Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 17.0s

Alternatives: 17

Speedup: N/A×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Alternative 1: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - e^{\frac{\pi}{s}}\\ \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(-\log \left(\frac{u}{t\_0} + \left(\frac{-1}{t\_0} + \frac{u}{1 + e^{\frac{\pi}{-s}}}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- -1.0 (exp (/ PI s)))))
   (*
    (- s)
    (log
     (expm1
      (-
       (log
        (+ (/ u t_0) (+ (/ -1.0 t_0) (/ u (+ 1.0 (exp (/ PI (- s))))))))))))))

C

float code(float u, float s) {
	float t_0 = -1.0f - expf((((float) M_PI) / s));
	return -s * logf(expm1f(-logf(((u / t_0) + ((-1.0f / t_0) + (u / (1.0f + expf((((float) M_PI) / -s)))))))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(-1.0) - exp(Float32(Float32(pi) / s)))
	return Float32(Float32(-s) * log(expm1(Float32(-log(Float32(Float32(u / t_0) + Float32(Float32(Float32(-1.0) / t_0) + Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))))))))))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \left(\frac{u}{-\left(1 + e^{\frac{\pi}{s}}\right)} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]
7. Step-by-step derivation

   1. lift-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   2. lift-neg.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   3. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   4. lift-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   5. lift-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{\color{blue}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   6. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   8. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   9. lift-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   10. lift-+.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\color{blue}{\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   11. lift-neg.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\color{blue}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

8. Applied rewrites 99.0%

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

10. Applied rewrites 99.0%

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

11. Final simplification 99.0%

    \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(-\log \left(\frac{u}{-1 - e^{\frac{\pi}{s}}} + \left(\frac{-1}{-1 - e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{\frac{\pi}{-s}}}\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 2: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{-s}}\\ t_1 := \frac{-1}{1 + t\_0}\\ t_2 := e^{\frac{\pi}{s}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(-1 + \frac{-1}{\frac{-1}{t\_2 + 1} + u \cdot \left(\frac{-1}{-1 - t\_2} + t\_1\right)}\right) \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\log \left(-1 + \frac{-1}{\frac{u}{-1 - t\_0}}\right) \cdot \frac{s \cdot s}{-s}\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(-1 + \frac{-1}{u \cdot \left(t\_1 + \frac{-1}{-1 + \left(-1 - \frac{\pi}{s}\right)}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI (- s))))
        (t_1 (/ -1.0 (+ 1.0 t_0)))
        (t_2 (exp (/ PI s))))
   (if (<=
        (*
         (- s)
         (log
          (+
           -1.0
           (/
            -1.0
            (+ (/ -1.0 (+ t_2 1.0)) (* u (+ (/ -1.0 (- -1.0 t_2)) t_1)))))))
        -1.9999999949504854e-6)
     (* (log (+ -1.0 (/ -1.0 (/ u (- -1.0 t_0))))) (/ (* s s) (- s)))
     (*
      (- s)
      (log
       (+ -1.0 (/ -1.0 (* u (+ t_1 (/ -1.0 (+ -1.0 (- -1.0 (/ PI s)))))))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / -s));
	float t_1 = -1.0f / (1.0f + t_0);
	float t_2 = expf((((float) M_PI) / s));
	float tmp;
	if ((-s * logf((-1.0f + (-1.0f / ((-1.0f / (t_2 + 1.0f)) + (u * ((-1.0f / (-1.0f - t_2)) + t_1))))))) <= -1.9999999949504854e-6f) {
		tmp = logf((-1.0f + (-1.0f / (u / (-1.0f - t_0))))) * ((s * s) / -s);
	} else {
		tmp = -s * logf((-1.0f + (-1.0f / (u * (t_1 + (-1.0f / (-1.0f + (-1.0f - (((float) M_PI) / s)))))))));
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / Float32(-s)))
	t_1 = Float32(Float32(-1.0) / Float32(Float32(1.0) + t_0))
	t_2 = exp(Float32(Float32(pi) / s))
	tmp = Float32(0.0)
	if (Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(Float32(Float32(-1.0) / Float32(t_2 + Float32(1.0))) + Float32(u * Float32(Float32(Float32(-1.0) / Float32(Float32(-1.0) - t_2)) + t_1))))))) <= Float32(-1.9999999949504854e-6))
		tmp = Float32(log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(u / Float32(Float32(-1.0) - t_0))))) * Float32(Float32(s * s) / Float32(-s)));
	else
		tmp = Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(u * Float32(t_1 + Float32(Float32(-1.0) / Float32(Float32(-1.0) + Float32(Float32(-1.0) - Float32(Float32(pi) / s))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	t_0 = exp((single(pi) / -s));
	t_1 = single(-1.0) / (single(1.0) + t_0);
	t_2 = exp((single(pi) / s));
	tmp = single(0.0);
	if ((-s * log((single(-1.0) + (single(-1.0) / ((single(-1.0) / (t_2 + single(1.0))) + (u * ((single(-1.0) / (single(-1.0) - t_2)) + t_1))))))) <= single(-1.9999999949504854e-6))
		tmp = log((single(-1.0) + (single(-1.0) / (u / (single(-1.0) - t_0))))) * ((s * s) / -s);
	else
		tmp = -s * log((single(-1.0) + (single(-1.0) / (u * (t_1 + (single(-1.0) / (single(-1.0) + (single(-1.0) - (single(pi) / s)))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.99999999e-6

   1. Initial program 98.5%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around inf

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      3. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      6. lower-exp.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{-1 \cdot s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      13. lower-neg.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)} - 1\right) \]

   5. Applied rewrites 89.7%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]
   6. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      3. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      5. lower--.f32 N/A

         \[\leadsto \frac{\color{blue}{0 - s \cdot s}}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{0 - \color{blue}{s \cdot s}}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      7. lower-+.f32 89.7

         \[\leadsto \frac{0 - s \cdot s}{\color{blue}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)} - 1\right) \]

   7. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\frac{0 - s \cdot s}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)} - 1\right) \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)}\right)}\right)} - 1\right) \]

      2. unsub-neg N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

      3. lower--.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}}\right)}\right)} - 1\right) \]

      5. +-commutative N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\color{blue}{\frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s} + -1 \cdot \mathsf{PI}\left(\right)}}{s}\right)}\right)} - 1\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{\mathsf{PI}\left(\right)}^{2}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}}{s}\right)}\right)} - 1\right) \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{\mathsf{PI}\left(\right)}^{2}}{s}}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      10. lower-PI.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}, \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}\right)}{s}\right)}\right)} - 1\right) \]

      13. lower-neg.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}, \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}\right)}{s}\right)}\right)} - 1\right) \]

      14. lower-PI.f32 81.7

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\color{blue}{\pi}\right)}{s}\right)}\right)} - 1\right) \]

   10. Applied rewrites 81.7%

       \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\pi\right)}{s}\right)}}\right)} - 1\right) \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       2. lower-+.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       3. lower-exp.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       4. associate-*r/ N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       5. lower-/.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       6. mul-1-neg N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

       7. lower-neg.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

       8. lower-PI.f32 89.7

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\color{blue}{\pi}}{s}}}} - 1\right) \]

   13. Applied rewrites 89.7%

       \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{-\pi}{s}}}}} - 1\right) \]

   if -1.99999999e-6 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 99.1%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around inf

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      3. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      6. lower-exp.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{-1 \cdot s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      13. lower-neg.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)} - 1\right) \]

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in s around inf

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}}\right)} - 1\right) \]
   7. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}}\right)} - 1\right) \]

      2. lower-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right)} - 1\right) \]

      3. lower-PI.f32 99.1

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

   8. Applied rewrites 99.1%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \color{blue}{\left(1 + \frac{\pi}{s}\right)}}\right)} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-s\right) \cdot \log \left(-1 + \frac{-1}{\frac{-1}{e^{\frac{\pi}{s}} + 1} + u \cdot \left(\frac{-1}{-1 - e^{\frac{\pi}{s}}} + \frac{-1}{1 + e^{\frac{\pi}{-s}}}\right)}\right) \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\log \left(-1 + \frac{-1}{\frac{u}{-1 - e^{\frac{\pi}{-s}}}}\right) \cdot \frac{s \cdot s}{-s}\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(-1 + \frac{-1}{u \cdot \left(\frac{-1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{-1 + \left(-1 - \frac{\pi}{s}\right)}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ (exp (/ PI s)) 1.0)))
   (*
    (- s)
    (log
     (+
      -1.0
      (/
       -1.0
       (+ (/ u t_0) (+ (/ -1.0 t_0) (/ u (- -1.0 (exp (/ PI (- s)))))))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s)) + 1.0f;
	return -s * logf((-1.0f + (-1.0f / ((u / t_0) + ((-1.0f / t_0) + (u / (-1.0f - expf((((float) M_PI) / -s)))))))));
}

Julia

function code(u, s)
	t_0 = Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0))
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(Float32(u / t_0) + Float32(Float32(Float32(-1.0) / t_0) + Float32(u / Float32(Float32(-1.0) - exp(Float32(Float32(pi) / Float32(-s)))))))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s)) + single(1.0);
	tmp = -s * log((single(-1.0) + (single(-1.0) / ((u / t_0) + ((single(-1.0) / t_0) + (u / (single(-1.0) - exp((single(pi) / -s)))))))));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \left(\frac{u}{-\left(1 + e^{\frac{\pi}{s}}\right)} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]
7. Step-by-step derivation

   1. lift-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   2. lift-neg.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   3. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   4. lift-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   5. lift-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{\color{blue}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   6. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   8. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   9. lift-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   10. lift-+.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\color{blue}{\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   11. lift-neg.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\color{blue}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

8. Applied rewrites 99.0%

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

9. Final simplification 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{-1}{\frac{u}{e^{\frac{\pi}{s}} + 1} + \left(\frac{-1}{e^{\frac{\pi}{s}} + 1} + \frac{u}{-1 - e^{\frac{\pi}{-s}}}\right)}\right) \]
10. Add Preprocessing

Alternative 4: 98.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     -1.0
     (+ (/ -1.0 (+ (exp (/ PI s)) 1.0)) (/ u (- -1.0 (exp (/ PI (- s)))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + (-1.0f / ((-1.0f / (expf((((float) M_PI) / s)) + 1.0f)) + (u / (-1.0f - expf((((float) M_PI) / -s))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(Float32(Float32(-1.0) / Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0))) + Float32(u / Float32(Float32(-1.0) - exp(Float32(Float32(pi) / Float32(-s))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(-1.0) / ((single(-1.0) / (exp((single(pi) / s)) + single(1.0))) + (u / (single(-1.0) - exp((single(pi) / -s))))))));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(1 + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}\right)}{s}}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(1 + \frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}\right)\right)}}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. distribute-lft-out N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(1 + \frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \left(\mathsf{PI}\left(\right) + \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}\right)}\right)}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(1 + \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) + \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}\right)\right)\right)}\right)}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. remove-double-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(1 + \frac{\color{blue}{\mathsf{PI}\left(\right) + \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \color{blue}{\left(1 + \frac{\mathsf{PI}\left(\right) + \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + \left(1 + \color{blue}{\frac{\mathsf{PI}\left(\right) + \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}}\right)}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

5. Applied rewrites 97.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + \color{blue}{\left(1 + \frac{\pi + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, \pi \cdot \left(\pi \cdot 0.5\right)\right)}{s}}{s}\right)}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

6. Taylor expanded in s around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. lower-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{-1 \cdot s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-1 \cdot s}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. lower-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{-1 \cdot s}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   10. lower-neg.f32 98.7

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{-1}{\frac{-1}{e^{\frac{\pi}{s}} + 1} + \frac{u}{-1 - e^{\frac{\pi}{-s}}}}\right) \]
10. Add Preprocessing

Alternative 5: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{-1}{u \cdot \left(\frac{-1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{-1 + \left(-1 - \frac{\pi + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, \left(\pi \cdot \pi\right) \cdot 0.5\right)}{s}}{s}\right)}\right)}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     -1.0
     (*
      u
      (+
       (/ -1.0 (+ 1.0 (exp (/ PI (- s)))))
       (/
        -1.0
        (+
         -1.0
         (-
          -1.0
          (/
           (+
            PI
            (/
             (fma 0.16666666666666666 (/ (* PI (* PI PI)) s) (* (* PI PI) 0.5))
             s))
           s)))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + (-1.0f / (u * ((-1.0f / (1.0f + expf((((float) M_PI) / -s)))) + (-1.0f / (-1.0f + (-1.0f - ((((float) M_PI) + (fmaf(0.16666666666666666f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) / s), ((((float) M_PI) * ((float) M_PI)) * 0.5f)) / s)) / s)))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(u * Float32(Float32(Float32(-1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(-1.0) / Float32(Float32(-1.0) + Float32(Float32(-1.0) - Float32(Float32(Float32(pi) + Float32(fma(Float32(0.16666666666666666), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) / s), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(0.5))) / s)) / s))))))))))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

   2. sub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

   3. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   6. lower-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   7. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   9. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   11. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{-1 \cdot s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   12. mul-1-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   13. lower-neg.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   14. distribute-neg-frac N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)} - 1\right) \]

5. Applied rewrites 97.6%

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

6. Taylor expanded in s around -inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)}\right)}\right)} - 1\right) \]

   2. unsub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

   3. lower--.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}}\right)}\right)} - 1\right) \]

8. Applied rewrites 97.1%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-\left(\pi + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, 0.5 \cdot \left(\pi \cdot \pi\right)\right)}{s}\right)}{s}\right)}}\right)} - 1\right) \]

9. Final simplification 97.1%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{-1}{u \cdot \left(\frac{-1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{-1 + \left(-1 - \frac{\pi + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, \left(\pi \cdot \pi\right) \cdot 0.5\right)}{s}}{s}\right)}\right)}\right) \]
10. Add Preprocessing

Alternative 6: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{-1}{u \cdot \left(\frac{-1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{-1 + \left(-1 + \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\pi\right)}{s}\right)}\right)}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     -1.0
     (*
      u
      (+
       (/ -1.0 (+ 1.0 (exp (/ PI (- s)))))
       (/
        -1.0
        (+ -1.0 (+ -1.0 (/ (fma -0.5 (/ (* PI PI) s) (- PI)) s)))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + (-1.0f / (u * ((-1.0f / (1.0f + expf((((float) M_PI) / -s)))) + (-1.0f / (-1.0f + (-1.0f + (fmaf(-0.5f, ((((float) M_PI) * ((float) M_PI)) / s), -((float) M_PI)) / s)))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(u * Float32(Float32(Float32(-1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(-1.0) / Float32(Float32(-1.0) + Float32(Float32(-1.0) + Float32(fma(Float32(-0.5), Float32(Float32(Float32(pi) * Float32(pi)) / s), Float32(-Float32(pi))) / s))))))))))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

   2. sub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

   3. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   6. lower-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   7. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   9. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   11. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{-1 \cdot s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   12. mul-1-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   13. lower-neg.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

   14. distribute-neg-frac N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)} - 1\right) \]

5. Applied rewrites 97.6%

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

6. Taylor expanded in s around -inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)}\right)}\right)} - 1\right) \]

   2. unsub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

   3. lower--.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}}\right)}\right)} - 1\right) \]

   5. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\color{blue}{\frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s} + -1 \cdot \mathsf{PI}\left(\right)}}{s}\right)}\right)} - 1\right) \]

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-PI.f32 96.4

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\color{blue}{\pi}\right)}{s}\right)}\right)} - 1\right) \]

8. Applied rewrites 96.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\pi\right)}{s}\right)}}\right)} - 1\right) \]

9. Final simplification 96.4%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{-1}{u \cdot \left(\frac{-1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{-1 + \left(-1 + \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\pi\right)}{s}\right)}\right)}\right) \]
10. Add Preprocessing

Alternative 7: 69.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{-s}}\\ \mathbf{if}\;s \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(-1 + \frac{-1}{u \cdot \left(\frac{-1}{1 + t\_0} + \frac{-1}{-1 + -1}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-1 + \frac{-1}{\frac{u}{-1 - t\_0}}\right) \cdot \frac{s \cdot s}{-s}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI (- s)))))
   (if (<= s 3.999999999279835e-23)
     (*
      (- s)
      (log
       (+
        -1.0
        (/ -1.0 (* u (+ (/ -1.0 (+ 1.0 t_0)) (/ -1.0 (+ -1.0 -1.0))))))))
     (* (log (+ -1.0 (/ -1.0 (/ u (- -1.0 t_0))))) (/ (* s s) (- s))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / -s));
	float tmp;
	if (s <= 3.999999999279835e-23f) {
		tmp = -s * logf((-1.0f + (-1.0f / (u * ((-1.0f / (1.0f + t_0)) + (-1.0f / (-1.0f + -1.0f)))))));
	} else {
		tmp = logf((-1.0f + (-1.0f / (u / (-1.0f - t_0))))) * ((s * s) / -s);
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / Float32(-s)))
	tmp = Float32(0.0)
	if (s <= Float32(3.999999999279835e-23))
		tmp = Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(u * Float32(Float32(Float32(-1.0) / Float32(Float32(1.0) + t_0)) + Float32(Float32(-1.0) / Float32(Float32(-1.0) + Float32(-1.0)))))))));
	else
		tmp = Float32(log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(u / Float32(Float32(-1.0) - t_0))))) * Float32(Float32(s * s) / Float32(-s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	t_0 = exp((single(pi) / -s));
	tmp = single(0.0);
	if (s <= single(3.999999999279835e-23))
		tmp = -s * log((single(-1.0) + (single(-1.0) / (u * ((single(-1.0) / (single(1.0) + t_0)) + (single(-1.0) / (single(-1.0) + single(-1.0))))))));
	else
		tmp = log((single(-1.0) + (single(-1.0) / (u / (single(-1.0) - t_0))))) * ((s * s) / -s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 4e-23

   1. Initial program 99.2%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around inf

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      3. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      6. lower-exp.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{-1 \cdot s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      13. lower-neg.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)} - 1\right) \]

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in s around inf

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{1}}\right)} - 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 37.4%

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

   if 4e-23 < s

   1. Initial program 98.8%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around inf

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      3. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      6. lower-exp.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{-1 \cdot s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      13. lower-neg.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)} - 1\right) \]

   5. Applied rewrites 96.3%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]
   6. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      3. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      5. lower--.f32 N/A

         \[\leadsto \frac{\color{blue}{0 - s \cdot s}}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{0 - \color{blue}{s \cdot s}}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      7. lower-+.f32 93.7

         \[\leadsto \frac{0 - s \cdot s}{\color{blue}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)} - 1\right) \]

   7. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\frac{0 - s \cdot s}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)} - 1\right) \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)}\right)}\right)} - 1\right) \]

      2. unsub-neg N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

      3. lower--.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}}\right)}\right)} - 1\right) \]

      5. +-commutative N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\color{blue}{\frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s} + -1 \cdot \mathsf{PI}\left(\right)}}{s}\right)}\right)} - 1\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{\mathsf{PI}\left(\right)}^{2}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}}{s}\right)}\right)} - 1\right) \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{\mathsf{PI}\left(\right)}^{2}}{s}}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      10. lower-PI.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}, \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}\right)}{s}\right)}\right)} - 1\right) \]

      13. lower-neg.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}, \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}\right)}{s}\right)}\right)} - 1\right) \]

      14. lower-PI.f32 91.5

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\color{blue}{\pi}\right)}{s}\right)}\right)} - 1\right) \]

   10. Applied rewrites 91.5%

       \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\pi\right)}{s}\right)}}\right)} - 1\right) \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       2. lower-+.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       3. lower-exp.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       4. associate-*r/ N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       5. lower-/.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       6. mul-1-neg N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

       7. lower-neg.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

       8. lower-PI.f32 93.7

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\color{blue}{\pi}}{s}}}} - 1\right) \]

   13. Applied rewrites 93.7%

       \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{-\pi}{s}}}}} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(-1 + \frac{-1}{u \cdot \left(\frac{-1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{-1 + -1}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-1 + \frac{-1}{\frac{u}{-1 - e^{\frac{\pi}{-s}}}}\right) \cdot \frac{s \cdot s}{-s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(1 - \frac{\mathsf{fma}\left(-1, \pi, 0.5 \cdot \frac{u \cdot \pi}{0.25}\right)}{s}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-1 + \frac{-1}{\frac{u}{-1 - e^{\frac{\pi}{-s}}}}\right) \cdot \frac{s \cdot s}{-s}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 3.999999999279835e-23)
   (* (- s) (log (- 1.0 (/ (fma -1.0 PI (* 0.5 (/ (* u PI) 0.25))) s))))
   (*
    (log (+ -1.0 (/ -1.0 (/ u (- -1.0 (exp (/ PI (- s))))))))
    (/ (* s s) (- s)))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 3.999999999279835e-23f) {
		tmp = -s * logf((1.0f - (fmaf(-1.0f, ((float) M_PI), (0.5f * ((u * ((float) M_PI)) / 0.25f))) / s)));
	} else {
		tmp = logf((-1.0f + (-1.0f / (u / (-1.0f - expf((((float) M_PI) / -s))))))) * ((s * s) / -s);
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(3.999999999279835e-23))
		tmp = Float32(Float32(-s) * log(Float32(Float32(1.0) - Float32(fma(Float32(-1.0), Float32(pi), Float32(Float32(0.5) * Float32(Float32(u * Float32(pi)) / Float32(0.25)))) / s))));
	else
		tmp = Float32(log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(u / Float32(Float32(-1.0) - exp(Float32(Float32(pi) / Float32(-s)))))))) * Float32(Float32(s * s) / Float32(-s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 4e-23

   1. Initial program 99.2%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.2%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \left(\frac{u}{-\left(1 + e^{\frac{\pi}{s}}\right)} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]
   7. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      2. lift-neg.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      3. lift-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      4. lift-exp.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      5. lift-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{\color{blue}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      6. lift-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      7. lift-PI.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      8. lift-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      9. lift-exp.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      10. lift-+.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\color{blue}{\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      11. lift-neg.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\color{blue}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in s around -inf

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \color{blue}{\left(\left(-1 \cdot \frac{\left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2}} + \frac{1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2}}\right) - \frac{-1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2}}}{s} + \frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)}\right) - 1\right)} \]

   11. Applied rewrites 22.2%

       \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(-1, \frac{\pi}{1}, \frac{u \cdot \pi}{0.25} \cdot 0.5\right)}{-s} + 1\right)} \]

   if 4e-23 < s

   1. Initial program 98.8%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around inf

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      3. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)}} - 1\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      6. lower-exp.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-1 \cdot s}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{-1 \cdot s}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      13. lower-neg.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)\right)} - 1\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}\right)} - 1\right) \]

   5. Applied rewrites 96.3%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]
   6. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      3. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      5. lower--.f32 N/A

         \[\leadsto \frac{\color{blue}{0 - s \cdot s}}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{0 - \color{blue}{s \cdot s}}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

      7. lower-+.f32 93.7

         \[\leadsto \frac{0 - s \cdot s}{\color{blue}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)} - 1\right) \]

   7. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\frac{0 - s \cdot s}{0 + s}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)} - 1\right) \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)}\right)}\right)} - 1\right) \]

      2. unsub-neg N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

      3. lower--.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right)} - 1\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right) + \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}}{s}}\right)}\right)} - 1\right) \]

      5. +-commutative N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\color{blue}{\frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s} + -1 \cdot \mathsf{PI}\left(\right)}}{s}\right)}\right)} - 1\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{\mathsf{PI}\left(\right)}^{2}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}}{s}\right)}\right)} - 1\right) \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{\mathsf{PI}\left(\right)}^{2}}{s}}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      10. lower-PI.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}}{s}, -1 \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)} - 1\right) \]

      12. mul-1-neg N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}, \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}\right)}{s}\right)}\right)} - 1\right) \]

      13. lower-neg.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}, \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}\right)}{s}\right)}\right)} - 1\right) \]

      14. lower-PI.f32 91.5

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\color{blue}{\pi}\right)}{s}\right)}\right)} - 1\right) \]

   10. Applied rewrites 91.5%

       \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{\pi \cdot \pi}{s}, -\pi\right)}{s}\right)}}\right)} - 1\right) \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       2. lower-+.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{\color{blue}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       3. lower-exp.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + \color{blue}{e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       4. associate-*r/ N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       5. lower-/.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

       6. mul-1-neg N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

       7. lower-neg.f32 N/A

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

       8. lower-PI.f32 93.7

          \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\color{blue}{\pi}}{s}}}} - 1\right) \]

   13. Applied rewrites 93.7%

       \[\leadsto \frac{0 - s \cdot s}{0 + s} \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{-\pi}{s}}}}} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(1 - \frac{\mathsf{fma}\left(-1, \pi, 0.5 \cdot \frac{u \cdot \pi}{0.25}\right)}{s}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-1 + \frac{-1}{\frac{u}{-1 - e^{\frac{\pi}{-s}}}}\right) \cdot \frac{s \cdot s}{-s}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 25.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(1 - \frac{\mathsf{fma}\left(-1, \pi, 0.5 \cdot \frac{u \cdot \pi}{0.25}\right)}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (- 1.0 (/ (fma -1.0 PI (* 0.5 (/ (* u PI) 0.25))) s)))))

C

float code(float u, float s) {
	return -s * logf((1.0f - (fmaf(-1.0f, ((float) M_PI), (0.5f * ((u * ((float) M_PI)) / 0.25f))) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(1.0) - Float32(fma(Float32(-1.0), Float32(pi), Float32(Float32(0.5) * Float32(Float32(u * Float32(pi)) / Float32(0.25)))) / s))))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \left(\frac{u}{-\left(1 + e^{\frac{\pi}{s}}\right)} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]
7. Step-by-step derivation

   1. lift-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   2. lift-neg.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   3. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   4. lift-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   5. lift-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{\color{blue}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   6. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{s}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   8. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + e^{\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   9. lift-exp.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\left(1 + \color{blue}{e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   10. lift-+.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\mathsf{neg}\left(\color{blue}{\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

   11. lift-neg.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \left(\frac{u}{\color{blue}{\mathsf{neg}\left(\left(1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right)}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

8. Applied rewrites 99.0%

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

9. Taylor expanded in s around -inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \color{blue}{\left(\left(-1 \cdot \frac{\left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2}} + \frac{1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2}}\right) - \frac{-1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2}}}{s} + \frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)}\right) - 1\right)} \]

11. Applied rewrites 24.6%

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(-1, \frac{\pi}{1}, \frac{u \cdot \pi}{0.25} \cdot 0.5\right)}{-s} + 1\right)} \]

12. Final simplification 24.6%

    \[\leadsto \left(-s\right) \cdot \log \left(1 - \frac{\mathsf{fma}\left(-1, \pi, 0.5 \cdot \frac{u \cdot \pi}{0.25}\right)}{s}\right) \]
13. Add Preprocessing

Alternative 10: 14.1% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(s \cdot s\right) \cdot \frac{\mathsf{fma}\left(u, \pi, -\mathsf{fma}\left(\pi, -u, \pi\right)\right)}{s}}{s} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (/ (* (* s s) (/ (fma u PI (- (fma PI (- u) PI))) s)) s))

C

float code(float u, float s) {
	return ((s * s) * (fmaf(u, ((float) M_PI), -fmaf(((float) M_PI), -u, ((float) M_PI))) / s)) / s;
}

Julia

function code(u, s)
	return Float32(Float32(Float32(s * s) * Float32(fma(u, Float32(pi), Float32(-fma(Float32(pi), Float32(-u), Float32(pi)))) / s)) / s)
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in s around inf

   \[\leadsto \color{blue}{s \cdot \left(-1 \cdot \log \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right) + -1 \cdot \frac{\left(\frac{-1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2} \cdot \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right)} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2} \cdot \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right)}\right) - \frac{1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2} \cdot \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right)}}{s}\right)} \]

9. Applied rewrites 11.5%

   \[\leadsto \color{blue}{s \cdot \frac{\mathsf{fma}\left(-1, \frac{\pi \cdot u}{1}, \mathsf{fma}\left(1, \frac{\pi}{1}, \frac{0.25 \cdot \left(\pi \cdot u\right)}{-0.25}\right)\right)}{-s}} \]

11. Applied rewrites 14.2%

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(u, \pi, -\mathsf{fma}\left(\pi, -u, \pi\right)\right)}{s} \cdot \left(s \cdot \left(-s\right)\right)}{-s}} \]

12. Final simplification 14.2%

    \[\leadsto \frac{\left(s \cdot s\right) \cdot \frac{\mathsf{fma}\left(u, \pi, -\mathsf{fma}\left(\pi, -u, \pi\right)\right)}{s}}{s} \]
13. Add Preprocessing

Alternative 11: 13.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{s} \cdot \left(\left(s \cdot s\right) \cdot \frac{-1}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (/ PI s) (* (* s s) (/ -1.0 s))))

C

float code(float u, float s) {
	return (((float) M_PI) / s) * ((s * s) * (-1.0f / s));
}

Julia

function code(u, s)
	return Float32(Float32(Float32(pi) / s) * Float32(Float32(s * s) * Float32(Float32(-1.0) / s)))
end

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) / s) * ((s * s) * (single(-1.0) / s));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

5. Taylor expanded in u around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 11.3

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

7. Applied rewrites 11.3%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\pi}{s}} \]
8. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{0 - \color{blue}{s \cdot s}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   5. +-lft-identity N/A

      \[\leadsto \frac{0 - s \cdot s}{\color{blue}{s}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   6. lift--.f32 N/A

      \[\leadsto \frac{\color{blue}{0 - s \cdot s}}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   7. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(0 - s \cdot s\right)\right)}{\mathsf{neg}\left(s\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   8. lift-neg.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(0 - s \cdot s\right)\right)}{\color{blue}{\mathsf{neg}\left(s\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   9. div-inv N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(0 - s \cdot s\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(s\right)}\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   10. lift--.f32 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(0 - s \cdot s\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(s\right)}\right) \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   11. sub0-neg N/A

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

   12. remove-double-neg N/A

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

   13. lower-*.f32 N/A

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

   14. frac-2neg N/A

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

   15. metadata-eval N/A

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

   16. lift-neg.f32 N/A

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

   17. remove-double-neg N/A

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

   18. lower-/.f32 14.0

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

9. Applied rewrites 14.0%

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

10. Final simplification 14.0%

    \[\leadsto \frac{\pi}{s} \cdot \left(\left(s \cdot s\right) \cdot \frac{-1}{s}\right) \]
11. Add Preprocessing

Alternative 12: 13.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{s} \cdot \frac{s \cdot s}{-s} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (/ PI s) (/ (* s s) (- s))))

C

float code(float u, float s) {
	return (((float) M_PI) / s) * ((s * s) / -s);
}

Julia

function code(u, s)
	return Float32(Float32(Float32(pi) / s) * Float32(Float32(s * s) / Float32(-s)))
end

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) / s) * ((s * s) / -s);
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

5. Taylor expanded in u around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 11.3

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

7. Applied rewrites 11.3%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\pi}{s}} \]
8. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{0 - \color{blue}{s \cdot s}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   5. +-lft-identity N/A

      \[\leadsto \frac{0 - s \cdot s}{\color{blue}{s}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   6. lift--.f32 N/A

      \[\leadsto \frac{\color{blue}{0 - s \cdot s}}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   7. lower-/.f32 14.0

      \[\leadsto \color{blue}{\frac{0 - s \cdot s}{s}} \cdot \frac{\pi}{s} \]

   8. lift--.f32 N/A

      \[\leadsto \frac{\color{blue}{0 - s \cdot s}}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   9. sub0-neg N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lift-neg.f32 N/A

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

   13. lower-*.f32 14.0

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

9. Applied rewrites 14.0%

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

10. Final simplification 14.0%

    \[\leadsto \frac{\pi}{s} \cdot \frac{s \cdot s}{-s} \]
11. Add Preprocessing

Alternative 13: 11.6% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \frac{s \cdot \mathsf{fma}\left(u, \pi, -\mathsf{fma}\left(\pi, -u, \pi\right)\right)}{s} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (/ (* s (fma u PI (- (fma PI (- u) PI)))) s))

C

float code(float u, float s) {
	return (s * fmaf(u, ((float) M_PI), -fmaf(((float) M_PI), -u, ((float) M_PI)))) / s;
}

Julia

function code(u, s)
	return Float32(Float32(s * fma(u, Float32(pi), Float32(-fma(Float32(pi), Float32(-u), Float32(pi))))) / s)
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in s around inf

   \[\leadsto \color{blue}{s \cdot \left(-1 \cdot \log \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right) + -1 \cdot \frac{\left(\frac{-1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2} \cdot \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right)} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2} \cdot \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right)}\right) - \frac{1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2} \cdot \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right)}}{s}\right)} \]

9. Applied rewrites 11.5%

   \[\leadsto \color{blue}{s \cdot \frac{\mathsf{fma}\left(-1, \frac{\pi \cdot u}{1}, \mathsf{fma}\left(1, \frac{\pi}{1}, \frac{0.25 \cdot \left(\pi \cdot u\right)}{-0.25}\right)\right)}{-s}} \]

11. Applied rewrites 11.5%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u, \pi, -\mathsf{fma}\left(\pi, -u, \pi\right)\right) \cdot s}{s}} \]

12. Final simplification 11.5%

    \[\leadsto \frac{s \cdot \mathsf{fma}\left(u, \pi, -\mathsf{fma}\left(\pi, -u, \pi\right)\right)}{s} \]
13. Add Preprocessing

Alternative 14: 11.6% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot -4 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (fma u (* PI -0.5) (* PI 0.25)) -4.0))

C

float code(float u, float s) {
	return fmaf(u, (((float) M_PI) * -0.5f), (((float) M_PI) * 0.25f)) * -4.0f;
}

Julia

function code(u, s)
	return Float32(fma(u, Float32(Float32(pi) * Float32(-0.5)), Float32(Float32(pi) * Float32(0.25))) * Float32(-4.0))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4} \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot -4 \]

   4. metadata-eval N/A

      \[\leadsto \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)\right) \cdot -4 \]

   5. lower-fma.f32 N/A

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

   6. distribute-rgt-out-- N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 11.5

       \[\leadsto \mathsf{fma}\left(u, \pi \cdot -0.5, \color{blue}{\pi} \cdot 0.25\right) \cdot -4 \]

7. Applied rewrites 11.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot -4} \]
8. Add Preprocessing

Alternative 15: 11.6% accurate, 42.5× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \mathsf{fma}\left(2, u, -1\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* PI (fma 2.0 u -1.0)))

C

float code(float u, float s) {
	return ((float) M_PI) * fmaf(2.0f, u, -1.0f);
}

Julia

function code(u, s)
	return Float32(Float32(pi) * fma(Float32(2.0), u, Float32(-1.0)))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in s around inf

   \[\leadsto \color{blue}{s \cdot \left(-1 \cdot \log \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right) + -1 \cdot \frac{\left(\frac{-1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2} \cdot \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right)} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2} \cdot \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right)}\right) - \frac{1}{4} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{{\left(\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)\right)}^{2} \cdot \left(\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot u + \frac{1}{2} \cdot u\right)} - 1\right)}}{s}\right)} \]

9. Applied rewrites 11.5%

   \[\leadsto \color{blue}{s \cdot \frac{\mathsf{fma}\left(-1, \frac{\pi \cdot u}{1}, \mathsf{fma}\left(1, \frac{\pi}{1}, \frac{0.25 \cdot \left(\pi \cdot u\right)}{-0.25}\right)\right)}{-s}} \]

10. Taylor expanded in s around 0

    \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{PI}\left(\right) + -2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto -1 \cdot \color{blue}{\left(-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right)} \]

    2. distribute-lft-in N/A

       \[\leadsto \color{blue}{-1 \cdot \left(-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right) + -1 \cdot \mathsf{PI}\left(\right)} \]

    3. associate-*r* N/A

       \[\leadsto \color{blue}{\left(-1 \cdot -2\right) \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)} + -1 \cdot \mathsf{PI}\left(\right) \]

    4. metadata-eval N/A

       \[\leadsto \color{blue}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \mathsf{PI}\left(\right) \]

    5. associate-*r* N/A

       \[\leadsto \color{blue}{\left(2 \cdot u\right) \cdot \mathsf{PI}\left(\right)} + -1 \cdot \mathsf{PI}\left(\right) \]

    6. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \left(2 \cdot u + -1\right)} \]

    7. lower-*.f32 N/A

       \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \left(2 \cdot u + -1\right)} \]

    8. lower-PI.f32 N/A

       \[\leadsto \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(2 \cdot u + -1\right) \]

    9. lower-fma.f32 11.5

       \[\leadsto \pi \cdot \color{blue}{\mathsf{fma}\left(2, u, -1\right)} \]

12. Applied rewrites 11.5%

    \[\leadsto \color{blue}{\pi \cdot \mathsf{fma}\left(2, u, -1\right)} \]
13. Add Preprocessing

Alternative 16: 11.4% accurate, 170.0× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- PI))

C

float code(float u, float s) {
	return -((float) M_PI);
}

Julia

function code(u, s)
	return Float32(-Float32(pi))
end

MATLAB

function tmp = code(u, s)
	tmp = -single(pi);
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   3. lower-PI.f32 11.3

      \[\leadsto -\color{blue}{\pi} \]

5. Applied rewrites 11.3%

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

Alternative 17: 10.3% accurate, 510.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. distribute-rgt-out N/A

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

   4. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{1}{2} + u \cdot \color{blue}{0}} - 1\right)\right)\right) \]

   5. mul0-rgt N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{1}{2} + \color{blue}{0}} - 1\right)\right)\right) \]

   6. metadata-eval N/A

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

   7. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\color{blue}{2} - 1\right)\right)\right) \]

   8. metadata-eval N/A

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

   9. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto s \cdot \color{blue}{0} \]

   11. lower-*.f32 10.6

       \[\leadsto \color{blue}{s \cdot 0} \]

9. Applied rewrites 10.6%

   \[\leadsto \color{blue}{s \cdot 0} \]
10. Step-by-step derivation

    1. mul0-rgt 10.6

       \[\leadsto \color{blue}{0} \]

11. Applied rewrites 10.6%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) 1.0))))