Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.4s

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%

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

5. Final simplification 98.9%

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

Alternative 2: 25.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * (s * u)) + ((s * (log(s) - log(single(pi)))) - ((s ^ single(2.0)) / 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%

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

5. Taylor expanded in s around inf 24.9%

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

   1. +-commutative 24.9%

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

   2. fma-def 24.9%

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

7. Simplified 24.9%

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

8. Taylor expanded in u around 0 25.1%

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

   1. +-commutative 25.1%

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

   2. mul-1-neg 25.1%

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

   3. unsub-neg 25.1%

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

   4. associate-*r/ 25.1%

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

   5. associate-*r* 25.1%

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

   6. log1p-def 25.1%

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

10. Simplified 25.1%

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

11. Taylor expanded in s around 0 25.1%

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

12. Taylor expanded in s around 0 25.1%

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

    1. *-commutative 25.1%

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

14. Simplified 25.1%

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

15. Final simplification 25.1%

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

Alternative 3: 25.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Taylor expanded in s around inf 24.9%

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

   1. +-commutative 24.9%

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

   2. fma-def 24.9%

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

7. Simplified 24.9%

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

8. Taylor expanded in u around 0 25.1%

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

   1. +-commutative 25.1%

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

   2. mul-1-neg 25.1%

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

   3. unsub-neg 25.1%

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

   4. associate-*r/ 25.1%

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

   5. associate-*r* 25.1%

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

   6. log1p-def 25.1%

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

10. Simplified 25.1%

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

11. Taylor expanded in s around 0 25.1%

    \[\leadsto \color{blue}{2 \cdot \left(s \cdot u\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
12. Step-by-step derivation

    1. *-commutative 25.1%

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

13. Simplified 25.1%

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

14. Final simplification 25.1%

    \[\leadsto 2 \cdot \left(s \cdot u\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
15. Add Preprocessing

Alternative 4: 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%

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

5. Taylor expanded in s around inf 24.9%

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

   1. +-commutative 24.9%

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

   2. fma-def 24.9%

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

7. Simplified 24.9%

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

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. mul-1-neg 25.0%

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

   2. distribute-rgt-neg-in 25.0%

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

   3. log1p-def 25.0%

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

10. Simplified 25.0%

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

11. Final simplification 25.0%

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

Alternative 5: 11.5% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

MATLAB

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

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

5. Taylor expanded in s around inf 11.7%

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

6. Final simplification 11.7%

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

Alternative 6: 11.5% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around -inf 11.7%

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

   1. *-commutative 11.7%

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

   2. associate-*l* 11.7%

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

   3. +-commutative 11.7%

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

   4. associate-*r* 11.7%

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

   5. distribute-rgt-out 11.7%

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

   6. *-commutative 11.7%

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

7. Simplified 11.7%

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

8. Final simplification 11.7%

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

Alternative 7: 11.5% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around inf 11.7%

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

   1. associate--r+ 11.7%

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

   2. cancel-sign-sub-inv 11.7%

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

   3. distribute-rgt-out-- 11.7%

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

   4. *-commutative 11.7%

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

   5. metadata-eval 11.7%

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

   6. metadata-eval 11.7%

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

   7. *-commutative 11.7%

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

7. Simplified 11.7%

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

8. Final simplification 11.7%

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

Alternative 8: 11.5% 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 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%

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

5. Taylor expanded in s around inf 11.7%

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

   1. associate--r+ 11.7%

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

   2. cancel-sign-sub-inv 11.7%

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

   3. distribute-rgt-out-- 11.7%

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

   4. *-commutative 11.7%

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

   5. metadata-eval 11.7%

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

   6. metadata-eval 11.7%

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

   7. *-commutative 11.7%

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

7. Simplified 11.7%

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

   1. associate-*l* 11.7%

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

   2. distribute-lft-out 11.7%

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

9. Applied egg-rr 11.7%

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

10. Final simplification 11.7%

    \[\leadsto 4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right) \]
11. 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.4%

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

   1. neg-mul-1 11.4%

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

7. Simplified 11.4%

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

8. Final simplification 11.4%

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

Reproduce

herbie shell --seed 2024026 
(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))))