Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 23.2s

Alternatives: 10

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, 0.8× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (exp (- (exp (log1p (/ PI s))) 1.0))))))
     -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((expf(log1pf((((float) M_PI) / s))) - 1.0f)))))) + -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(exp(log1p(Float32(Float32(pi) / s))) - Float32(1.0))))))) + Float32(-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. Step-by-step derivation

   1. expm1-log1p-u 98.9%

      \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}}}} + -1\right)\right) \]

   2. expm1-undefine 98.9%

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

6. Applied egg-rr 98.9%

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

Alternative 2: 98.9% 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 98.9%

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

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. Add Preprocessing

Alternative 3: 25.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-1.0) * ((single(4.0) * ((s * (u * ((single(-0.25) * single(pi)) - (single(0.25) * single(pi))))) / single(pi))) + (s * (log(single(pi)) + (single(-1.0) * log(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 24.7%

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

6. Taylor expanded in s around 0 24.8%

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

7. Taylor expanded in u around 0 25.0%

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

Alternative 4: 25.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-1.0) * (s * (log(single(pi)) + (single(-1.0) * log(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 24.7%

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

6. Taylor expanded in s around 0 24.8%

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

7. Taylor expanded in u around 0 25.0%

   \[\leadsto -1 \cdot \color{blue}{\left(s \cdot \left(\log \pi + -1 \cdot \log s\right)\right)} \]
8. Add Preprocessing

Alternative 5: 25.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\pi}{s}\\ -4 \cdot \frac{s \cdot \frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}{s}}{t\_0} + -1 \cdot \left(s \cdot \log t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ PI s))))
   (+
    (* -4.0 (/ (* s (/ (* u (- (* -0.25 PI) (* 0.25 PI))) s)) t_0))
    (* -1.0 (* s (log t_0))))))

C

float code(float u, float s) {
	float t_0 = 1.0f + (((float) M_PI) / s);
	return (-4.0f * ((s * ((u * ((-0.25f * ((float) M_PI)) - (0.25f * ((float) M_PI)))) / s)) / t_0)) + (-1.0f * (s * logf(t_0)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + Float32(Float32(pi) / s))
	return Float32(Float32(Float32(-4.0) * Float32(Float32(s * Float32(Float32(u * Float32(Float32(Float32(-0.25) * Float32(pi)) - Float32(Float32(0.25) * Float32(pi)))) / s)) / t_0)) + Float32(Float32(-1.0) * Float32(s * log(t_0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + (single(pi) / s);
	tmp = (single(-4.0) * ((s * ((u * ((single(-0.25) * single(pi)) - (single(0.25) * single(pi)))) / s)) / t_0)) + (single(-1.0) * (s * log(t_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. Taylor expanded in s around -inf 24.7%

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

6. Taylor expanded in u around 0 24.9%

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

7. Taylor expanded in s around 0 24.9%

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

Alternative 6: 25.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(1.0) + (single(4.0) * (((single(-0.25) * (u * single(pi))) - (single(-0.25) * single(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 24.7%

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

6. Taylor expanded in u around 0 24.9%

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

Alternative 7: 25.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-1.0) * (s * log((single(1.0) + (single(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 24.7%

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

6. Taylor expanded in u around 0 24.9%

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

Alternative 8: 11.6% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(0.25) * (u * single(pi))) - (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 11.3%

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

   1. metadata-eval 11.3%

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

   2. cancel-sign-sub-inv 11.3%

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

   3. associate-*r* 11.3%

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

   4. distribute-rgt-out-- 11.3%

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

7. Applied egg-rr 11.3%

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

Alternative 9: 11.6% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (((single(pi) * u) * single(0.5)) + (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 11.3%

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

   1. associate--r+ 11.3%

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

   2. cancel-sign-sub-inv 11.3%

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

   3. distribute-rgt-out-- 11.3%

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

   4. *-commutative 11.3%

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

   5. metadata-eval 11.3%

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

   6. metadata-eval 11.3%

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

   7. *-commutative 11.3%

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

7. Applied egg-rr 11.3%

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

Alternative 10: 11.3% accurate, 144.3× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \pi \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-1.0) * 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 u around 0 11.0%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024034 -o generate:simplify
(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))))