Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.2s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \frac{1}{1 + e^{-\frac{x}{s}}} \]
4. Add Preprocessing

Alternative 2: 48.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 2.0) 0.5 (/ -1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 2.0f) {
		tmp = 0.5f;
	} else {
		tmp = -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 (exp(-(x / s)) <= 2.0e0) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(-Float32(x / s))) <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp(-(x / s)) <= single(2.0))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 2

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 53.2%

      \[\leadsto \color{blue}{0.5} \]

   if 2 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

      3. --lowering--.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

      4. /-lowering-/.f32 39.5

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   5. Simplified 39.5%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{s}{x}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{s}{\mathsf{neg}\left(x\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{s}{\mathsf{neg}\left(x\right)}} \]

      4. neg-lowering-neg.f32 35.1

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

   8. Simplified 35.1%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(x\right)}{s}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{\mathsf{neg}\left(x\right)}{s}} \]

      3. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\mathsf{neg}\left(\frac{x}{s}\right)}} \]

      4. frac-2neg N/A

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

      5. /-lowering-/.f32 N/A

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

      6. /-lowering-/.f32 39.5

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

   10. Applied egg-rr 39.5%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 2.0) 0.5 (* s (/ -1.0 x))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 2.0f) {
		tmp = 0.5f;
	} else {
		tmp = s * (-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 (exp(-(x / s)) <= 2.0e0) then
        tmp = 0.5e0
    else
        tmp = s * ((-1.0e0) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(-Float32(x / s))) <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(s * Float32(Float32(-1.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp(-(x / s)) <= single(2.0))
		tmp = single(0.5);
	else
		tmp = s * (single(-1.0) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 2

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 53.2%

      \[\leadsto \color{blue}{0.5} \]

   if 2 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

      3. --lowering--.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

      4. /-lowering-/.f32 39.5

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   5. Simplified 39.5%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{s}{x}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{s}{\mathsf{neg}\left(x\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{s}{\mathsf{neg}\left(x\right)}} \]

      4. neg-lowering-neg.f32 35.1

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

   8. Simplified 35.1%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(x\right)}{s}}} \]

      2. associate-/r/ N/A

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

      3. *-lowering-*.f32 N/A

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

      4. metadata-eval N/A

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

      5. frac-2neg N/A

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

      6. /-lowering-/.f32 35.1

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

   10. Applied egg-rr 35.1%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 2.0) 0.5 (/ s (- x))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 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 (exp(-(x / s)) <= 2.0e0) then
        tmp = 0.5e0
    else
        tmp = s / -x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(-Float32(x / s))) <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(s / Float32(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp(-(x / s)) <= single(2.0))
		tmp = single(0.5);
	else
		tmp = s / -x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 2

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 53.2%

      \[\leadsto \color{blue}{0.5} \]

   if 2 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

      3. --lowering--.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

      4. /-lowering-/.f32 39.5

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   5. Simplified 39.5%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{s}{x}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{s}{\mathsf{neg}\left(x\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{s}{\mathsf{neg}\left(x\right)}} \]

      4. neg-lowering-neg.f32 35.1

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

   8. Simplified 35.1%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 4.999999898305949e-32)
   0.5
   (/
    1.0
    (fma
     x
     (fma (/ x (* s s)) (fma -0.16666666666666666 (/ x s) 0.5) (/ -1.0 s))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999898305949e-32f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, fmaf((x / (s * s)), fmaf(-0.16666666666666666f, (x / s), 0.5f), (-1.0f / s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(4.999999898305949e-32))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, fma(Float32(x / Float32(s * s)), fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)), Float32(Float32(-1.0) / s)), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 4.9999999e-32

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 47.3%

      \[\leadsto \color{blue}{0.5} \]

   if 4.9999999e-32 < (neg.f32 x)

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   5. Simplified 86.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 65.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}, x \cdot x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 5.0)
   0.5
   (/ 1.0 (fma (* x (/ -0.16666666666666666 (* s (* s s)))) (* x x) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 5.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf((x * (-0.16666666666666666f / (s * (s * s)))), (x * x), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(5.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(Float32(x * Float32(Float32(-0.16666666666666666) / Float32(s * Float32(s * s)))), Float32(x * x), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 53.0%

      \[\leadsto \color{blue}{0.5} \]

   if 5 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

   5. Simplified 75.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{2}}}}{s}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{2}}}}{s}} \]

      2. cube-mult N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \frac{\color{blue}{x \cdot \left(x \cdot x\right)}}{{s}^{2}}}{s}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \frac{x \cdot \color{blue}{{x}^{2}}}{{s}^{2}}}{s}} \]

      4. associate-/l* N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot \frac{{x}^{2}}{{s}^{2}}\right)}}{s}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot \frac{{x}^{2}}{{s}^{2}}\right)}}{s}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)}{s}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{x}{{s}^{2}}\right)}\right)}{s}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{x}{{s}^{2}}\right)}\right)}{s}} \]

      9. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \left(x \cdot \color{blue}{\frac{x}{{s}^{2}}}\right)\right)}{s}} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \left(x \cdot \frac{x}{\color{blue}{s \cdot s}}\right)\right)}{s}} \]

      11. *-lowering-*.f32 81.7

          \[\leadsto \frac{1}{2 + \frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \frac{x}{\color{blue}{s \cdot s}}\right)\right)}{s}} \]

   8. Simplified 81.7%

      \[\leadsto \frac{1}{2 + \frac{\color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \frac{x}{s \cdot s}\right)\right)}}{s}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6} \cdot \left(x \cdot \left(x \cdot \frac{x}{s \cdot s}\right)\right)}{s} + 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot \left(x \cdot \frac{x}{s \cdot s}\right)\right) \cdot \frac{-1}{6}}}{s} + 2} \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(x \cdot \frac{x}{s \cdot s}\right)\right) \cdot \frac{\frac{-1}{6}}{s}} + 2} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{1}{\left(x \cdot \color{blue}{\frac{x \cdot x}{s \cdot s}}\right) \cdot \frac{\frac{-1}{6}}{s} + 2} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(x \cdot x\right)}{s \cdot s}} \cdot \frac{\frac{-1}{6}}{s} + 2} \]

      6. times-frac N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}}{\left(s \cdot s\right) \cdot s}} + 2} \]

      7. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}}{\color{blue}{s \cdot \left(s \cdot s\right)}} + 2} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{-1}{6}}{s \cdot \left(s \cdot s\right)}} + 2} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6}}{s \cdot \left(s \cdot s\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)} + 2} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\frac{\frac{-1}{6}}{s \cdot \left(s \cdot s\right)} \cdot x\right) \cdot \left(x \cdot x\right)} + 2} \]

      11. accelerator-lowering-fma.f32 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{\frac{-1}{6}}{s \cdot \left(s \cdot s\right)} \cdot x, x \cdot x, 2\right)}} \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{6}}{s \cdot \left(s \cdot s\right)} \cdot x}, x \cdot x, 2\right)} \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{6}}{s \cdot \left(s \cdot s\right)}} \cdot x, x \cdot x, 2\right)} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{-1}{6}}{\color{blue}{s \cdot \left(s \cdot s\right)}} \cdot x, x \cdot x, 2\right)} \]

      15. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{-1}{6}}{s \cdot \color{blue}{\left(s \cdot s\right)}} \cdot x, x \cdot x, 2\right)} \]

      16. *-lowering-*.f32 84.9

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)} \cdot x, \color{blue}{x \cdot x}, 2\right)} \]

   10. Applied egg-rr 84.9%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)} \cdot x, x \cdot x, 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}, x \cdot x, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 50000.0)
   0.5
   (/ 1.0 (* (/ -0.16666666666666666 (* s (* s s))) (* x (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 50000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((-0.16666666666666666f / (s * (s * s))) * (x * (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 / s) <= 50000.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((-0.16666666666666666e0) / (s * (s * s))) * (x * (x * x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(50000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(-0.16666666666666666) / Float32(s * Float32(s * s))) * Float32(x * Float32(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(50000.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(-0.16666666666666666) / (s * (s * s))) * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e4

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 51.8%

      \[\leadsto \color{blue}{0.5} \]

   if 5e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

   5. Simplified 78.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{{x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{{s}^{2} \cdot x} - \frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{{s}^{2} \cdot x} - \frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)}} \]

      2. cube-mult N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{{s}^{2} \cdot x} - \frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{{s}^{2} \cdot x} - \frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{{s}^{2} \cdot x} - \frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{{s}^{2} \cdot x} - \frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)} \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{{s}^{2} \cdot x} - \frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{s}^{2} \cdot x} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)}} \]

      8. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{s}^{2} \cdot x} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{{s}^{2} \cdot x}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{{s}^{2} \cdot x} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)} \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{\frac{1}{2}}{{s}^{2} \cdot x}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{\color{blue}{\left(s \cdot s\right)} \cdot x} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{\color{blue}{s \cdot \left(s \cdot x\right)}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{\color{blue}{s \cdot \left(s \cdot x\right)}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \color{blue}{\left(x \cdot s\right)}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)} \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \color{blue}{\left(x \cdot s\right)}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)} \]

      17. associate-*r/ N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \left(x \cdot s\right)} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{6} \cdot 1}{{s}^{3}}}\right)\right)\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \left(x \cdot s\right)} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{6}}}{{s}^{3}}\right)\right)\right)} \]

      19. distribute-neg-frac N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \left(x \cdot s\right)} + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{6}\right)}{{s}^{3}}}\right)} \]

      20. metadata-eval N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \left(x \cdot s\right)} + \frac{\color{blue}{\frac{-1}{6}}}{{s}^{3}}\right)} \]

      21. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \left(x \cdot s\right)} + \color{blue}{\frac{\frac{-1}{6}}{{s}^{3}}}\right)} \]

      22. cube-mult N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \left(x \cdot s\right)} + \frac{\frac{-1}{6}}{\color{blue}{s \cdot \left(s \cdot s\right)}}\right)} \]

      23. unpow2 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \left(x \cdot s\right)} + \frac{\frac{-1}{6}}{s \cdot \color{blue}{{s}^{2}}}\right)} \]

      24. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \left(x \cdot s\right)} + \frac{\frac{-1}{6}}{\color{blue}{s \cdot {s}^{2}}}\right)} \]

      25. unpow2 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\frac{1}{2}}{s \cdot \left(x \cdot s\right)} + \frac{\frac{-1}{6}}{s \cdot \color{blue}{\left(s \cdot s\right)}}\right)} \]

      26. *-lowering-*.f32 85.3

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{0.5}{s \cdot \left(x \cdot s\right)} + \frac{-0.16666666666666666}{s \cdot \color{blue}{\left(s \cdot s\right)}}\right)} \]

   8. Simplified 85.3%

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{0.5}{s \cdot \left(x \cdot s\right)} + \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{\frac{-1}{6}}{{s}^{3}}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{\frac{-1}{6}}{{s}^{3}}}} \]

       2. cube-mult N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{-1}{6}}{\color{blue}{s \cdot \left(s \cdot s\right)}}} \]

       3. unpow2 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{-1}{6}}{s \cdot \color{blue}{{s}^{2}}}} \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{-1}{6}}{\color{blue}{s \cdot {s}^{2}}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{-1}{6}}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

       6. *-lowering-*.f32 85.3

          \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-0.16666666666666666}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

   11. Simplified 85.3%

       \[\leadsto \frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.2500000392179937 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot \left(x \cdot -0.16666666666666666\right), \frac{x}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.2500000392179937e-23)
   0.5
   (/ 1.0 (fma (* x (* x -0.16666666666666666)) (/ x (* s (* s s))) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.2500000392179937e-23f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf((x * (x * -0.16666666666666666f)), (x / (s * (s * s))), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.2500000392179937e-23))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(Float32(x * Float32(x * Float32(-0.16666666666666666))), Float32(x / Float32(s * Float32(s * s))), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.25000004e-23

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 48.7%

      \[\leadsto \color{blue}{0.5} \]

   if 1.25000004e-23 < (neg.f32 x)

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

   5. Simplified 80.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{2}}}}{s}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{2}}}}{s}} \]

      2. cube-mult N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \frac{\color{blue}{x \cdot \left(x \cdot x\right)}}{{s}^{2}}}{s}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \frac{x \cdot \color{blue}{{x}^{2}}}{{s}^{2}}}{s}} \]

      4. associate-/l* N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot \frac{{x}^{2}}{{s}^{2}}\right)}}{s}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot \frac{{x}^{2}}{{s}^{2}}\right)}}{s}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)}{s}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{x}{{s}^{2}}\right)}\right)}{s}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{x}{{s}^{2}}\right)}\right)}{s}} \]

      9. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \left(x \cdot \color{blue}{\frac{x}{{s}^{2}}}\right)\right)}{s}} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \left(x \cdot \left(x \cdot \frac{x}{\color{blue}{s \cdot s}}\right)\right)}{s}} \]

      11. *-lowering-*.f32 81.8

          \[\leadsto \frac{1}{2 + \frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \frac{x}{\color{blue}{s \cdot s}}\right)\right)}{s}} \]

   8. Simplified 81.8%

      \[\leadsto \frac{1}{2 + \frac{\color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \frac{x}{s \cdot s}\right)\right)}}{s}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{2}}}}{s}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3}}{{s}^{2}}}}{s}} \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3}}{{s}^{2}}}}{s}} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{2 + \frac{\frac{\color{blue}{{x}^{3} \cdot \frac{-1}{6}}}{{s}^{2}}}{s}} \]

       4. cube-mult N/A

          \[\leadsto \frac{1}{2 + \frac{\frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{-1}{6}}{{s}^{2}}}{s}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{2 + \frac{\frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{-1}{6}}{{s}^{2}}}{s}} \]

       6. associate-*l* N/A

          \[\leadsto \frac{1}{2 + \frac{\frac{\color{blue}{x \cdot \left({x}^{2} \cdot \frac{-1}{6}\right)}}{{s}^{2}}}{s}} \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{2 + \frac{\frac{\color{blue}{x \cdot \left({x}^{2} \cdot \frac{-1}{6}\right)}}{{s}^{2}}}{s}} \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{2 + \frac{\frac{x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)}}{{s}^{2}}}{s}} \]

       9. unpow2 N/A

          \[\leadsto \frac{1}{2 + \frac{\frac{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}\right)}{{s}^{2}}}{s}} \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \frac{1}{2 + \frac{\frac{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}\right)}{{s}^{2}}}{s}} \]

       11. unpow2 N/A

           \[\leadsto \frac{1}{2 + \frac{\frac{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot s}}}{s}} \]

       12. *-lowering-*.f32 79.9

           \[\leadsto \frac{1}{2 + \frac{\frac{x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)}{\color{blue}{s \cdot s}}}{s}} \]

   11. Simplified 79.9%

       \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)}{s \cdot s}}}{s}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)}{s \cdot s}}{s} + 2}} \]

       2. associate-/l/ N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)}{s \cdot \left(s \cdot s\right)}} + 2} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x}}{s \cdot \left(s \cdot s\right)} + 2} \]

       4. associate-/l* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot \frac{x}{s \cdot \left(s \cdot s\right)}} + 2} \]

       5. accelerator-lowering-fma.f32 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{-1}{6}, \frac{x}{s \cdot \left(s \cdot s\right)}, 2\right)}} \]

       6. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, \frac{x}{s \cdot \left(s \cdot s\right)}, 2\right)} \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, \frac{x}{s \cdot \left(s \cdot s\right)}, 2\right)} \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot \frac{-1}{6}\right)}, \frac{x}{s \cdot \left(s \cdot s\right)}, 2\right)} \]

       9. /-lowering-/.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(x \cdot \frac{-1}{6}\right), \color{blue}{\frac{x}{s \cdot \left(s \cdot s\right)}}, 2\right)} \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(x \cdot \frac{-1}{6}\right), \frac{x}{\color{blue}{s \cdot \left(s \cdot s\right)}}, 2\right)} \]

       11. *-lowering-*.f32 87.7

           \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot \left(x \cdot -0.16666666666666666\right), \frac{x}{s \cdot \color{blue}{\left(s \cdot s\right)}}, 2\right)} \]

   13. Applied egg-rr 87.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot -0.16666666666666666\right), \frac{x}{s \cdot \left(s \cdot s\right)}, 2\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 50000.0) 0.5 (/ (* (* s (* s s)) -6.0) (* x (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 50000.0f) {
		tmp = 0.5f;
	} else {
		tmp = ((s * (s * s)) * -6.0f) / (x * (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 / s) <= 50000.0e0) then
        tmp = 0.5e0
    else
        tmp = ((s * (s * s)) * (-6.0e0)) / (x * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(50000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(s * Float32(s * s)) * Float32(-6.0)) / Float32(x * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(50000.0))
		tmp = single(0.5);
	else
		tmp = ((s * (s * s)) * single(-6.0)) / (x * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e4

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 51.8%

      \[\leadsto \color{blue}{0.5} \]

   if 5e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

   5. Simplified 78.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-6 \cdot \frac{{s}^{3}}{{x}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-6 \cdot {s}^{3}}{{x}^{3}}} \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-6 \cdot {s}^{3}}{{x}^{3}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{s}^{3} \cdot -6}}{{x}^{3}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{{s}^{3} \cdot -6}}{{x}^{3}} \]

      5. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(s \cdot \left(s \cdot s\right)\right)} \cdot -6}{{x}^{3}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(s \cdot \color{blue}{{s}^{2}}\right) \cdot -6}{{x}^{3}} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(s \cdot {s}^{2}\right)} \cdot -6}{{x}^{3}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(s \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot -6}{{x}^{3}} \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \frac{\left(s \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot -6}{{x}^{3}} \]

      10. cube-mult N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{{x}^{2}}} \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{\color{blue}{x \cdot {x}^{2}}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

      14. *-lowering-*.f32 83.6

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 50000.0) 0.5 (/ 2.0 (/ (* x x) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 50000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f / ((x * x) / (s * s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 50000.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 / ((x * x) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(50000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) / Float32(Float32(x * x) / Float32(s * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(50000.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) / ((x * x) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e4

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 51.8%

      \[\leadsto \color{blue}{0.5} \]

   if 5e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. accelerator-lowering-fma.f32 N/A

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

   5. Simplified 70.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(0.5, \frac{x}{s}, -1\right), 2\right)}} \]

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f32 N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f32 N/A

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

      5. /-lowering-/.f32 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 78.6

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

   8. Simplified 78.6%

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. /-lowering-/.f32 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f32 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. *-lowering-*.f32 76.4

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

   10. Applied egg-rr 76.4%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{x \cdot x}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 50000.0) 0.5 (/ (* (* s s) 2.0) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 50000.0f) {
		tmp = 0.5f;
	} else {
		tmp = ((s * s) * 2.0f) / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 50000.0e0) then
        tmp = 0.5e0
    else
        tmp = ((s * s) * 2.0e0) / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(50000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(s * s) * Float32(2.0)) / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(50000.0))
		tmp = single(0.5);
	else
		tmp = ((s * s) * single(2.0)) / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e4

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 51.8%

      \[\leadsto \color{blue}{0.5} \]

   if 5e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. accelerator-lowering-fma.f32 N/A

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

   5. Simplified 70.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(0.5, \frac{x}{s}, -1\right), 2\right)}} \]

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {s}^{2}}{{x}^{2}}} \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {s}^{2}}{{x}^{2}}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {s}^{2}}}{{x}^{2}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 73.6

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

   8. Simplified 73.6%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 50000.0) 0.5 (* (* s s) (/ 2.0 (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 50000.0f) {
		tmp = 0.5f;
	} else {
		tmp = (s * s) * (2.0f / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 50000.0e0) then
        tmp = 0.5e0
    else
        tmp = (s * s) * (2.0e0 / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(50000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * s) * Float32(Float32(2.0) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(50000.0))
		tmp = single(0.5);
	else
		tmp = (s * s) * (single(2.0) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e4

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 51.8%

      \[\leadsto \color{blue}{0.5} \]

   if 5e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. accelerator-lowering-fma.f32 N/A

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

   5. Simplified 70.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(0.5, \frac{x}{s}, -1\right), 2\right)}} \]

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f32 N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f32 N/A

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

      5. /-lowering-/.f32 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 78.6

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

   8. Simplified 78.6%

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-/r/ N/A

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

      5. *-lowering-*.f32 N/A

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

      6. /-lowering-/.f32 N/A

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

      7. *-lowering-*.f32 N/A

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

      8. *-lowering-*.f32 73.6

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

   10. Applied egg-rr 73.6%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -5.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -5.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - (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 / s) <= (-5.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(-5.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(-5.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -5

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if -5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

      3. --lowering--.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

      4. /-lowering-/.f32 61.2

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   5. Simplified 61.2%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.2% accurate, 128.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(0.5)
end

MATLAB

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

Derivation

1. Initial program 99.9%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Simplified 35.5%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))