Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 13.5s

Alternatives: 9

Speedup: N/A×

• Rival profile vector overflowed, profile may not be complete

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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

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%

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

Alternative 2: 25.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log(((single(4.0) * (((single(-0.25) * (u * single(pi))) - (single(pi) * single(-0.25))) / 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. Taylor expanded in s around -inf 24.7%

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

6. Taylor expanded in u around 0 25.1%

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

   1. *-commutative 25.1%

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

8. Simplified 25.1%

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

9. Final simplification 25.1%

   \[\leadsto s \cdot \left(-\log \left(4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \pi \cdot -0.25}{s} + 1\right)\right) \]
10. 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 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 24.7%

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

6. Taylor expanded in u around 0 25.0%

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

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

8. Simplified 25.0%

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

Alternative 4: 11.7% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around -inf 11.6%

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

6. Final simplification 11.6%

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

Alternative 5: 11.7% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(u * Float32(pi)) * Float32(0.25)) - Float32(Float32(Float32(-0.25) * Float32(u * Float32(pi))) + 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) * (u * single(pi))) + (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 11.6%

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

6. Final simplification 11.6%

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

Alternative 6: 11.7% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around -inf 11.6%

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

   1. associate--r+ 11.6%

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

   2. cancel-sign-sub-inv 11.6%

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

   3. cancel-sign-sub-inv 11.6%

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

   4. metadata-eval 11.6%

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

   5. associate-*r* 11.6%

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

   6. distribute-rgt-out 11.6%

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

   7. metadata-eval 11.6%

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

   8. *-commutative 11.6%

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

   9. *-commutative 11.6%

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

   10. associate-*l* 11.6%

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

7. Simplified 11.6%

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

8. Taylor expanded in u around inf 11.6%

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

   1. clear-num 11.6%

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

   2. inv-pow 11.6%

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

10. Applied egg-rr 11.6%

    \[\leadsto \left(-s\right) \cdot \left(4 \cdot \frac{u \cdot \left(-0.5 \cdot \pi + 0.25 \cdot \color{blue}{{\left(\frac{u}{\pi}\right)}^{-1}}\right)}{s}\right) \]
11. Step-by-step derivation

    1. unpow-1 11.6%

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

12. Simplified 11.6%

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

13. Taylor expanded in s around 0 11.6%

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

    1. associate-*r/ 11.6%

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

    2. associate-*l/ 11.6%

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

    3. distribute-rgt-out 11.6%

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

15. Simplified 11.6%

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

Alternative 7: 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 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 11.6%

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

   1. associate--r+ 11.6%

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

   2. cancel-sign-sub-inv 11.6%

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

   3. cancel-sign-sub-inv 11.6%

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

   4. metadata-eval 11.6%

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

   5. associate-*r* 11.6%

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

   6. distribute-rgt-out 11.6%

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

   7. metadata-eval 11.6%

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

   8. *-commutative 11.6%

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

   9. *-commutative 11.6%

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

   10. associate-*l* 11.6%

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

7. Simplified 11.6%

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

8. Taylor expanded in u around inf 11.6%

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

   1. +-commutative 11.6%

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

   2. mul-1-neg 11.6%

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

   3. unsub-neg 11.6%

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

   4. *-commutative 11.6%

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

10. Simplified 11.6%

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

Alternative 8: 11.7% 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 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 11.6%

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

   1. associate--r+ 11.6%

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

   2. cancel-sign-sub-inv 11.6%

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

   3. cancel-sign-sub-inv 11.6%

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

   4. metadata-eval 11.6%

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

   5. associate-*r* 11.6%

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

   6. distribute-rgt-out 11.6%

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

   7. metadata-eval 11.6%

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

   8. *-commutative 11.6%

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

   9. *-commutative 11.6%

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

   10. associate-*l* 11.6%

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

7. Simplified 11.6%

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

8. Taylor expanded in u around 0 11.6%

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

   1. neg-mul-1 11.6%

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

   2. +-commutative 11.6%

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

   3. associate-*r* 11.6%

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

   4. neg-mul-1 11.6%

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

   5. distribute-rgt-out 11.6%

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

10. Simplified 11.6%

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

11. Final simplification 11.6%

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

Alternative 9: 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 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u around 0 11.3%

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

   1. neg-mul-1 11.3%

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

7. Simplified 11.3%

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

Reproduce

herbie shell --seed 2024155 
(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))))