Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 21.0s

Alternatives: 11

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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := \frac{u}{1 + e^{\frac{\pi}{-s}}}\\ t_2 := t\_1 + \frac{1 - u}{1 + t\_0}\\ \left(-s\right) \cdot \log \left(\frac{-1 + {t\_2}^{-3}}{{t\_2}^{-2} + \left(1 + \frac{-1}{\frac{1 - u}{-1 - t\_0} - t\_1}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s)))
        (t_1 (/ u (+ 1.0 (exp (/ PI (- s))))))
        (t_2 (+ t_1 (/ (- 1.0 u) (+ 1.0 t_0)))))
   (*
    (- s)
    (log
     (/
      (+ -1.0 (pow t_2 -3.0))
      (+
       (pow t_2 -2.0)
       (+ 1.0 (/ -1.0 (- (/ (- 1.0 u) (- -1.0 t_0)) t_1)))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = u / (1.0f + expf((((float) M_PI) / -s)));
	float t_2 = t_1 + ((1.0f - u) / (1.0f + t_0));
	return -s * logf(((-1.0f + powf(t_2, -3.0f)) / (powf(t_2, -2.0f) + (1.0f + (-1.0f / (((1.0f - u) / (-1.0f - t_0)) - t_1))))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s)))))
	t_2 = Float32(t_1 + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + t_0)))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(-1.0) + (t_2 ^ Float32(-3.0))) / Float32((t_2 ^ Float32(-2.0)) + Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(1.0) - u) / Float32(Float32(-1.0) - t_0)) - t_1)))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = u / (single(1.0) + exp((single(pi) / -s)));
	t_2 = t_1 + ((single(1.0) - u) / (single(1.0) + t_0));
	tmp = -s * log(((single(-1.0) + (t_2 ^ single(-3.0))) / ((t_2 ^ single(-2.0)) + (single(1.0) + (single(-1.0) / (((single(1.0) - u) / (single(-1.0) - t_0)) - t_1))))));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

   1. clear-num 98.6%

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

   2. associate-/r/ 98.6%

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

6. Applied egg-rr 98.6%

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

8. Applied egg-rr 98.8%

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

   1. +-commutative 98.8%

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

   2. associate-*l/ 98.8%

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

   3. metadata-eval 98.8%

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

10. Simplified 98.8%

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

11. Final simplification 98.8%

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

Alternative 2: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + {\left(\sqrt{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}^{-2}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (pow
     (sqrt
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))))
     -2.0)))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + powf(sqrtf(((u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))))), -2.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + (sqrt(Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) ^ Float32(-2.0)))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (sqrt(((u / (single(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))))) ^ single(-2.0))));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

   1. clear-num 98.6%

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

   2. associate-/r/ 98.6%

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

6. Applied egg-rr 98.6%

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

   1. inv-pow 98.6%

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

   2. add-sqr-sqrt 98.6%

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

   3. unpow-prod-down 98.6%

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

8. Applied egg-rr 98.6%

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

   1. pow-sqr 98.7%

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

   2. metadata-eval 98.7%

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

10. Simplified 98.7%

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

11. Final simplification 98.7%

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

Alternative 3: 99.0% accurate, 1.3× speedup

Math

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

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 (/ 1.0 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) * (1.0f / s)))))))));
}

Julia

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

   1. clear-num 98.6%

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

   2. associate-/r/ 98.6%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 4: 99.0% accurate, 1.3× speedup

Math

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

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(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-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 98.6%

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

3. Simplified 98.6%

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

5. Final simplification 98.6%

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

Alternative 5: 97.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in s around inf 86.0%

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

6. Taylor expanded in s around 0 97.1%

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

   1. mul-1-neg 97.1%

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

   2. distribute-frac-neg2 97.1%

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

8. Simplified 97.1%

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

9. Final simplification 97.1%

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

Alternative 6: 25.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{s \cdot \left(u \cdot \left(\frac{\pi}{s} \cdot 0.25 - \frac{\pi}{s} \cdot -0.25\right)\right)}{-1 + \frac{\pi}{s}} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (-
  (*
   -4.0
   (/ (* s (* u (- (* (/ PI s) 0.25) (* (/ PI s) -0.25)))) (+ -1.0 (/ PI s))))
  (* s (log (+ 1.0 (/ PI s))))))

C

float code(float u, float s) {
	return (-4.0f * ((s * (u * (((((float) M_PI) / s) * 0.25f) - ((((float) M_PI) / s) * -0.25f)))) / (-1.0f + (((float) M_PI) / s)))) - (s * logf((1.0f + (((float) M_PI) / s))));
}

Julia

function code(u, s)
	return Float32(Float32(Float32(-4.0) * Float32(Float32(s * Float32(u * Float32(Float32(Float32(Float32(pi) / s) * Float32(0.25)) - Float32(Float32(Float32(pi) / s) * Float32(-0.25))))) / Float32(Float32(-1.0) + Float32(Float32(pi) / s)))) - Float32(s * log(Float32(Float32(1.0) + Float32(Float32(pi) / s)))))
end

MATLAB

function tmp = code(u, s)
	tmp = (single(-4.0) * ((s * (u * (((single(pi) / s) * single(0.25)) - ((single(pi) / s) * single(-0.25))))) / (single(-1.0) + (single(pi) / s)))) - (s * log((single(1.0) + (single(pi) / s))));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in s around -inf 25.4%

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

6. Taylor expanded in u around 0 25.5%

   \[\leadsto \color{blue}{-4 \cdot \frac{s \cdot \left(u \cdot \left(-0.25 \cdot \frac{\pi}{s} - 0.25 \cdot \frac{\pi}{s}\right)\right)}{1 + \frac{\pi}{s}} + -1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 25.5%

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

   2. sqrt-unprod 25.5%

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

   3. sqr-neg 25.5%

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

   4. sqrt-unprod -0.0%

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

   5. add-sqr-sqrt 25.5%

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

   6. distribute-frac-neg2 25.5%

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

8. Applied egg-rr 25.5%

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

9. Final simplification 25.5%

   \[\leadsto -4 \cdot \frac{s \cdot \left(u \cdot \left(\frac{\pi}{s} \cdot 0.25 - \frac{\pi}{s} \cdot -0.25\right)\right)}{-1 + \frac{\pi}{s}} - s \cdot \log \left(1 + \frac{\pi}{s}\right) \]
10. Add Preprocessing

Alternative 7: 25.0% 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 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in s around -inf 25.4%

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

6. Taylor expanded in u around 0 25.5%

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

   1. log1p-define 25.5%

      \[\leadsto -1 \cdot \left(s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)}\right) \]

   2. associate-*r* 25.5%

      \[\leadsto \color{blue}{\left(-1 \cdot s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

   3. neg-mul-1 25.5%

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

8. Simplified 25.5%

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

9. Final simplification 25.5%

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

Alternative 8: 11.7% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(\pi \cdot 2 - \frac{\pi}{u}\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = u * ((single(pi) * single(2.0)) - (single(pi) / u));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in s around inf 11.7%

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

   1. associate--r+ 11.7%

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

   2. cancel-sign-sub-inv 11.7%

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

   3. distribute-rgt-out-- 11.7%

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

   4. *-commutative 11.7%

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

   5. metadata-eval 11.7%

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

   6. metadata-eval 11.7%

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

   7. *-commutative 11.7%

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

7. Simplified 11.7%

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

8. Taylor expanded in u around inf 11.7%

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

   1. +-commutative 11.7%

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

   2. mul-1-neg 11.7%

      \[\leadsto u \cdot \left(2 \cdot \pi + \color{blue}{\left(-\frac{\pi}{u}\right)}\right) \]

   3. unsub-neg 11.7%

      \[\leadsto u \cdot \color{blue}{\left(2 \cdot \pi - \frac{\pi}{u}\right)} \]

   4. *-commutative 11.7%

      \[\leadsto u \cdot \left(\color{blue}{\pi \cdot 2} - \frac{\pi}{u}\right) \]

10. Simplified 11.7%

    \[\leadsto \color{blue}{u \cdot \left(\pi \cdot 2 - \frac{\pi}{u}\right)} \]
11. Add Preprocessing

Alternative 9: 11.7% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * (u * single(pi))) - single(pi);
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in s around inf 11.7%

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

   1. associate--r+ 11.7%

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

   2. cancel-sign-sub-inv 11.7%

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

   3. distribute-rgt-out-- 11.7%

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

   4. *-commutative 11.7%

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

   5. metadata-eval 11.7%

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

   6. metadata-eval 11.7%

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

   7. *-commutative 11.7%

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

7. Simplified 11.7%

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

8. Taylor expanded in u around 0 11.7%

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

   1. neg-mul-1 11.7%

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

   2. +-commutative 11.7%

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

   3. unsub-neg 11.7%

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

   4. *-commutative 11.7%

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

   5. *-commutative 11.7%

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

10. Simplified 11.7%

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

11. Final simplification 11.7%

    \[\leadsto 2 \cdot \left(u \cdot \pi\right) - \pi \]
12. Add Preprocessing

Alternative 10: 11.5% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in u around 0 11.5%

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

   1. associate-*r/ 11.6%

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

7. Applied egg-rr 11.6%

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

8. Final simplification 11.6%

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

Alternative 11: 11.5% 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 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in u around 0 11.6%

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

   1. neg-mul-1 11.6%

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

7. Simplified 11.6%

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

Reproduce

herbie shell --seed 2024132 
(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))))