-commutative 97.8%

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

   4. +-commutative 97.8%

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

   5. associate-+r+ 97.8%

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

   6. +-commutative 97.8%

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

   7. associate-*r/ 97.8%

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

   8. mul-1-neg 97.8%

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

10. Simplified 97.8%

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

11. Taylor expanded in s around 0 97.8%

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

    1. +-commutative 97.8%

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

    2. associate-+r+ 97.8%

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

    3. mul-1-neg 97.8%

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

    4. distribute-frac-neg 97.8%

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

    5. +-commutative 97.8%

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

    6. distribute-frac-neg 97.8%

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

    7. *-lft-identity 97.8%

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

    8. *-lft-identity 97.8%

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

    9. unpow1 97.8%

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

    10. sqr-pow 47.9%

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

    11. fabs-sqr 47.9%

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

    12. sqr-pow 99.5%

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

    13. unpow1 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 3: 65.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. neg-mul-1 99.5%

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

   3. times-frac 99.5%

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

   4. metadata-eval 99.5%

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

   5. metadata-eval 99.5%

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

   6. times-frac 99.5%

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

   7. neg-mul-1 99.5%

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

   8. *-un-lft-identity 99.5%

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

   9. distribute-frac-neg 99.5%

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

   10. rec-exp 99.5%

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

   11. div-inv 99.5%

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

   12. exp-prod 97.1%

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

   13. add-sqr-sqrt 97.1%

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

   14. sqrt-unprod 97.1%

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

   15. sqr-neg 97.1%

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

   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 96.8%

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

   18. exp-prod 96.0%

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

   19. div-inv 96.0%

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

5. Applied egg-rr 97.9%

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

   1. rec-exp 97.9%

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

   2. distribute-neg-frac 97.9%

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

7. Simplified 97.9%

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

8. Taylor expanded in s around 0 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+r+ 97.8%

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

   3. +-commutative 97.8%

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

   4. +-commutative 97.8%

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

   5. associate-+r+ 97.8%

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

   6. +-commutative 97.8%

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

   7. associate-*r/ 97.8%

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

   8. mul-1-neg 97.8%

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

10. Simplified 97.8%

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

11. Taylor expanded in x around 0 97.3%

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

    1. fma-def 97.3%

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

    2. *-lft-identity 97.3%

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

    3. *-lft-identity 97.3%

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

    4. unpow1 97.3%

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

    5. sqr-pow 47.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, 3 + e^{\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}}, -1 \cdot x\right)} \]

    6. fabs-sqr 47.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, 3 + e^{\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}}, -1 \cdot x\right)} \]

    7. sqr-pow 62.2%

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

    8. unpow1 62.2%

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

    9. neg-mul-1 62.2%

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

13. Simplified 62.2%

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

14. Final simplification 62.2%

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

Alternative 4: 81.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.9999999996399175e-23)
   (/ 0.25 s)
   (/ (/ 1.0 s) (+ 4.0 (* 0.5 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.9999999996399175e-23f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / s) / (4.0f + (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 (abs(x) <= 1.9999999996399175e-23) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / s) / (4.0e0 + (0.5e0 * ((x * x) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + 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 (abs(x) <= single(1.9999999996399175e-23))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / s) / (single(4.0) + (single(0.5) * ((x * x) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 2e-23

   1. Initial program 97.7%

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

      1. /-rgt-identity 97.7%

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

      2. associate-/l/ 97.7%

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

      3. *-lft-identity 97.7%

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

      4. +-commutative 97.7%

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

      5. distribute-rgt-in 97.8%

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

      6. *-lft-identity 97.8%

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

      7. +-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. *-lft-identity 97.8%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in s around inf 76.4%

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

   if 2e-23 < (fabs.f32 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 64.3%

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

      1. mul-1-neg 64.3%

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

      2. unsub-neg 64.3%

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

   6. Simplified 64.3%

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

      1. associate-+r- 64.3%

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

      2. add-sqr-sqrt 30.2%

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

      3. fabs-sqr 30.2%

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

      4. add-sqr-sqrt 55.9%

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

      5. add-sqr-sqrt 30.3%

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

      6. fabs-sqr 30.3%

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

      7. add-sqr-sqrt 56.2%

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

   8. Applied egg-rr 56.2%

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

   9. Taylor expanded in x around 0 85.0%

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

       1. unpow2 85.0%

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

       2. unpow2 85.0%

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

   11. Simplified 85.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 5: 62.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 97.0%

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

   1. expm1-log1p-u 95.8%

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

   2. expm1-udef 95.1%

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

   3. clear-num 95.1%

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

   4. div-inv 95.1%

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

   5. add-sqr-sqrt 45.7%

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

   6. fabs-sqr 45.7%

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

   7. add-sqr-sqrt 57.7%

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

   8. clear-num 57.7%

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

   9. /-rgt-identity 57.7%

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

6. Applied egg-rr 57.7%

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

   1. expm1-def 58.1%

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

   2. expm1-log1p 59.3%

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

   3. associate-/r* 59.3%

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

   4. *-lft-identity 59.3%

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

   5. associate-*l/ 59.3%

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

   6. associate-*r/ 59.3%

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

   7. *-rgt-identity 59.3%

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

   8. +-commutative 59.3%

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

8. Simplified 59.3%

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

9. Final simplification 59.3%

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

Alternative 6: 40.1% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8500001008069376 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.8500001008069376e-6) (/ 0.25 s) (/ (/ 1.0 s) (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.8500001008069376e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / s) / (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 <= 2.8500001008069376e-6) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / s) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.8500001008069376e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.8500001008069376e-6))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / s) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.8500001e-6

   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. /-rgt-identity 99.2%

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

      2. associate-/l/ 99.2%

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

      3. *-lft-identity 99.2%

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

      4. +-commutative 99.2%

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

      5. distribute-rgt-in 99.3%

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

      6. *-lft-identity 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-rgt-in 99.3%

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

      9. *-lft-identity 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 32.6%

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

   if 2.8500001e-6 < 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%

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

   4. Taylor expanded in s around inf 100.0%

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

   5. Taylor expanded in s around inf 54.1%

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

      1. +-commutative 54.1%

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

   7. Simplified 54.1%

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

   8. Taylor expanded in s around 0 54.1%

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

      1. unpow1 54.1%

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

      2. sqr-pow 54.1%

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

      3. fabs-sqr 54.1%

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

      4. sqr-pow 54.1%

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

      5. unpow1 54.1%

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

   10. Simplified 54.1%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8500001008069376 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s}}\\ \end{array} \]

Alternative 7: 51.3% accurate, 68.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 97.0%

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

5. Taylor expanded in s around inf 52.4%

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

   1. +-commutative 52.4%

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

7. Simplified 52.4%

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

8. Taylor expanded in x around 0 52.4%

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

   1. associate-/r* 52.4%

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

   2. +-commutative 52.4%

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

   3. *-lft-identity 52.4%

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

   4. *-lft-identity 52.4%

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

   5. unpow1 52.4%

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

   6. sqr-pow 25.0%

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

   7. fabs-sqr 25.0%

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

   8. sqr-pow 52.3%

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

   9. unpow1 52.3%

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

10. Simplified 52.3%

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

11. Final simplification 52.3%

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

Alternative 8: 29.5% accurate, 121.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8500001008069376 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.8500001008069376e-6) (/ 0.25 s) (/ 1.0 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.8500001008069376e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / 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 <= 2.8500001008069376e-6) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.8500001008069376e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.8500001008069376e-6))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.8500001e-6

   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. /-rgt-identity 99.2%

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

      2. associate-/l/ 99.2%

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

      3. *-lft-identity 99.2%

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

      4. +-commutative 99.2%

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

      5. distribute-rgt-in 99.3%

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

      6. *-lft-identity 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-rgt-in 99.3%

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

      9. *-lft-identity 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 32.6%

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

   if 2.8500001e-6 < 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%

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

   4. Taylor expanded in s around inf 100.0%

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

   5. Taylor expanded in s around inf 54.1%

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

      1. +-commutative 54.1%

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

   7. Simplified 54.1%

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

   8. Taylor expanded in s around 0 10.4%

      \[\leadsto \color{blue}{\frac{1}{\left|x\right|}} \]
   9. Step-by-step derivation

      1. unpow1 10.4%

         \[\leadsto \frac{1}{\left|\color{blue}{{x}^{1}}\right|} \]

      2. sqr-pow 10.4%

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

      3. fabs-sqr 10.4%

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

      4. sqr-pow 10.4%

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

      5. unpow1 10.4%

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

   10. Simplified 10.4%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8500001008069376 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 9: 28.0% 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. Step-by-step derivation

   1. /-rgt-identity 99.4%

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

   2. associate-/l/ 99.4%

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

   3. *-lft-identity 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-rgt-in 99.5%

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

   6. *-lft-identity 99.5%

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

   7. +-commutative 99.5%

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

   8. distribute-rgt-in 99.5%

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

   9. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 25.4%

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

5. Final simplification 25.4%

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

Reproduce

herbie shell --seed 2023215 
(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))))))