Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 19.3s

Alternatives: 10

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{1 + e^{\frac{\left|x\right|}{-s}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (/ 1.0 (+ 1.0 (exp (/ (fabs x) (- s)))))
  (expm1 (log1p (fma s (exp (/ (fabs x) s)) s)))))

C

float code(float x, float s) {
	return (1.0f / (1.0f + expf((fabsf(x) / -s)))) / expm1f(log1pf(fmaf(s, expf((fabsf(x) / s)), s)));
}

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(abs(x) / Float32(-s))))) / expm1(log1p(fma(s, exp(Float32(abs(x) / s)), s))))
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{\frac{t_0}{s}}{{\left(1 + t_0\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s)))) (/ (/ t_0 s) (pow (+ 1.0 t_0) 2.0))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return (t_0 / s) / powf((1.0f + t_0), 2.0f);
}

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 / s) / ((1.0e0 + t_0) ** 2.0e0)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(Float32(t_0 / s) / (Float32(Float32(1.0) + t_0) ^ Float32(2.0)))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = (t_0 / s) / ((single(1.0) + t_0) ^ single(2.0));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 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. Taylor expanded in s around 0 99.4%

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

   1. associate-/r* 99.4%

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

   2. mul-1-neg 99.4%

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

   3. distribute-frac-neg 99.4%

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

   4. rec-exp 99.4%

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

   5. mul-1-neg 99.4%

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

   6. unpow2 99.4%

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

   7. mul-1-neg 99.4%

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

   8. distribute-frac-neg 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 62.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.5%

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

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\frac{1}{1 + e^{\frac{\left|x\right|}{-s}}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)\right)\right)}} \]
6. Step-by-step derivation

   1. fma-def 99.4%

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

   2. expm1-log1p-u 99.4%

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

   3. *-commutative 99.4%

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

   4. fma-def 99.4%

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

   5. add-sqr-sqrt 47.7%

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

   6. fabs-sqr 47.7%

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

   7. add-sqr-sqrt 62.7%

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

7. Applied egg-rr 62.7%

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

8. Final simplification 62.7%

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

Alternative 4: 62.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / (single(1.0) + exp((abs(x) / -s)))) / (s + (s * 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 99.4%

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

   1. expm1-log1p-u 99.5%

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

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\frac{1}{1 + e^{\frac{\left|x\right|}{-s}}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)\right)\right)}} \]
6. Step-by-step derivation

   1. fma-def 99.4%

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

   2. expm1-log1p-u 99.4%

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

   3. add-sqr-sqrt 47.6%

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

   4. fabs-sqr 47.6%

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

   5. add-sqr-sqrt 62.7%

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

7. Applied egg-rr 62.7%

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

8. Final simplification 62.7%

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

Alternative 5: 55.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.0999999986962872 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{\mathsf{expm1}\left(\frac{x}{s}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.0999999986962872e-16)
   (/ 1.0 (+ (* s 4.0) (* x (/ x s))))
   (/ (/ (/ 1.0 s) (expm1 (/ x s))) 2.0)))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.0999999986962872e-16f) {
		tmp = 1.0f / ((s * 4.0f) + (x * (x / s)));
	} else {
		tmp = ((1.0f / s) / expm1f((x / s))) / 2.0f;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.0999999986962872e-16))
		tmp = Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(Float32(1.0) / s) / expm1(Float32(x / s))) / Float32(2.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1.1e-16

   1. Initial program 98.1%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in s around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. mul-1-neg 98.3%

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

      3. distribute-frac-neg 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in s around inf 76.8%

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

      1. +-commutative 76.8%

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

      2. +-commutative 76.8%

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

      3. associate-+r+ 76.8%

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

      4. metadata-eval 76.8%

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

      5. associate-*r* 76.8%

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

      6. *-commutative 76.8%

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

      7. metadata-eval 76.8%

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

      8. associate-*r* 76.8%

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

      9. *-commutative 76.8%

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

      10. associate-+r+ 76.8%

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

   9. Simplified 76.8%

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

   10. Taylor expanded in x around 0 76.8%

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

       1. unpow2 76.8%

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

       2. associate-*l/ 77.7%

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

       3. *-commutative 77.7%

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

   12. Simplified 77.7%

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

   if 1.1e-16 < (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%

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

      1. expm1-log1p-u 99.7%

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

      2. expm1-udef 99.3%

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

      3. associate-/l/ 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.9%

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

      3. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 99.2%

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

   9. Applied egg-rr 50.2%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{s \cdot \mathsf{expm1}\left(\frac{x}{s}\right)}\right)} - 1}}{1 + e^{\frac{\left|x\right|}{-s}}} \]
   10. Step-by-step derivation

       1. expm1-def 50.2%

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

       2. expm1-log1p 52.3%

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

       3. associate-/r* 51.8%

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

   11. Simplified 51.8%

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

   12. Taylor expanded in s around inf 51.8%

       \[\leadsto \frac{\frac{\frac{1}{s}}{\mathsf{expm1}\left(\frac{x}{s}\right)}}{\color{blue}{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.0999999986962872 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{\mathsf{expm1}\left(\frac{x}{s}\right)}}{2}\\ \end{array} \]

Alternative 6: 95.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s + (s * exp((abs(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 99.4%

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

   1. fma-udef 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in s around inf 96.0%

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

7. Final simplification 96.0%

   \[\leadsto \frac{0.5}{s + s \cdot e^{\frac{\left|x\right|}{s}}} \]

Alternative 7: 94.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around inf 95.7%

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

5. Final simplification 95.7%

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

Alternative 8: 66.5% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(4.0)) + (x * (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 99.4%

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

4. Taylor expanded in s around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. mul-1-neg 99.4%

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

   3. distribute-frac-neg 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in s around inf 27.6%

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

   1. +-commutative 27.6%

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

   2. +-commutative 27.6%

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

   3. associate-+r+ 27.6%

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

   4. metadata-eval 27.6%

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

   5. associate-*r* 27.6%

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

   6. *-commutative 27.6%

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

   7. metadata-eval 27.6%

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

   8. associate-*r* 27.6%

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

   9. *-commutative 27.6%

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

   10. associate-+r+ 27.6%

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

9. Simplified 66.0%

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

10. Taylor expanded in x around 0 66.0%

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

    1. unpow2 66.0%

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

    2. associate-*l/ 66.3%

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

    3. *-commutative 66.3%

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

12. Simplified 66.3%

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

13. Final simplification 66.3%

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

Alternative 9: 45.6% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000233721948e-7) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.0000000233721948e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.0000000233721948e-7) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.0000000233721948e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.0000000233721948e-7))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000002e-7

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around inf 37.1%

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

   if 2.00000002e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in s around inf 4.4%

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

      1. +-commutative 4.4%

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

      2. +-commutative 4.4%

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

      3. associate-+r+ 4.4%

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

      4. metadata-eval 4.4%

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

      5. associate-*r* 4.4%

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

      6. *-commutative 4.4%

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

      7. metadata-eval 4.4%

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

      8. associate-*r* 4.4%

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

      9. *-commutative 4.4%

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

      10. associate-+r+ 4.4%

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

   9. Simplified 76.4%

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

   10. Taylor expanded in s around 0 73.3%

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

       1. unpow2 73.3%

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

   12. Simplified 73.3%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 10: 28.1% 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%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around inf 26.8%

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

5. Final simplification 26.8%

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

Reproduce

herbie shell --seed 2023279 
(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))))))