Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 7.4s

Alternatives: 9

Speedup: 0.5×

Specification

\[\left(\frac{2328306437}{10000000000000000000} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq \frac{10651631}{10000000}\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1 (+ 1 (exp (/ PI s))))))
  (*
   (- s)
   (log
    (-
     (/ 1 (+ (* u (- (/ 1 (+ 1 (exp (/ (- PI) s)))) t_0)) t_0))
     1)))))

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1 (+ 1 (exp (/ PI s))))))
  (*
   (- s)
   (log
    (-
     (/ 1 (+ (* u (- (/ 1 (+ 1 (exp (/ (- PI) s)))) t_0)) t_0))
     1)))))

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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := \frac{u}{e^{\frac{-\pi}{s}} - -1}\\ t_2 := \frac{u}{-1 - t\_0}\\ \left(-s\right) \cdot \log \left(\frac{1}{\left(t\_2 + t\_1\right) \cdot \left(\left(t\_1 - t\_2\right) \cdot \frac{-1}{t\_2 - t\_1}\right) + \frac{1}{1 + t\_0}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (exp (/ PI s)))
       (t_1 (/ u (- (exp (/ (- PI) s)) -1)))
       (t_2 (/ u (- -1 t_0))))
  (*
   (- s)
   (log
    (-
     (/
      1
      (+
       (* (+ t_2 t_1) (* (- t_1 t_2) (/ -1 (- t_2 t_1))))
       (/ 1 (+ 1 t_0))))
     1)))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = u / (expf((-((float) M_PI) / s)) - -1.0f);
	float t_2 = u / (-1.0f - t_0);
	return -s * logf(((1.0f / (((t_2 + t_1) * ((t_1 - t_2) * (-1.0f / (t_2 - t_1)))) + (1.0f / (1.0f + t_0)))) - 1.0f));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(u / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0)))
	t_2 = Float32(u / Float32(Float32(-1.0) - t_0))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(t_2 + t_1) * Float32(Float32(t_1 - t_2) * Float32(Float32(-1.0) / Float32(t_2 - t_1)))) + Float32(Float32(1.0) / Float32(Float32(1.0) + t_0)))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = u / (exp((-single(pi) / s)) - single(-1.0));
	t_2 = u / (single(-1.0) - t_0);
	tmp = -s * log(((single(1.0) / (((t_2 + t_1) * ((t_1 - t_2) * (single(-1.0) / (t_2 - t_1)))) + (single(1.0) / (single(1.0) + t_0)))) - single(1.0)));
end

Derivation

1. Initial program 99.0%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-lft-in N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f32 N/A

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 98.9%

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

Alternative 2: 97.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  (*
 (- s)
 (log
  (-
   (/
    1
    (*
     u
     (-
      (/ 1 (+ 1 (exp (* -1 (/ PI s)))))
      (/ 1 (+ 1 (exp (/ PI s)))))))
   1))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 97.6%

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

Alternative 3: 86.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  (*
 (- s)
 (log
  (-
   (/
    1
    (+
     (+ (/ u (- (exp (/ (- PI) s)) -1)) (* (/ -1 (- (/ PI s) -2)) u))
     (/ 1 (+ 2 (/ PI s)))))
   1))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 95.2%

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

4. Applied rewrites 95.2%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 86.4%

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

7. Applied rewrites 86.4%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-rgt-in N/A

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

9. Applied rewrites 86.3%

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

Alternative 4: 86.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2 + \frac{\pi}{s}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1 (+ 2 (/ PI s)))))
  (*
   (- s)
   (log
    (-
     (/ 1 (+ (* u (- (/ 1 (+ 1 (exp (/ (- PI) s)))) t_0)) t_0))
     1)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (2.0f + (((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(2.0) + 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(2.0) + (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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 95.2%

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

4. Applied rewrites 95.2%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 86.4%

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

7. Applied rewrites 86.4%

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

Alternative 5: 85.8% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  (*
 (- s)
 (log (- (/ (- (/ PI s) -2) (- (* (/ (* 1/2 PI) s) (+ u u)) -1)) 1))))

C

float code(float u, float s) {
	return -s * logf(((((((float) M_PI) / s) - -2.0f) / ((((0.5f * ((float) M_PI)) / s) * (u + u)) - -1.0f)) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(Float32(Float32(pi) / s) - Float32(-2.0)) / Float32(Float32(Float32(Float32(Float32(0.5) * Float32(pi)) / s) * Float32(u + u)) - Float32(-1.0))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((((single(pi) / s) - single(-2.0)) / ((((single(0.5) * single(pi)) / s) * (u + u)) - single(-1.0))) - single(1.0)));
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 95.2%

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

4. Applied rewrites 95.2%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 86.4%

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

7. Applied rewrites 86.4%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. add-to-fraction N/A

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

9. Applied rewrites 85.8%

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

10. Taylor expanded in s around inf

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\color{blue}{2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s}} - -1} - 1\right) \]
11. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{2 \cdot \color{blue}{\frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} - -1} - 1\right) \]

    2. lower-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{s}} - -1} - 1\right) \]

    3. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s} - -1} - 1\right) \]

    4. lower--.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s} - -1} - 1\right) \]

    5. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s} - -1} - 1\right) \]

    6. lower-PI.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s} - -1} - 1\right) \]

    7. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s} - -1} - 1\right) \]

    8. lower-PI.f32 85.8%

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s} - -1} - 1\right) \]

12. Applied rewrites 85.8%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\color{blue}{2 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s}} - -1} - 1\right) \]
13. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{2 \cdot \color{blue}{\frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s}} - -1} - 1\right) \]

    2. count-2-rev N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\left(\frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s} + \color{blue}{\frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s}}\right) - -1} - 1\right) \]

    3. lift-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\left(\frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s} + \frac{\color{blue}{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}}{s}\right) - -1} - 1\right) \]

    4. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\left(\frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s} + \frac{\color{blue}{u} \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s}\right) - -1} - 1\right) \]

    5. associate-/l* N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\left(u \cdot \frac{\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi}{s} + \frac{\color{blue}{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}}{s}\right) - -1} - 1\right) \]

    6. lift-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\left(u \cdot \frac{\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi}{s} + \frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{\color{blue}{s}}\right) - -1} - 1\right) \]

    7. lift-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\left(u \cdot \frac{\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi}{s} + \frac{u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right)}{s}\right) - -1} - 1\right) \]

    8. associate-/l* N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\left(u \cdot \frac{\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi}{s} + u \cdot \color{blue}{\frac{\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi}{s}}\right) - -1} - 1\right) \]

    9. distribute-rgt-out N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\frac{\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi}{s} \cdot \color{blue}{\left(u + u\right)} - -1} - 1\right) \]

    10. lower-*.f32 N/A

        \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\pi}{s} - -2}{\frac{\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi}{s} \cdot \color{blue}{\left(u + u\right)} - -1} - 1\right) \]

14. Applied rewrites 85.8%

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

Alternative 6: 25.0% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  (* (- s) (log (+ 1 (/ PI s)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in s around -inf

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

4. Applied rewrites 24.8%

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

5. Taylor expanded in u around 0

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right) \]

   3. lower-PI.f32 25.0%

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

7. Applied rewrites 25.0%

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

Alternative 7: 11.5% accurate, 15.9× speedup

Math

\[4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \pi\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  (* 4 (- (* u (- (* 1/4 PI) (* -1/4 PI))) (* 1/4 PI))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

      \[\leadsto 4 \cdot \color{blue}{\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

   2. lower--.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   6. lower-PI.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   8. lower-PI.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   10. lower-PI.f32 11.5%

       \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \pi\right) \]

4. Applied rewrites 11.5%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \pi\right)} \]
5. Add Preprocessing

Alternative 8: 11.3% accurate, 170.0× speedup

Math

\[-\pi \]

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%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

      \[\leadsto -1 \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

   2. lower-PI.f32 11.3%

      \[\leadsto -1 \cdot \pi \]

4. Applied rewrites 11.3%

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

   1. lift-*.f32 N/A

      \[\leadsto -1 \cdot \color{blue}{\pi} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\pi\right) \]

   3. lift-neg.f32 11.3%

      \[\leadsto -\pi \]

6. Applied rewrites 11.3%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2328306437/10000000000000000000 u) (<= u 1)) (and (<= 0 s) (<= s 10651631/10000000)))
  (* (- s) (log (- (/ 1 (+ (* u (- (/ 1 (+ 1 (exp (/ (- PI) s)))) (/ 1 (+ 1 (exp (/ PI s)))))) (/ 1 (+ 1 (exp (/ PI s)))))) 1))))