Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 16.8s

Alternatives: 11

Speedup: 1.3×

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: 98.9% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(pi) / s)))))) + Float32(-1.0)))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log(((single(1.0) / ((u / (single(1.0) + exp((-single(pi) / s)))) + ((single(1.0) - u) / (single(1.0) + (single(2.71828182845904523536) ^ (single(pi) / s)))))) + single(-1.0)));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.0%

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

   2. associate-/r/ 99.0%

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

6. Applied egg-rr 99.0%

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

   1. associate-*l/ 99.0%

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

   2. *-un-lft-identity 99.0%

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

   3. *-un-lft-identity 99.0%

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

   4. exp-prod 99.0%

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

8. Applied egg-rr 99.0%

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

   1. exp-1-e 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/
      1.0
      (+
       (/ u (+ 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 / ((u / (1.0f + expf((-((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s))))))));
}

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / ((u / (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%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 84.0%

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

   1. +-commutative 84.0%

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

7. Simplified 84.0%

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

8. Taylor expanded in s around 0 97.1%

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

   1. associate-*r* 97.1%

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

   2. neg-mul-1 97.1%

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

   3. associate--l+ 97.1%

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

   4. neg-mul-1 97.1%

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

   5. distribute-neg-frac2 97.1%

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

10. Simplified 97.1%

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

11. Final simplification 97.1%

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

Alternative 4: 25.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s (- (log1p (/ PI s)))))

C

float code(float u, float s) {
	return s * -log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 84.0%

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

   1. +-commutative 84.0%

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

7. Simplified 84.0%

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

8. Taylor expanded in u around 0 25.2%

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

   1. associate-*r* 25.2%

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

   2. neg-mul-1 25.2%

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

   3. log1p-define 25.2%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Simplified 25.2%

    \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

11. Final simplification 25.2%

    \[\leadsto s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]
12. Add Preprocessing

Alternative 5: 11.6% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(\left(u \cdot \pi\right) \cdot -0.25 + \pi \cdot 0.25\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(0.25) * (u * single(pi))) - (((u * single(pi)) * single(-0.25)) + (single(pi) * single(0.25))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 12.1%

   \[\leadsto \color{blue}{4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right)} \]

6. Final simplification 12.1%

   \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(\left(u \cdot \pi\right) \cdot -0.25 + \pi \cdot 0.25\right)\right) \]
7. Add Preprocessing

Alternative 6: 16.2% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(u \cdot -4\right) \cdot \left(\frac{1}{-1 + \left(\frac{\pi}{s} + -1\right)} - -0.5\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (* (* u -4.0) (- (/ 1.0 (+ -1.0 (+ (/ PI s) -1.0))) -0.5))))

C

float code(float u, float s) {
	return s * ((u * -4.0f) * ((1.0f / (-1.0f + ((((float) M_PI) / s) + -1.0f))) - -0.5f));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(u * Float32(-4.0)) * Float32(Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(Float32(pi) / s) + Float32(-1.0)))) - Float32(-0.5))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * ((u * single(-4.0)) * ((single(1.0) / (single(-1.0) + ((single(pi) / s) + single(-1.0)))) - single(-0.5)));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 9.1%

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

6. Taylor expanded in u around 0 8.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \left(u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - 0.5\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 8.9%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - 0.5\right)\right)} \]

   2. sub-neg 8.9%

      \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \color{blue}{\left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} + \left(-0.5\right)\right)}\right) \]

   3. neg-mul-1 8.9%

      \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{1 + e^{\color{blue}{-\frac{\pi}{s}}}} + \left(-0.5\right)\right)\right) \]

   4. distribute-neg-frac2 8.9%

      \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{1 + e^{\color{blue}{\frac{\pi}{-s}}}} + \left(-0.5\right)\right)\right) \]

   5. metadata-eval 8.9%

      \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \color{blue}{-0.5}\right)\right) \]

8. Simplified 8.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + -0.5\right)\right)} \]

9. Taylor expanded in s around inf 16.3%

   \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{\pi}{s}\right)}} + -0.5\right)\right) \]
10. Step-by-step derivation

    1. neg-mul-1 16.3%

       \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{1 + \left(1 + \color{blue}{\left(-\frac{\pi}{s}\right)}\right)} + -0.5\right)\right) \]

    2. unsub-neg 16.3%

       \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{1 + \color{blue}{\left(1 - \frac{\pi}{s}\right)}} + -0.5\right)\right) \]

11. Simplified 16.3%

    \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{1 + \color{blue}{\left(1 - \frac{\pi}{s}\right)}} + -0.5\right)\right) \]

12. Final simplification 16.3%

    \[\leadsto s \cdot \left(\left(u \cdot -4\right) \cdot \left(\frac{1}{-1 + \left(\frac{\pi}{s} + -1\right)} - -0.5\right)\right) \]
13. Add Preprocessing

Alternative 7: 11.6% accurate, 28.9× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * ((single(pi) * (single(0.25) + (u * single(-0.25)))) + (u * (single(pi) * single(-0.25))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around -inf 12.1%

   \[\leadsto \color{blue}{-4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate--r+ 12.1%

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

   2. cancel-sign-sub-inv 12.1%

      \[\leadsto -4 \cdot \color{blue}{\left(\left(-0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right)} \]

   3. cancel-sign-sub-inv 12.1%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(-0.25 \cdot \left(u \cdot \pi\right) + \left(--0.25\right) \cdot \pi\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   4. metadata-eval 12.1%

      \[\leadsto -4 \cdot \left(\left(-0.25 \cdot \left(u \cdot \pi\right) + \color{blue}{0.25} \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   5. +-commutative 12.1%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(0.25 \cdot \pi + -0.25 \cdot \left(u \cdot \pi\right)\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   6. associate-*r* 12.1%

      \[\leadsto -4 \cdot \left(\left(0.25 \cdot \pi + \color{blue}{\left(-0.25 \cdot u\right) \cdot \pi}\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   7. distribute-rgt-out 12.1%

      \[\leadsto -4 \cdot \left(\color{blue}{\pi \cdot \left(0.25 + -0.25 \cdot u\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   8. *-commutative 12.1%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + \color{blue}{u \cdot -0.25}\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   9. metadata-eval 12.1%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + \color{blue}{-0.25} \cdot \left(u \cdot \pi\right)\right) \]

   10. *-commutative 12.1%

       \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + \color{blue}{\left(u \cdot \pi\right) \cdot -0.25}\right) \]

   11. associate-*l* 12.1%

       \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + \color{blue}{u \cdot \left(\pi \cdot -0.25\right)}\right) \]

7. Simplified 12.1%

   \[\leadsto \color{blue}{-4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)} \]

8. Final simplification 12.1%

   \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \]
9. Add Preprocessing

Alternative 8: 11.3% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \left(u \cdot \left(\pi \cdot \frac{0.25}{u}\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * (u * (single(pi) * (single(0.25) / u)));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around -inf 12.1%

   \[\leadsto \color{blue}{-4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate--r+ 12.1%

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

   2. cancel-sign-sub-inv 12.1%

      \[\leadsto -4 \cdot \color{blue}{\left(\left(-0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right)} \]

   3. cancel-sign-sub-inv 12.1%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(-0.25 \cdot \left(u \cdot \pi\right) + \left(--0.25\right) \cdot \pi\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   4. metadata-eval 12.1%

      \[\leadsto -4 \cdot \left(\left(-0.25 \cdot \left(u \cdot \pi\right) + \color{blue}{0.25} \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   5. +-commutative 12.1%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(0.25 \cdot \pi + -0.25 \cdot \left(u \cdot \pi\right)\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   6. associate-*r* 12.1%

      \[\leadsto -4 \cdot \left(\left(0.25 \cdot \pi + \color{blue}{\left(-0.25 \cdot u\right) \cdot \pi}\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   7. distribute-rgt-out 12.1%

      \[\leadsto -4 \cdot \left(\color{blue}{\pi \cdot \left(0.25 + -0.25 \cdot u\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   8. *-commutative 12.1%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + \color{blue}{u \cdot -0.25}\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   9. metadata-eval 12.1%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + \color{blue}{-0.25} \cdot \left(u \cdot \pi\right)\right) \]

   10. *-commutative 12.1%

       \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + \color{blue}{\left(u \cdot \pi\right) \cdot -0.25}\right) \]

   11. associate-*l* 12.1%

       \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + \color{blue}{u \cdot \left(\pi \cdot -0.25\right)}\right) \]

7. Simplified 12.1%

   \[\leadsto \color{blue}{-4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)} \]

8. Taylor expanded in u around inf 12.1%

   \[\leadsto -4 \cdot \color{blue}{\left(u \cdot \left(-0.5 \cdot \pi + 0.25 \cdot \frac{\pi}{u}\right)\right)} \]

9. Taylor expanded in u around 0 11.8%

   \[\leadsto -4 \cdot \left(u \cdot \color{blue}{\left(0.25 \cdot \frac{\pi}{u}\right)}\right) \]
10. Step-by-step derivation

    1. *-commutative 11.8%

       \[\leadsto -4 \cdot \left(u \cdot \color{blue}{\left(\frac{\pi}{u} \cdot 0.25\right)}\right) \]

    2. *-rgt-identity 11.8%

       \[\leadsto -4 \cdot \left(u \cdot \left(\frac{\color{blue}{\pi \cdot 1}}{u} \cdot 0.25\right)\right) \]

    3. associate-*r/ 11.8%

       \[\leadsto -4 \cdot \left(u \cdot \left(\color{blue}{\left(\pi \cdot \frac{1}{u}\right)} \cdot 0.25\right)\right) \]

    4. associate-*l* 11.8%

       \[\leadsto -4 \cdot \left(u \cdot \color{blue}{\left(\pi \cdot \left(\frac{1}{u} \cdot 0.25\right)\right)}\right) \]

    5. *-commutative 11.8%

       \[\leadsto -4 \cdot \left(u \cdot \left(\pi \cdot \color{blue}{\left(0.25 \cdot \frac{1}{u}\right)}\right)\right) \]

    6. associate-*r/ 11.8%

       \[\leadsto -4 \cdot \left(u \cdot \left(\pi \cdot \color{blue}{\frac{0.25 \cdot 1}{u}}\right)\right) \]

    7. metadata-eval 11.8%

       \[\leadsto -4 \cdot \left(u \cdot \left(\pi \cdot \frac{\color{blue}{0.25}}{u}\right)\right) \]

11. Simplified 11.8%

    \[\leadsto -4 \cdot \left(u \cdot \color{blue}{\left(\pi \cdot \frac{0.25}{u}\right)}\right) \]

12. Final simplification 11.8%

    \[\leadsto -4 \cdot \left(u \cdot \left(\pi \cdot \frac{0.25}{u}\right)\right) \]
13. Add Preprocessing

Alternative 9: 11.6% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * (single(pi) * (single(0.25) + (u * single(-0.5))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around -inf 12.1%

   \[\leadsto \color{blue}{-4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate--r+ 12.1%

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

   2. cancel-sign-sub-inv 12.1%

      \[\leadsto -4 \cdot \color{blue}{\left(\left(-0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right)} \]

   3. cancel-sign-sub-inv 12.1%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(-0.25 \cdot \left(u \cdot \pi\right) + \left(--0.25\right) \cdot \pi\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   4. metadata-eval 12.1%

      \[\leadsto -4 \cdot \left(\left(-0.25 \cdot \left(u \cdot \pi\right) + \color{blue}{0.25} \cdot \pi\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   5. +-commutative 12.1%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(0.25 \cdot \pi + -0.25 \cdot \left(u \cdot \pi\right)\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   6. associate-*r* 12.1%

      \[\leadsto -4 \cdot \left(\left(0.25 \cdot \pi + \color{blue}{\left(-0.25 \cdot u\right) \cdot \pi}\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   7. distribute-rgt-out 12.1%

      \[\leadsto -4 \cdot \left(\color{blue}{\pi \cdot \left(0.25 + -0.25 \cdot u\right)} + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   8. *-commutative 12.1%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + \color{blue}{u \cdot -0.25}\right) + \left(-0.25\right) \cdot \left(u \cdot \pi\right)\right) \]

   9. metadata-eval 12.1%

      \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + \color{blue}{-0.25} \cdot \left(u \cdot \pi\right)\right) \]

   10. *-commutative 12.1%

       \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + \color{blue}{\left(u \cdot \pi\right) \cdot -0.25}\right) \]

   11. associate-*l* 12.1%

       \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + \color{blue}{u \cdot \left(\pi \cdot -0.25\right)}\right) \]

7. Simplified 12.1%

   \[\leadsto \color{blue}{-4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)} \]

8. Taylor expanded in u around 0 12.1%

   \[\leadsto -4 \cdot \color{blue}{\left(-0.5 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)} \]
9. Step-by-step derivation

   1. +-commutative 12.1%

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

   2. associate-*r* 12.1%

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

   3. distribute-rgt-out 12.1%

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

10. Simplified 12.1%

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

11. Final simplification 12.1%

    \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.5\right)\right) \]
12. Add Preprocessing

Alternative 10: 11.3% accurate, 61.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\frac{s}{\pi}}{s}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

6. Applied egg-rr 6.2%

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

   1. neg-sub0 6.2%

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

8. Simplified 6.2%

   \[\leadsto \color{blue}{-\log \left({\left(-1 + \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + 0}\right)}^{s}\right)} \]
9. Step-by-step derivation

   1. log-pow 6.2%

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

   2. inv-pow 6.2%

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

   3. +-rgt-identity 6.2%

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

   4. inv-pow 6.2%

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

   5. pow-pow 6.2%

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

   6. metadata-eval 6.2%

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

   7. pow1 6.2%

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

10. Applied egg-rr 6.2%

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

    1. +-commutative 6.2%

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

    2. metadata-eval 6.2%

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

    3. sub-neg 6.2%

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

    4. associate--l+ 6.2%

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

    5. expm1-undefine 6.2%

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

    6. log1p-define 6.2%

       \[\leadsto -s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\pi}{s}\right)\right)} \]

    7. log1p-expm1 11.8%

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

12. Simplified 11.8%

    \[\leadsto -\color{blue}{s \cdot \frac{\pi}{s}} \]
13. Step-by-step derivation

    1. associate-*r/ 11.8%

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

14. Applied egg-rr 11.8%

    \[\leadsto -\color{blue}{\frac{s \cdot \pi}{s}} \]
15. Step-by-step derivation

    1. clear-num 11.8%

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

    2. inv-pow 11.8%

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

    3. *-commutative 11.8%

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

16. Applied egg-rr 11.8%

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

    1. unpow-1 11.8%

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

    2. associate-/r* 11.8%

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

18. Simplified 11.8%

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

19. Final simplification 11.8%

    \[\leadsto \frac{-1}{\frac{\frac{s}{\pi}}{s}} \]
20. Add Preprocessing

Alternative 11: 11.3% accurate, 216.5× 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%

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

3. Simplified 99.0%

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

5. Taylor expanded in u around 0 11.8%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
6. Step-by-step derivation

   1. neg-mul-1 11.8%

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

7. Simplified 11.8%

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

8. Final simplification 11.8%

   \[\leadsto -\pi \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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))))