Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 18.9s

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, 0.8× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ PI (- s)))))
      (/ (- 1.0 u) (+ 1.0 (exp (+ (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(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(exp(log1p(Float32(Float32(pi) / s))) + Float32(-1.0))))))) + Float32(-1.0))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. expm1-log1p-u 99.1%

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

   2. expm1-udef 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + exp((single(pi) / s));
	tmp = -s * log((single(-1.0) + (single(1.0) / ((single(1.0) / t_0) + (u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) + (single(-1.0) / t_0)))))));
end

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 3: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 4: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 38.5%

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

6. Final simplification 38.5%

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

Alternative 5: 24.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. expm1-log1p-u 99.1%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in s around -inf 25.1%

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

8. Final simplification 25.1%

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

Alternative 6: 11.6% 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 11.4%

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

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. cancel-sign-sub-inv 11.4%

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

   4. metadata-eval 11.4%

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

   5. associate-*r* 11.4%

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

   6. distribute-rgt-out 11.4%

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

   7. metadata-eval 11.4%

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

   8. *-commutative 11.4%

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

   9. associate-*r* 11.4%

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

   10. *-commutative 11.4%

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

7. Simplified 11.4%

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

   1. +-commutative 11.4%

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

   2. fma-def 11.4%

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

   3. fma-def 11.4%

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

9. Applied egg-rr 11.4%

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

10. Taylor expanded in u around 0 11.4%

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

    1. +-commutative 11.4%

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

    2. associate-*r* 11.4%

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

    3. distribute-rgt-out 11.4%

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

12. Simplified 11.4%

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

13. Final simplification 11.4%

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

Alternative 7: 11.3% accurate, 54.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. add-log-exp 23.6%

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

   2. *-commutative 23.6%

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

   3. neg-log 23.6%

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

   4. exp-to-pow 23.6%

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

6. Applied egg-rr 23.6%

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

7. Taylor expanded in u around 0 5.9%

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

   1. log-pow 5.9%

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

   2. rec-exp 6.0%

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

   3. mul-1-neg 6.0%

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

   4. rem-log-exp 11.3%

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

   5. mul-1-neg 11.3%

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

   6. distribute-neg-frac 11.3%

      \[\leadsto s \cdot \color{blue}{\frac{-\pi}{s}} \]

9. Simplified 11.3%

   \[\leadsto \color{blue}{s \cdot \frac{-\pi}{s}} \]
10. Step-by-step derivation

    1. expm1-log1p-u -0.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \frac{-\pi}{s}\right)\right)} \]

    2. expm1-udef -0.0%

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

    3. associate-*r/ -0.0%

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

11. Applied egg-rr -0.0%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{s \cdot \left(-\pi\right)}{s}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def -0.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{s \cdot \left(-\pi\right)}{s}\right)\right)} \]

    2. expm1-log1p 11.4%

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

13. Simplified 11.4%

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

    1. clear-num 11.4%

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

    2. inv-pow 11.4%

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

15. Applied egg-rr 11.4%

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

    1. unpow-1 11.4%

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

17. Simplified 11.4%

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

18. Final simplification 11.4%

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

Alternative 8: 11.3% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. add-log-exp 23.6%

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

   2. *-commutative 23.6%

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

   3. neg-log 23.6%

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

   4. exp-to-pow 23.6%

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

6. Applied egg-rr 23.6%

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

7. Taylor expanded in u around 0 5.9%

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

   1. log-pow 5.9%

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

   2. rec-exp 6.0%

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

   3. mul-1-neg 6.0%

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

   4. rem-log-exp 11.3%

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

   5. mul-1-neg 11.3%

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

   6. distribute-neg-frac 11.3%

      \[\leadsto s \cdot \color{blue}{\frac{-\pi}{s}} \]

9. Simplified 11.3%

   \[\leadsto \color{blue}{s \cdot \frac{-\pi}{s}} \]
10. Step-by-step derivation

    1. expm1-log1p-u -0.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \frac{-\pi}{s}\right)\right)} \]

    2. expm1-udef -0.0%

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

    3. associate-*r/ -0.0%

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

11. Applied egg-rr -0.0%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{s \cdot \left(-\pi\right)}{s}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def -0.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{s \cdot \left(-\pi\right)}{s}\right)\right)} \]

    2. expm1-log1p 11.4%

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

13. Simplified 11.4%

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

14. Final simplification 11.4%

    \[\leadsto \frac{s \cdot \left(-\pi\right)}{s} \]
15. 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 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
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 2024021 
(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))))