Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 12.1s

Alternatives: 12

Speedup: 1.0×

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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{\left(t\_0 + 1\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)} \]
6. Add Preprocessing

Alternative 2: 60.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {e}^{\left(\frac{x}{s}\right)}} \cdot \frac{0.5}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (* (/ 1.0 (+ 1.0 (pow E (/ x s)))) (/ 0.5 s)))

C

float code(float x, float s) {
	return (1.0f / (1.0f + powf(((float) M_E), (x / s)))) * (0.5f / s);
}

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(x / s)))) * Float32(Float32(0.5) / s))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (x / s)))) * (single(0.5) / s);
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. times-frac 99.5%

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

6. Applied egg-rr 60.0%

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

7. Taylor expanded in x around 0 59.5%

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

   1. *-un-lft-identity 59.5%

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

   2. exp-prod 59.5%

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

9. Applied egg-rr 59.5%

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

    1. exp-1-e 59.5%

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

11. Simplified 59.5%

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

12. Final simplification 59.5%

    \[\leadsto \frac{1}{1 + {e}^{\left(\frac{x}{s}\right)}} \cdot \frac{0.5}{s} \]
13. Add Preprocessing

Alternative 3: 60.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* (/ 0.5 s) (/ 1.0 (+ 1.0 (exp (/ x s))))))

C

float code(float x, float s) {
	return (0.5f / s) * (1.0f / (1.0f + expf((x / s))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.5e0 / s) * (1.0e0 / (1.0e0 + exp((x / s))))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.5) / s) * Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.5) / s) * (single(1.0) / (single(1.0) + exp((x / s))));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. times-frac 99.5%

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

6. Applied egg-rr 60.0%

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

7. Taylor expanded in x around 0 59.5%

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

8. Final simplification 59.5%

   \[\leadsto \frac{0.5}{s} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]
9. Add Preprocessing

Alternative 4: 60.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ 1.0 (exp (/ x s))))))

C

float code(float x, float s) {
	return 0.5f / (s * (1.0f + expf((x / s))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * (1.0e0 + exp((x / s))))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(s * Float32(Float32(1.0) + exp(Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s * (single(1.0) + exp((x / s))));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. times-frac 99.5%

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

6. Applied egg-rr 60.0%

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

7. Taylor expanded in x around 0 59.5%

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

   1. clear-num 59.5%

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

   2. frac-times 59.5%

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

   3. metadata-eval 59.5%

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

   4. /-rgt-identity 59.5%

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

   5. +-commutative 59.5%

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

9. Applied egg-rr 59.5%

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

10. Final simplification 59.5%

    \[\leadsto \frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \]
11. Add Preprocessing

Alternative 5: 39.4% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.029999999329447746:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.029999999329447746) (/ 0.25 s) (* (/ 0.5 s) (/ 1.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.029999999329447746f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / s) * (1.0f / (x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.029999999329447746e0) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / s) * (1.0e0 / (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.029999999329447746))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) * Float32(Float32(1.0) / Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.029999999329447746))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.5) / s) * (single(1.0) / (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0299999993

   1. Initial program 99.3%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in s around inf 30.6%

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

   if 0.0299999993 < 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%

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

      1. *-un-lft-identity 100.0%

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

      2. times-frac 100.0%

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

   6. Applied egg-rr -0.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   8. Taylor expanded in x around 0 47.5%

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

   9. Taylor expanded in x around inf 47.5%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.029999999329447746:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.7% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.029999999329447746:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.029999999329447746) (/ 0.25 s) (* (/ 0.5 s) (/ s x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.029999999329447746f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / s) * (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.029999999329447746e0) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / s) * (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.029999999329447746))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) * Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.029999999329447746))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.5) / s) * (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0299999993

   1. Initial program 99.3%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in s around inf 30.6%

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

   if 0.0299999993 < 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%

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

      1. *-un-lft-identity 100.0%

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

      2. times-frac 100.0%

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

   6. Applied egg-rr -0.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   8. Taylor expanded in x around 0 47.5%

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

   9. Taylor expanded in x around inf 32.7%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.029999999329447746:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.2% accurate, 56.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s} \cdot \frac{0.5}{\frac{x}{s} + 2} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* (/ 1.0 s) (/ 0.5 (+ (/ x s) 2.0))))

C

float code(float x, float s) {
	return (1.0f / s) * (0.5f / ((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) * (0.5e0 / ((x / s) + 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) * Float32(Float32(0.5) / Float32(Float32(x / s) + Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / s) * (single(0.5) / ((x / s) + single(2.0)));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. times-frac 99.5%

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

6. Applied egg-rr 60.0%

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

7. Taylor expanded in x around 0 59.5%

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

8. Taylor expanded in x around 0 47.9%

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

   1. associate-*r/ 47.9%

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

   2. clear-num 47.9%

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

   3. +-commutative 47.9%

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

10. Applied egg-rr 47.9%

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

    1. associate-/r/ 47.9%

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

    2. associate-*l/ 47.9%

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

    3. metadata-eval 47.9%

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

    4. +-commutative 47.9%

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

12. Simplified 47.9%

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

13. Final simplification 47.9%

    \[\leadsto \frac{1}{s} \cdot \frac{0.5}{\frac{x}{s} + 2} \]
14. Add Preprocessing

Alternative 8: 50.2% accurate, 56.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s} \cdot \frac{1}{\frac{x}{s} + 2} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* (/ 0.5 s) (/ 1.0 (+ (/ x s) 2.0))))

C

float code(float x, float s) {
	return (0.5f / s) * (1.0f / ((x / s) + 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.5e0 / s) * (1.0e0 / ((x / s) + 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.5) / s) * Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.5) / s) * (single(1.0) / ((x / s) + single(2.0)));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. times-frac 99.5%

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

6. Applied egg-rr 60.0%

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

7. Taylor expanded in x around 0 59.5%

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

8. Taylor expanded in x around 0 47.9%

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

9. Final simplification 47.9%

   \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\frac{x}{s} + 2} \]
10. Add Preprocessing

Alternative 9: 50.3% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot \left(\frac{x}{s} + 2\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ (/ x s) 2.0))))

C

float code(float x, float s) {
	return 0.5f / (s * ((x / s) + 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * ((x / s) + 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(s * Float32(Float32(x / s) + Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s * ((x / s) + single(2.0)));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. times-frac 99.5%

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

6. Applied egg-rr 60.0%

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

7. Taylor expanded in x around 0 59.5%

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

8. Taylor expanded in x around 0 47.9%

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

   1. frac-times 47.9%

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

   2. metadata-eval 47.9%

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

   3. +-commutative 47.9%

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

10. Applied egg-rr 47.9%

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

11. Final simplification 47.9%

    \[\leadsto \frac{0.5}{s \cdot \left(\frac{x}{s} + 2\right)} \]
12. Add Preprocessing

Alternative 10: 50.2% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{s}}{\frac{x}{s} + 2} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 0.5 s) (+ (/ x s) 2.0)))

C

float code(float x, float s) {
	return (0.5f / s) / ((x / s) + 2.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.5e0 / s) / ((x / s) + 2.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.5) / s) / Float32(Float32(x / s) + Float32(2.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.5) / s) / ((x / s) + single(2.0));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. times-frac 99.5%

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

6. Applied egg-rr 60.0%

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

7. Taylor expanded in x around 0 59.5%

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

8. Taylor expanded in x around 0 47.9%

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

   1. associate-*l/ 47.9%

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

   2. *-un-lft-identity 47.9%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{2 + \frac{x}{s}} \]

   3. +-commutative 47.9%

      \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

10. Applied egg-rr 47.9%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{s}}{\frac{x}{s} + 2}} \]

11. Final simplification 47.9%

    \[\leadsto \frac{\frac{0.5}{s}}{\frac{x}{s} + 2} \]
12. Add Preprocessing

Alternative 11: 28.2% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.029999999329447746:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.029999999329447746) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.029999999329447746f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.029999999329447746e0) then
        tmp = 0.25e0 / s
    else
        tmp = 0.5e0 / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.029999999329447746))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.029999999329447746))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0299999993

   1. Initial program 99.3%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in s around inf 30.6%

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

   if 0.0299999993 < 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%

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

      1. *-un-lft-identity 100.0%

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

      2. times-frac 100.0%

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

   6. Applied egg-rr -0.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   8. Taylor expanded in x around 0 47.5%

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

   9. Taylor expanded in x around inf 10.9%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.029999999329447746:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.4% 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 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 22.7%

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

6. Final simplification 22.7%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))))