Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 8.8s

Alternatives: 14

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := \frac{1}{t\_0 + 1}\\ t_2 := \frac{1}{1 + t\_0}\\ \left(-s\right) \cdot \log \left(\frac{{\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - t\_2, t\_2\right)\right)}^{-2} - 1}{\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\pi}{s}} + 1} - t\_1, u, t\_1\right)} + 1}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s)))
        (t_1 (/ 1.0 (+ t_0 1.0)))
        (t_2 (/ 1.0 (+ 1.0 t_0))))
   (*
    (- s)
    (log
     (/
      (-
       (pow (fma u (- (/ 1.0 (+ 1.0 (pow (exp -1.0) (/ PI s)))) t_2) t_2) -2.0)
       1.0)
      (+
       (/ 1.0 (fma (- (/ 1.0 (+ (exp (/ (- PI) s)) 1.0)) t_1) u t_1))
       1.0))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = 1.0f / (t_0 + 1.0f);
	float t_2 = 1.0f / (1.0f + t_0);
	return -s * logf(((powf(fmaf(u, ((1.0f / (1.0f + powf(expf(-1.0f), (((float) M_PI) / s)))) - t_2), t_2), -2.0f) - 1.0f) / ((1.0f / fmaf(((1.0f / (expf((-((float) M_PI) / s)) + 1.0f)) - t_1), u, t_1)) + 1.0f)));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(Float32(1.0) / Float32(t_0 + Float32(1.0)))
	t_2 = Float32(Float32(1.0) / Float32(Float32(1.0) + t_0))
	return Float32(Float32(-s) * log(Float32(Float32((fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-1.0)) ^ Float32(Float32(pi) / s)))) - t_2), t_2) ^ Float32(-2.0)) - Float32(1.0)) / Float32(Float32(Float32(1.0) / fma(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) + Float32(1.0))) - t_1), u, t_1)) + 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. Add Preprocessing

4. Applied rewrites 98.9%

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

5. Taylor expanded in s around 0

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

   1. pow-flip N/A

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

   2. metadata-eval N/A

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

   3. lower-pow.f32 N/A

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

7. Applied rewrites 99.0%

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

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

4. Applied rewrites 99.0%

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

3. Taylor expanded in u around inf

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

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

5. Applied rewrites 97.7%

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

Alternative 4: 92.1% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+
      (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 2.0 (/ PI s)))))
      (/ 1.0 (+ 1.0 (+ 1.0 (fma 0.5 (/ (* PI PI) (* s s)) (/ PI s)))))))
    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 / (2.0f + (((float) M_PI) / s))))) + (1.0f / (1.0f + (1.0f + fmaf(0.5f, ((((float) M_PI) * ((float) M_PI)) / (s * s)), (((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) + exp(Float32(Float32(-Float32(pi)) / s)))) - Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(pi) / s))))) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + fma(Float32(0.5), Float32(Float32(Float32(pi) * Float32(pi)) / Float32(s * s)), 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. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
4. 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. lift-/.f32 N/A

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

   3. lift-PI.f32 94.9

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

5. Applied rewrites 94.9%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 92.1

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

8. Applied rewrites 92.1%

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

Alternative 5: 92.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
4. 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. lift-/.f32 N/A

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

   3. lift-PI.f32 94.9

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

5. Applied rewrites 94.9%

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 92.1

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

10. Applied rewrites 92.1%

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

Alternative 6: 92.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
4. 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. lift-/.f32 N/A

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

   3. lift-PI.f32 94.9

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

5. Applied rewrites 94.9%

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in s around 0

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 94.9

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

10. Applied rewrites 94.9%

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

11. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-fma.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f32 N/A

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

    6. lift-PI.f32 N/A

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

    7. lift-PI.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lift-/.f32 N/A

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

    11. lift-PI.f32 92.1

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

13. Applied rewrites 92.1%

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

Alternative 7: 85.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

3. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
4. 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. lift-/.f32 N/A

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

   3. lift-PI.f32 94.9

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

5. Applied rewrites 94.9%

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in s around 0

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 94.9

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

10. Applied rewrites 94.9%

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

11. Taylor expanded in s around inf

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

    1. lift-/.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-+.f32 85.9

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

13. Applied rewrites 85.9%

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

Alternative 8: 37.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf(((1.0f / fmaf(u, (0.5f - (1.0f / (2.0f + (((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) / fma(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) + 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. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
4. 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. lift-/.f32 N/A

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

   3. lift-PI.f32 94.9

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

5. Applied rewrites 94.9%

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.7%

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

Alternative 9: 37.7% accurate, 1.8× speedup

Math

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

FPCore

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

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. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
4. 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. lift-/.f32 N/A

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

   3. lift-PI.f32 94.9

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

5. Applied rewrites 94.9%

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in s around 0

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 94.9

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

10. Applied rewrites 94.9%

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

11. Taylor expanded in s around inf

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

13. Applied rewrites 37.7%

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

Alternative 10: 25.2% 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. Add Preprocessing

3. Taylor expanded in s around inf

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

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

5. Applied rewrites 25.0%

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

6. Taylor expanded in u around 0

   \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right) \]
7. 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 25.2

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

8. Applied rewrites 25.2%

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

Alternative 11: 11.6% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(4, u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right), 4 \cdot \mathsf{fma}\left(-0.0625, \mathsf{fma}\left(4, \pi, 8 \cdot \pi\right), 0.5 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (fma
  4.0
  (* u (- (* 0.25 PI) (* -0.25 PI)))
  (* 4.0 (fma -0.0625 (fma 4.0 PI (* 8.0 PI)) (* 0.5 PI)))))

C

float code(float u, float s) {
	return fmaf(4.0f, (u * ((0.25f * ((float) M_PI)) - (-0.25f * ((float) M_PI)))), (4.0f * fmaf(-0.0625f, fmaf(4.0f, ((float) M_PI), (8.0f * ((float) M_PI))), (0.5f * ((float) M_PI)))));
}

Julia

function code(u, s)
	return fma(Float32(4.0), Float32(u * Float32(Float32(Float32(0.25) * Float32(pi)) - Float32(Float32(-0.25) * Float32(pi)))), Float32(Float32(4.0) * fma(Float32(-0.0625), fma(Float32(4.0), Float32(pi), Float32(Float32(8.0) * Float32(pi))), Float32(Float32(0.5) * 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. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
4. 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. lift-/.f32 N/A

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

   3. lift-PI.f32 94.9

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

5. Applied rewrites 94.9%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-/.f32 N/A

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

7. Applied rewrites 94.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right)\right)}^{3} + {\left(\frac{1}{1 + e^{\frac{\pi}{s}}}\right)}^{3}}{\mathsf{fma}\left(u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right), u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right), \frac{1}{1 + e^{\frac{\pi}{s}}} \cdot \frac{1}{1 + e^{\frac{\pi}{s}}} - \left(u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right)\right) \cdot \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}}} - 1\right) \]

8. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) + 4 \cdot \left(\frac{-1}{16} \cdot \left(4 \cdot \mathsf{PI}\left(\right) + 8 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 11.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right), 4 \cdot \mathsf{fma}\left(-0.0625, \mathsf{fma}\left(4, \pi, 8 \cdot \pi\right), 0.5 \cdot \pi\right)\right)} \]
11. Add Preprocessing

Alternative 12: 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. Add Preprocessing

3. Taylor expanded in s around inf

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

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

5. Applied rewrites 11.6%

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

Alternative 13: 11.6% accurate, 30.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 (fma -1.0 PI (* 2.0 (* u PI))))

C

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

Julia

function code(u, s)
	return fma(Float32(-1.0), Float32(pi), Float32(Float32(2.0) * Float32(u * 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. Add Preprocessing

3. Taylor expanded in s around inf

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

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

5. Applied rewrites 11.6%

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

6. Taylor expanded in u around 0

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

   1. lower-fma.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 11.6

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

8. Applied rewrites 11.6%

   \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\pi}, 2 \cdot \left(u \cdot \pi\right)\right) \]
9. Add Preprocessing

Alternative 14: 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. Add Preprocessing

3. Taylor expanded in u around 0

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

      \[\leadsto -\pi \]

5. Applied rewrites 11.4%

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

Reproduce

herbie shell --seed 2025091 
(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))))