Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 5.8s

Alternatives: 15

Speedup: 1.0×

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: 99.0% 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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in u around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-+.f32 99.0

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

7. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0)))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / fma(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) + Float32(1.0))) - t_0), u, t_0)) - Float32(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. Applied rewrites 99.0%

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

Alternative 3: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 97.7%

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

Alternative 4: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}\right) + \frac{1}{1 + \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot \frac{\pi}{s}, -0.5, -\pi\right)}{s}, -1, 1\right)}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+
      (*
       u
       (-
        (/ 1.0 (+ 1.0 (exp (/ (- PI) s))))
        (/ 1.0 (fma (/ (fma (/ (* PI PI) s) -0.5 (- PI)) s) -1.0 2.0))))
      (/ 1.0 (+ 1.0 (fma (/ (fma (* PI (/ PI s)) -0.5 (- PI)) s) -1.0 1.0)))))
    1.0))))

C

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

Julia

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 97.3%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

5. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

7. Applied rewrites 92.7%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}\right) + \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot \frac{\pi}{s}, -0.5, -\pi\right)}{s}, -1, 1\right)}}} - 1\right) \]
8. Add Preprocessing

Alternative 5: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}\right) + \frac{1}{1 + \left(\frac{\pi}{s} + 1\right)}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+
      (*
       u
       (-
        (/ 1.0 (+ 1.0 (exp (/ (- PI) s))))
        (/ 1.0 (fma (/ (fma (/ (* PI PI) s) -0.5 (- PI)) s) -1.0 2.0))))
      (/ 1.0 (+ 1.0 (+ (/ PI s) 1.0)))))
    1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - (1.0f / fmaf((fmaf(((((float) M_PI) * ((float) M_PI)) / s), -0.5f, -((float) M_PI)) / s), -1.0f, 2.0f)))) + (1.0f / (1.0f + ((((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(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - Float32(Float32(1.0) / fma(Float32(fma(Float32(Float32(Float32(pi) * Float32(pi)) / s), Float32(-0.5), Float32(-Float32(pi))) / s), Float32(-1.0), Float32(2.0))))) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(pi) / s) + Float32(1.0)))))) - Float32(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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 97.3%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 86.2

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}\right) + \frac{1}{1 + \left(\frac{\pi}{s} + 1\right)}} - 1\right) \]

7. Applied rewrites 86.2%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}\right) + \frac{1}{1 + \color{blue}{\left(\frac{\pi}{s} + 1\right)}}} - 1\right) \]
8. Add Preprocessing

Alternative 6: 37.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + 1} - \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+
      (*
       u
       (-
        (/ 1.0 (+ 1.0 1.0))
        (/ 1.0 (fma (/ (fma (/ (* PI PI) s) -0.5 (- PI)) s) -1.0 2.0))))
      (/ 1.0 (+ 1.0 (exp (/ PI s))))))
    1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + 1.0f)) - (1.0f / fmaf((fmaf(((((float) M_PI) * ((float) M_PI)) / s), -0.5f, -((float) M_PI)) / s), -1.0f, 2.0f)))) + (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(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(1.0))) - Float32(Float32(1.0) / fma(Float32(fma(Float32(Float32(Float32(pi) * Float32(pi)) / s), Float32(-0.5), Float32(-Float32(pi))) / s), Float32(-1.0), Float32(2.0))))) + Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) - Float32(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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 97.3%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

5. Taylor expanded in s around inf

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

7. Applied rewrites 37.7%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + \color{blue}{1}} - \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\pi \cdot \pi}{s}, -0.5, -\pi\right)}{s}, -1, 2\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
8. Add Preprocessing

Alternative 7: 37.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(0.5 - \frac{1}{2 + \frac{\pi}{s}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (fma (- 0.5 (/ 1.0 (+ 2.0 (/ PI s)))) u (/ 1.0 (+ (exp (/ PI s)) 1.0))))
    1.0))))

C

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

Julia

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

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.0

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

4. Applied rewrites 4.0%

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

5. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\pi}{s}, \frac{1}{2}\right) - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.0

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\pi}{s}, 0.5\right) - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

7. Applied rewrites 4.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\pi}{s}, 0.5\right) - \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.7%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
11. Step-by-step derivation

    1. lift-+.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lift-/.f32 N/A

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

    5. lift-+.f32 N/A

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

    6. lift-exp.f32 N/A

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

    7. lift-PI.f32 N/A

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

12. Applied rewrites 37.7%

    \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\mathsf{fma}\left(0.5 - \frac{1}{2 + \frac{\pi}{s}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)}} - 1\right) \]
13. Add Preprocessing

Alternative 8: 37.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{\frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/ 1.0 (+ (* u (- 0.5 (/ 1.0 (/ PI s)))) (/ 1.0 (+ 1.0 (exp (/ PI s))))))
    1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((u * (0.5f - (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(Float32(u * Float32(Float32(0.5) - 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(0.5) - (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 s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.0

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

4. Applied rewrites 4.0%

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

5. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\pi}{s}, \frac{1}{2}\right) - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.0

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\pi}{s}, 0.5\right) - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

7. Applied rewrites 4.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\pi}{s}, 0.5\right) - \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.7%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

11. Taylor expanded in s around 0

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

    1. lift-/.f32 N/A

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

    2. lift-PI.f32 37.7

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{\frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

13. Applied rewrites 37.7%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{\frac{\pi}{\color{blue}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
14. Add Preprocessing

Alternative 9: 36.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot \frac{\pi}{s}, -0.5, -\pi\right)}{s}, -1, 1\right)}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+
      (* u (- 0.5 (/ 1.0 (+ 2.0 (/ PI s)))))
      (/ 1.0 (+ 1.0 (fma (/ (fma (* PI (/ PI s)) -0.5 (- PI)) s) -1.0 1.0)))))
    1.0))))

C

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

Julia

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

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.0

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

4. Applied rewrites 4.0%

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

5. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\pi}{s}, \frac{1}{2}\right) - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.0

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\pi}{s}, 0.5\right) - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

7. Applied rewrites 4.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\pi}{s}, 0.5\right) - \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.7%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

11. Taylor expanded in s around -inf

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

    1. +-commutative N/A

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

13. Applied rewrites 36.8%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot \frac{\pi}{s}, -0.5, -\pi\right)}{s}, -1, 1\right)}}} - 1\right) \]
14. Add Preprocessing

Alternative 10: 36.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + \left(\frac{\pi}{s} + 1\right)}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+
      (* u (- 0.5 (/ 1.0 (+ 2.0 (/ PI s)))))
      (/ 1.0 (+ 1.0 (+ (/ PI s) 1.0)))))
    1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((u * (0.5f - (1.0f / (2.0f + (((float) M_PI) / s))))) + (1.0f / (1.0f + ((((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(0.5) - Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(pi) / s))))) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(pi) / s) + Float32(1.0)))))) - Float32(1.0))))
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 4.0

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

4. Applied rewrites 4.0%

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

5. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(\frac{1}{4}, \frac{\pi}{s}, \frac{1}{2}\right) - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 4.0

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\pi}{s}, 0.5\right) - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

7. Applied rewrites 4.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\mathsf{fma}\left(0.25, \frac{\pi}{s}, 0.5\right) - \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.7%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

11. Taylor expanded in s around inf

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. lift-PI.f32 36.3

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + \left(\frac{\pi}{s} + 1\right)}} - 1\right) \]

13. Applied rewrites 36.3%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{2 + \frac{\pi}{s}}\right) + \frac{1}{1 + \color{blue}{\left(\frac{\pi}{s} + 1\right)}}} - 1\right) \]
14. Add Preprocessing

Alternative 11: 25.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 24.9%

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

5. Taylor expanded in u around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{u \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{u} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 1\right)\right) \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 24.9

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

7. Applied rewrites 24.9%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 25.1

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

10. Applied rewrites 25.1%

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

Alternative 12: 25.1% accurate, 4.2× speedup

Math

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

FPCore

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 24.9%

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

5. Taylor expanded in u around 0

   \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right) \]
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. lift-/.f32 N/A

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

   3. lift-PI.f32 25.1

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

7. Applied rewrites 25.1%

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

Alternative 13: 11.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 11.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\pi \cdot 0.5, u, -0.25 \cdot \pi\right) \cdot 4} \]
5. Add Preprocessing

Alternative 14: 11.6% accurate, 36.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi \cdot u, 2, -\pi\right) \end{array} \]

FPCore

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

C

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

Julia

function code(u, s)
	return fma(Float32(Float32(pi) * u), Float32(2.0), Float32(-Float32(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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 11.6%

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

5. Taylor expanded in u around 0

   \[\leadsto -1 \cdot \mathsf{PI}\left(\right) + \color{blue}{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(u \cdot \mathsf{PI}\left(\right)\right) \cdot 2 + -1 \cdot \mathsf{PI}\left(\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 2, -1 \cdot \mathsf{PI}\left(\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 2, -1 \cdot \mathsf{PI}\left(\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 2, -1 \cdot \mathsf{PI}\left(\right)\right) \]

   6. lift-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\pi \cdot u, 2, -1 \cdot \mathsf{PI}\left(\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\pi \cdot u, 2, \mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \]

   8. lift-neg.f32 N/A

      \[\leadsto \mathsf{fma}\left(\pi \cdot u, 2, -\mathsf{PI}\left(\right)\right) \]

   9. lift-PI.f32 11.6

      \[\leadsto \mathsf{fma}\left(\pi \cdot u, 2, -\pi\right) \]

7. Applied rewrites 11.6%

   \[\leadsto \mathsf{fma}\left(\pi \cdot u, \color{blue}{2}, -\pi\right) \]
8. Add Preprocessing

Alternative 15: 11.4% accurate, 170.0× 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%

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

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lift-neg.f32 N/A

      \[\leadsto -\mathsf{PI}\left(\right) \]

   3. lift-PI.f32 11.4

      \[\leadsto -\pi \]

4. Applied rewrites 11.4%

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

Reproduce

herbie shell --seed 2025105 
(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))))