Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 16.4s

Alternatives: 10

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (exp (/ 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 + expf((((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(pi) / Float32(-s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(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) + exp((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. Final simplification 99.0%

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

Alternative 3: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / (((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))) + (u / single(2.0))))));
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. Taylor expanded in s around inf 37.5%

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

6. Final simplification 37.5%

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

Alternative 4: 11.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \left(-1 + e^{\mathsf{log1p}\left(\frac{\pi}{u}\right)}\right) + \pi \cdot 0.25\right) - \pi \cdot -0.25\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  4.0
  (*
   u
   (-
    (+ (* -0.25 (+ -1.0 (exp (log1p (/ PI u))))) (* PI 0.25))
    (* PI -0.25)))))

C

float code(float u, float s) {
	return 4.0f * (u * (((-0.25f * (-1.0f + expf(log1pf((((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(u * Float32(Float32(Float32(Float32(-0.25) * Float32(Float32(-1.0) + exp(log1p(Float32(Float32(pi) / u))))) + Float32(Float32(pi) * Float32(0.25))) - Float32(Float32(pi) * Float32(-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 11.0%

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

6. Taylor expanded in u around inf 11.0%

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

   1. expm1-log1p-u 11.0%

      \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{u}\right)\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

   2. expm1-undefine 11.0%

      \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{u}\right)} - 1\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

8. Applied egg-rr 11.0%

   \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{u}\right)} - 1\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

9. Final simplification 11.0%

   \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \left(-1 + e^{\mathsf{log1p}\left(\frac{\pi}{u}\right)}\right) + \pi \cdot 0.25\right) - \pi \cdot -0.25\right)\right) \]
10. Add Preprocessing

Alternative 5: 11.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(u \cdot \left(\left(\pi \cdot 0.25 + -0.25 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{u}\right)\right)\right) - \pi \cdot -0.25\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  4.0
  (* u (- (+ (* PI 0.25) (* -0.25 (expm1 (log1p (/ PI u))))) (* PI -0.25)))))

C

float code(float u, float s) {
	return 4.0f * (u * (((((float) M_PI) * 0.25f) + (-0.25f * expm1f(log1pf((((float) M_PI) / u))))) - (((float) M_PI) * -0.25f)));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(u * Float32(Float32(Float32(Float32(pi) * Float32(0.25)) + Float32(Float32(-0.25) * expm1(log1p(Float32(Float32(pi) / u))))) - Float32(Float32(pi) * Float32(-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 11.0%

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

6. Taylor expanded in u around inf 11.0%

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

   1. expm1-log1p-u 11.0%

      \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{u}\right)\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

   2. expm1-undefine 11.0%

      \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{u}\right)} - 1\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

8. Applied egg-rr 11.0%

   \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{u}\right)} - 1\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]
9. Step-by-step derivation

   1. expm1-define 11.0%

      \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{u}\right)\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

10. Simplified 11.0%

    \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{u}\right)\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

11. Final simplification 11.0%

    \[\leadsto 4 \cdot \left(u \cdot \left(\left(\pi \cdot 0.25 + -0.25 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{u}\right)\right)\right) - \pi \cdot -0.25\right)\right) \]
12. Add Preprocessing

Alternative 6: 11.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (u * (((single(pi) * single(0.25)) + (single(-0.25) * (single(pi) * (single(1.0) / 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 11.0%

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

6. Taylor expanded in u around inf 11.0%

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

   1. div-inv 11.0%

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

8. Applied egg-rr 11.0%

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

9. Final simplification 11.0%

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

Alternative 7: 11.4% accurate, 25.5× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return 4.0f * (u * (((((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(u * Float32(Float32(Float32(Float32(pi) * Float32(Float32(-0.25) + Float32(u * Float32(0.25)))) / u) - Float32(Float32(pi) * Float32(-0.25)))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (u * (((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 11.0%

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

6. Taylor expanded in u around inf 11.0%

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

   1. expm1-log1p-u 11.0%

      \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{u}\right)\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

   2. expm1-undefine 11.0%

      \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{u}\right)} - 1\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

8. Applied egg-rr 11.0%

   \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{u}\right)} - 1\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]
9. Step-by-step derivation

   1. expm1-define 11.0%

      \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{u}\right)\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

10. Simplified 11.0%

    \[\leadsto 4 \cdot \left(u \cdot \left(\left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{u}\right)\right)} + 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right)\right) \]

11. Taylor expanded in u around 0 11.0%

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

    1. +-commutative 11.0%

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

    2. associate-*r* 11.0%

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

    3. distribute-rgt-out 11.0%

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

13. Simplified 11.0%

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

14. Final simplification 11.0%

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

Alternative 8: 11.4% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (u * (((single(pi) * single(0.25)) + (single(-0.25) * (single(pi) / 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 11.0%

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

6. Taylor expanded in u around inf 11.0%

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

7. Final simplification 11.0%

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

Alternative 9: 11.4% 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. Step-by-step derivation

   1. add-exp-log 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in s around -inf 11.0%

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

9. Simplified 11.0%

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

10. Final simplification 11.0%

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

Alternative 10: 11.2% 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 10.7%

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

   1. mul-1-neg 10.7%

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

7. Simplified 10.7%

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

Reproduce

herbie shell --seed 2024098 
(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))))