Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 11.2s

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, 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: 25.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = ((s * log(s)) - (s * log(single(pi)))) + (single(2.0) * ((u * single(pi)) / ((single(pi) / 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 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)} \]
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 25.1%

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

9. Taylor expanded in s around 0 25.2%

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

    1. mul-1-neg 25.2%

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

    2. unsub-neg 25.2%

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

11. Simplified 25.2%

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

    1. sub-neg 25.2%

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

    2. distribute-rgt-in 25.2%

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

13. Applied egg-rr 25.2%

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

14. Final simplification 25.2%

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

Alternative 3: 25.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * ((u * single(pi)) / ((single(pi) / s) + single(1.0)))) + (s * (log(s) - log(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 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 25.1%

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

9. Taylor expanded in s around 0 25.2%

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

    1. mul-1-neg 25.2%

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

    2. unsub-neg 25.2%

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

11. Simplified 25.2%

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

12. Final simplification 25.2%

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

Alternative 4: 25.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (s * (log(s) - log(single(pi)))) + (single(2.0) * (s * 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 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 25.1%

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

9. Taylor expanded in s around 0 25.2%

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

    1. mul-1-neg 25.2%

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

    2. unsub-neg 25.2%

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

11. Simplified 25.2%

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

12. Taylor expanded in s around 0 25.1%

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

13. Final simplification 25.1%

    \[\leadsto s \cdot \left(\log s - \log \pi\right) + 2 \cdot \left(s \cdot u\right) \]
14. Add Preprocessing

Alternative 5: 25.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * ((u * single(pi)) / ((single(pi) / s) + single(1.0)))) - (s * log((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 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 25.1%

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

9. Taylor expanded in s around 0 25.2%

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

    1. mul-1-neg 25.2%

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

    2. unsub-neg 25.2%

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

11. Simplified 25.2%

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

    1. pow1 25.2%

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

    2. diff-log 25.1%

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

13. Applied egg-rr 25.1%

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

    1. unpow1 25.1%

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

15. Simplified 25.1%

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

16. Final simplification 25.1%

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

Alternative 6: 25.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * (s * u)) - (s * log(((single(pi) / 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 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)} \]
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 25.1%

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

9. Taylor expanded in s around 0 25.1%

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

10. Final simplification 25.1%

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

Alternative 7: 25.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log(((single(pi) / 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 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)} \]
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. Final simplification 25.0%

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

Alternative 8: 25.2% accurate, 4.1× speedup

Math

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

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(s * Float32(-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 24.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)} \]
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. mul-1-neg 25.0%

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

   2. log1p-define 25.0%

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

   3. *-commutative 25.0%

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

   4. distribute-rgt-neg-in 25.0%

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

10. Simplified 25.0%

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

11. Final simplification 25.0%

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

Alternative 9: 12.2% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * ((u * single(pi)) * single(0.5));
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 11.9%

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

   1. associate--r+ 11.9%

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

   2. cancel-sign-sub-inv 11.9%

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

   3. metadata-eval 11.9%

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

   4. cancel-sign-sub-inv 11.9%

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

   5. associate-*r* 11.9%

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

   6. distribute-rgt-out 11.9%

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

   7. *-commutative 11.9%

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

   8. metadata-eval 11.9%

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

   9. *-commutative 11.9%

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

   10. associate-*l* 11.9%

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

7. Simplified 11.9%

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

8. Taylor expanded in u around inf 5.1%

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

   1. associate-*r* 5.1%

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

10. Simplified 5.1%

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

    1. add-sqr-sqrt -0.0%

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

    2. sqrt-unprod 12.1%

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

    3. associate-*l* 12.1%

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

    4. associate-*l* 12.1%

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

    5. swap-sqr 12.1%

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

    6. metadata-eval 12.1%

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

    7. pow2 12.1%

       \[\leadsto -4 \cdot \sqrt{0.25 \cdot \color{blue}{{\left(u \cdot \pi\right)}^{2}}} \]

12. Applied egg-rr 12.1%

    \[\leadsto -4 \cdot \color{blue}{\sqrt{0.25 \cdot {\left(u \cdot \pi\right)}^{2}}} \]
13. Step-by-step derivation

    1. *-commutative 12.1%

       \[\leadsto -4 \cdot \sqrt{\color{blue}{{\left(u \cdot \pi\right)}^{2} \cdot 0.25}} \]

    2. sqrt-prod 12.1%

       \[\leadsto -4 \cdot \color{blue}{\left(\sqrt{{\left(u \cdot \pi\right)}^{2}} \cdot \sqrt{0.25}\right)} \]

    3. sqrt-pow1 12.1%

       \[\leadsto -4 \cdot \left(\color{blue}{{\left(u \cdot \pi\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{0.25}\right) \]

    4. metadata-eval 12.1%

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

    5. pow1 12.1%

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

    6. *-commutative 12.1%

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

    7. metadata-eval 12.1%

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

14. Applied egg-rr 12.1%

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

15. Final simplification 12.1%

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

Alternative 10: 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 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 11.6%

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

   1. neg-mul-1 11.6%

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

7. Simplified 11.6%

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

Reproduce

herbie shell --seed 2024106 
(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))))