Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.1s

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: 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

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

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(Float32(-s) * 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 98.9%

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

3. Simplified 98.9%

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

Alternative 2: 25.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (- (* 2.0 (* u (/ PI (+ 1.0 (/ PI s))))) (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return (2.0f * (u * (((float) M_PI) / (1.0f + (((float) M_PI) / s))))) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(Float32(2.0) * Float32(u * Float32(Float32(pi) / Float32(Float32(1.0) + Float32(Float32(pi) / s))))) - Float32(s * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-define 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in u around 0 25.0%

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

   1. +-commutative 25.0%

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

   2. mul-1-neg 25.0%

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

   3. unsub-neg 25.0%

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

   4. associate-/l* 25.0%

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

   5. log1p-define 25.0%

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

10. Simplified 25.0%

    \[\leadsto \color{blue}{2 \cdot \left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]
11. Add Preprocessing

Alternative 3: 25.1% accurate, 4.1× speedup

Math

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

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(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-define 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in u around 0 25.0%

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

   1. associate-*r* 25.0%

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

   2. neg-mul-1 25.0%

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

   3. log1p-define 25.0%

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

10. Simplified 25.0%

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

Alternative 4: 11.8% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. add-sqr-sqrt 98.0%

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

   2. distribute-rgt-neg-in 98.0%

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

6. Applied egg-rr 98.0%

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

   1. distribute-rgt-neg-out 98.0%

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

   2. add-sqr-sqrt 98.9%

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

   3. pow1 98.9%

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

   4. metadata-eval 98.9%

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

   5. pow-div 61.8%

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

   6. pow1 61.8%

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

   7. distribute-frac-neg 61.8%

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

   8. associate-*l/ 61.5%

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

8. Applied egg-rr 61.5%

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

9. Taylor expanded in s around inf 12.3%

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

10. Final simplification 12.3%

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

Alternative 5: 11.8% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. add-sqr-sqrt 98.0%

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

   2. distribute-rgt-neg-in 98.0%

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

6. Applied egg-rr 98.0%

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

   1. distribute-rgt-neg-out 98.0%

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

   2. add-sqr-sqrt 98.9%

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

   3. pow1 98.9%

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

   4. metadata-eval 98.9%

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

   5. pow-div 61.8%

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

   6. pow1 61.8%

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

   7. distribute-frac-neg 61.8%

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

   8. associate-*l/ 61.5%

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

8. Applied egg-rr 61.5%

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

9. Taylor expanded in s around inf 12.3%

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

10. Taylor expanded in u around inf 12.3%

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

    1. +-commutative 12.3%

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

    2. mul-1-neg 12.3%

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

    3. unsub-neg 12.3%

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

    4. distribute-rgt-out-- 12.3%

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

    5. metadata-eval 12.3%

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

    6. associate-/l* 12.3%

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

12. Simplified 12.3%

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

Alternative 6: 11.8% 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 12.3%

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

   1. associate--r+ 12.3%

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

   2. cancel-sign-sub-inv 12.3%

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

   3. distribute-rgt-out-- 12.3%

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

   4. *-commutative 12.3%

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

   5. metadata-eval 12.3%

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

   6. metadata-eval 12.3%

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

   7. *-commutative 12.3%

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

7. Simplified 12.3%

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

8. Taylor expanded in u around inf 11.8%

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

9. Taylor expanded in u around inf 12.3%

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

    1. +-commutative 12.3%

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

    2. mul-1-neg 12.3%

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

    3. unsub-neg 12.3%

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

11. Simplified 12.3%

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

12. Final simplification 12.3%

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

Alternative 7: 11.8% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(pi) * (single(-1.0) + (u * single(2.0)));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. add-sqr-sqrt 98.0%

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

   2. distribute-rgt-neg-in 98.0%

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

6. Applied egg-rr 98.0%

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

7. Taylor expanded in s around inf 12.3%

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

   1. associate--r+ 12.3%

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

   2. cancel-sign-sub-inv 12.3%

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

   3. distribute-rgt-out-- 12.3%

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

   4. metadata-eval 12.3%

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

   5. *-commutative 12.3%

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

   6. metadata-eval 12.3%

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

   7. +-commutative 12.3%

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

   8. *-commutative 12.3%

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

   9. *-commutative 12.3%

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

   10. associate-*r* 12.3%

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

   11. *-commutative 12.3%

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

   12. fma-undefine 12.3%

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

   13. *-lft-identity 12.3%

       \[\leadsto 4 \cdot \color{blue}{\left(1 \cdot \mathsf{fma}\left(\pi, -0.25, \pi \cdot \left(u \cdot 0.5\right)\right)\right)} \]

   14. *-lft-identity 12.3%

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

   15. fma-undefine 12.3%

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

9. Simplified 12.3%

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

10. Final simplification 12.3%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]
11. Add Preprocessing

Alternative 8: 11.6% 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 12.1%

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

   1. neg-mul-1 12.1%

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

7. Simplified 12.1%

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

Alternative 9: 10.3% accurate, 433.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 10.2%

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

6. Taylor expanded in s around 0 10.2%

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

Reproduce

herbie shell --seed 2024144 
(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))))