Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 33.9s

Alternatives: 9

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: 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

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

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(s * Float32(-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 98.8%

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

3. Final simplification 98.8%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + u \cdot \left(\frac{-1}{-1 - e^{\frac{\pi}{-s}}} + \frac{1}{-1 - e^{\frac{\pi}{s}}}\right)} + -1\right)\right) \]
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 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 3: 13.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(u \cdot \pi\right)\\ s \cdot \left(4 \cdot \left(-\mathsf{log1p}\left(\frac{t\_0 + \left(\pi \cdot 0.25 + \frac{{\left(t\_0 + \pi \cdot 0.25\right)}^{2}}{s} \cdot 0.5\right)}{s}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* -0.5 (* u PI))))
   (*
    s
    (*
     4.0
     (-
      (log1p
       (/
        (+ t_0 (+ (* PI 0.25) (* (/ (pow (+ t_0 (* PI 0.25)) 2.0) s) 0.5)))
        s)))))))

C

float code(float u, float s) {
	float t_0 = -0.5f * (u * ((float) M_PI));
	return s * (4.0f * -log1pf(((t_0 + ((((float) M_PI) * 0.25f) + ((powf((t_0 + (((float) M_PI) * 0.25f)), 2.0f) / s) * 0.5f))) / s)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(-0.5) * Float32(u * Float32(pi)))
	return Float32(s * Float32(Float32(4.0) * Float32(-log1p(Float32(Float32(t_0 + Float32(Float32(Float32(pi) * Float32(0.25)) + Float32(Float32((Float32(t_0 + Float32(Float32(pi) * Float32(0.25))) ^ Float32(2.0)) / s) * Float32(0.5)))) / s)))))
end

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf 11.6%

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

   1. log1p-expm1-u 6.4%

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

   2. cancel-sign-sub-inv 6.4%

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

   3. metadata-eval 6.4%

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

   4. fma-define 6.4%

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

   5. distribute-rgt-out-- 6.4%

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

   6. metadata-eval 6.4%

      \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(u, \pi \cdot \color{blue}{-0.5}, 0.25 \cdot \pi\right)}{s}\right)\right)\right) \]

   7. *-commutative 6.4%

      \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(u, \pi \cdot -0.5, \color{blue}{\pi \cdot 0.25}\right)}{s}\right)\right)\right) \]

5. Applied egg-rr 6.4%

   \[\leadsto \left(-s\right) \cdot \left(4 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)}{s}\right)\right)}\right) \]

6. Taylor expanded in s around inf 13.2%

   \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-0.5 \cdot \left(u \cdot \pi\right) + \left(0.25 \cdot \pi + 0.5 \cdot \frac{{\left(-0.5 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}^{2}}{s}\right)}{s}}\right)\right) \]

7. Final simplification 13.2%

   \[\leadsto s \cdot \left(4 \cdot \left(-\mathsf{log1p}\left(\frac{-0.5 \cdot \left(u \cdot \pi\right) + \left(\pi \cdot 0.25 + \frac{{\left(-0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot 0.25\right)}^{2}}{s} \cdot 0.5\right)}{s}\right)\right)\right) \]
8. Add Preprocessing

Alternative 4: 13.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot 0.25\\ s \cdot \left(\mathsf{log1p}\left(\frac{t\_0 - -0.5 \cdot \frac{{t\_0}^{2}}{s}}{s}\right) \cdot \left(-4\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ (* -0.5 (* u PI)) (* PI 0.25))))
   (* s (* (log1p (/ (- t_0 (* -0.5 (/ (pow t_0 2.0) s))) s)) (- 4.0)))))

C

float code(float u, float s) {
	float t_0 = (-0.5f * (u * ((float) M_PI))) + (((float) M_PI) * 0.25f);
	return s * (log1pf(((t_0 - (-0.5f * (powf(t_0, 2.0f) / s))) / s)) * -4.0f);
}

Julia

function code(u, s)
	t_0 = Float32(Float32(Float32(-0.5) * Float32(u * Float32(pi))) + Float32(Float32(pi) * Float32(0.25)))
	return Float32(s * Float32(log1p(Float32(Float32(t_0 - Float32(Float32(-0.5) * Float32((t_0 ^ Float32(2.0)) / s))) / s)) * Float32(-Float32(4.0))))
end

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf 11.6%

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

   1. log1p-expm1-u 6.4%

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

   2. cancel-sign-sub-inv 6.4%

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

   3. metadata-eval 6.4%

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

   4. fma-define 6.4%

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

   5. distribute-rgt-out-- 6.4%

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

   6. metadata-eval 6.4%

      \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(u, \pi \cdot \color{blue}{-0.5}, 0.25 \cdot \pi\right)}{s}\right)\right)\right) \]

   7. *-commutative 6.4%

      \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(u, \pi \cdot -0.5, \color{blue}{\pi \cdot 0.25}\right)}{s}\right)\right)\right) \]

5. Applied egg-rr 6.4%

   \[\leadsto \left(-s\right) \cdot \left(4 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)}{s}\right)\right)}\right) \]

6. Taylor expanded in s around -inf 13.2%

   \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\color{blue}{-1 \cdot \frac{-1 \cdot \left(-0.5 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right) + -0.5 \cdot \frac{{\left(-0.5 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}^{2}}{s}}{s}}\right)\right) \]

7. Final simplification 13.2%

   \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\left(-0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot 0.25\right) - -0.5 \cdot \frac{{\left(-0.5 \cdot \left(u \cdot \pi\right) + \pi \cdot 0.25\right)}^{2}}{s}}{s}\right) \cdot \left(-4\right)\right) \]
8. Add Preprocessing

Alternative 5: 13.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.25 + u \cdot -0.5\right)\\ s \cdot \left(4 \cdot \left(-\mathsf{log1p}\left(\frac{t\_0 + 0.5 \cdot \frac{{t\_0}^{2}}{s}}{s}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* PI (+ 0.25 (* u -0.5)))))
   (* s (* 4.0 (- (log1p (/ (+ t_0 (* 0.5 (/ (pow t_0 2.0) s))) s)))))))

C

float code(float u, float s) {
	float t_0 = ((float) M_PI) * (0.25f + (u * -0.5f));
	return s * (4.0f * -log1pf(((t_0 + (0.5f * (powf(t_0, 2.0f) / s))) / s)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(pi) * Float32(Float32(0.25) + Float32(u * Float32(-0.5))))
	return Float32(s * Float32(Float32(4.0) * Float32(-log1p(Float32(Float32(t_0 + Float32(Float32(0.5) * Float32((t_0 ^ Float32(2.0)) / s))) / s)))))
end

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf 11.6%

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

   1. log1p-expm1-u 6.4%

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

   2. cancel-sign-sub-inv 6.4%

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

   3. metadata-eval 6.4%

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

   4. fma-define 6.4%

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

   5. distribute-rgt-out-- 6.4%

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

   6. metadata-eval 6.4%

      \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(u, \pi \cdot \color{blue}{-0.5}, 0.25 \cdot \pi\right)}{s}\right)\right)\right) \]

   7. *-commutative 6.4%

      \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(u, \pi \cdot -0.5, \color{blue}{\pi \cdot 0.25}\right)}{s}\right)\right)\right) \]

5. Applied egg-rr 6.4%

   \[\leadsto \left(-s\right) \cdot \left(4 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)}{s}\right)\right)}\right) \]

6. Taylor expanded in s around inf 13.2%

   \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-0.5 \cdot \left(u \cdot \pi\right) + \left(0.25 \cdot \pi + 0.5 \cdot \frac{{\left(-0.5 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}^{2}}{s}\right)}{s}}\right)\right) \]
7. Step-by-step derivation

8. Simplified 13.2%

   \[\leadsto \left(-s\right) \cdot \left(4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{0.5 \cdot \frac{{\left(\pi \cdot \left(-0.5 \cdot u + 0.25\right)\right)}^{2}}{s} + \pi \cdot \left(-0.5 \cdot u + 0.25\right)}{s}}\right)\right) \]

9. Final simplification 13.2%

   \[\leadsto s \cdot \left(4 \cdot \left(-\mathsf{log1p}\left(\frac{\pi \cdot \left(0.25 + u \cdot -0.5\right) + 0.5 \cdot \frac{{\left(\pi \cdot \left(0.25 + u \cdot -0.5\right)\right)}^{2}}{s}}{s}\right)\right)\right) \]
10. Add Preprocessing

Alternative 6: 11.6% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. clear-num 98.8%

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

   2. inv-pow 98.8%

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

6. Applied egg-rr 98.8%

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

   1. unpow-1 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in s around -inf 11.6%

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

11. Simplified 11.6%

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

12. Final simplification 11.6%

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

Alternative 7: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around -inf 11.6%

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

4. Taylor expanded in u around 0 11.6%

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

   1. neg-mul-1 11.6%

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

   2. unsub-neg 11.6%

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

   3. *-commutative 11.6%

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

   4. distribute-rgt-out-- 11.6%

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

   5. metadata-eval 11.6%

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

   6. associate-*r* 11.6%

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

   7. *-commutative 11.6%

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

   8. metadata-eval 11.6%

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

   9. distribute-rgt-out-- 11.6%

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

   10. distribute-rgt-out-- 11.6%

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

   11. metadata-eval 11.6%

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

   12. *-commutative 11.6%

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

   13. associate-*r* 11.6%

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

   14. metadata-eval 11.6%

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

6. Simplified 11.6%

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

7. Final simplification 11.6%

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

Alternative 8: 11.3% accurate, 72.2× speedup

Math

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

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(pi)) / Float32(-s))
end

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 11.3%

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

   1. associate-*r/ 11.3%

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

7. Applied egg-rr 11.3%

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

8. Final simplification 11.3%

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

Alternative 9: 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 11.3%

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

   1. mul-1-neg 11.3%

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

7. Simplified 11.3%

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

8. Final simplification 11.3%

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

Reproduce

herbie shell --seed 2024053 
(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))))