Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 23.3s

Alternatives: 8

Speedup: N/A×

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

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

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 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%

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

Alternative 2: 25.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(-log(fma(Float32(-4.0), Float32(Float32(Float32(pi) / s) * Float32(-0.25)), 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. Taylor expanded in s around inf 24.9%

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

   1. +-commutative 24.9%

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

   2. fma-def 24.9%

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

7. Simplified 24.9%

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

8. Taylor expanded in u around 0 25.0%

   \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(-4, \color{blue}{-0.25 \cdot \frac{\pi}{s}}, 1\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative 25.0%

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

10. Simplified 25.0%

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

11. Final simplification 25.0%

    \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(-4, \frac{\pi}{s} \cdot -0.25, 1\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 24.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return s * -logf((1.0f + (4.0f * (((-0.25f * (u * ((float) M_PI))) - (((u * ((float) M_PI)) * 0.25f) + (((float) M_PI) * -0.25f))) / 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(Float32(u * Float32(pi)) * Float32(0.25)) + Float32(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))) - (((u * single(pi)) * single(0.25)) + (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. Taylor expanded in s around -inf 24.9%

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

6. Final simplification 24.9%

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

Alternative 4: 11.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-4.0) * Float32(Float32(pi) * Float32(Float32(u * Float32(-0.25)) - fma(u, Float32(0.25), Float32(-0.25)))))
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 11.3%

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

   1. *-commutative 11.3%

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

   2. associate-*l* 11.3%

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

   3. +-commutative 11.3%

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

   4. associate-*r* 11.3%

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

   5. distribute-rgt-out 11.3%

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

   6. *-commutative 11.3%

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

7. Simplified 11.3%

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

   1. sub-neg 11.3%

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

   2. fma-def 11.3%

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

9. Applied egg-rr 11.3%

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

    1. sub-neg 11.3%

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

    2. *-commutative 11.3%

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

    3. associate-*r* 11.3%

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

    4. *-commutative 11.3%

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

    5. associate-*r* 11.3%

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

    6. distribute-rgt-out-- 11.3%

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

    7. *-commutative 11.3%

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

11. Simplified 11.3%

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

12. Final simplification 11.3%

    \[\leadsto -4 \cdot \left(\pi \cdot \left(u \cdot -0.25 - \mathsf{fma}\left(u, 0.25, -0.25\right)\right)\right) \]
13. Add Preprocessing

Alternative 5: 11.6% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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 11.3%

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

   1. *-commutative 11.3%

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

   2. associate-*l* 11.3%

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

   3. +-commutative 11.3%

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

   4. associate-*r* 11.3%

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

   5. distribute-rgt-out 11.3%

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

   6. *-commutative 11.3%

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

7. Simplified 11.3%

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

8. Final simplification 11.3%

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

Alternative 6: 11.6% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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 11.3%

   \[\leadsto \color{blue}{4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right)} \]
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. Simplified 11.3%

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

8. Final simplification 11.3%

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

Alternative 7: 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 11.3%

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

   1. *-commutative 11.3%

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

   2. associate-*l* 11.3%

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

   3. +-commutative 11.3%

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

   4. associate-*r* 11.3%

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

   5. distribute-rgt-out 11.3%

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

   6. *-commutative 11.3%

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

7. Simplified 11.3%

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

   1. sub-neg 11.3%

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

   2. fma-def 11.3%

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

9. Applied egg-rr 11.3%

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

    1. sub-neg 11.3%

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

    2. *-commutative 11.3%

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

    3. associate-*r* 11.3%

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

    4. *-commutative 11.3%

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

    5. associate-*r* 11.3%

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

    6. distribute-rgt-out-- 11.3%

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

    7. *-commutative 11.3%

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

11. Simplified 11.3%

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

12. Taylor expanded in u around 0 11.3%

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

    1. +-commutative 11.3%

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

    2. associate-*r* 11.3%

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

    3. distribute-rgt-out 11.3%

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

14. Simplified 11.3%

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

15. Final simplification 11.3%

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

Alternative 8: 11.4% 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 11.0%

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

   1. neg-mul-1 11.0%

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

7. Simplified 11.0%

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

8. Final simplification 11.0%

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

Reproduce

herbie shell --seed 2024027 
(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))))