Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 4.9s

Alternatives: 17

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

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

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%

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

Alternative 2: 97.4% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 97.4

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

6. Applied rewrites 97.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\pi}{s}} + 1} - \frac{1}{\color{blue}{2 + \mathsf{fma}\left(0.5, \frac{\pi \cdot \pi}{s \cdot s}, \frac{\pi}{s}\right)}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)} - 1\right) \]
7. 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: 96.7% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (*
      (-
       (/ 1.0 (+ (exp (/ (- PI) s)) 1.0))
       (/ 1.0 (+ 2.0 (/ (fma 0.5 (* PI PI) (* s PI)) (* s s)))))
      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 / (2.0f + (fmaf(0.5f, (((float) M_PI) * ((float) M_PI)), (s * ((float) M_PI))) / (s * s))))) * 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(Float32(2.0) + Float32(fma(Float32(0.5), Float32(Float32(pi) * Float32(pi)), Float32(s * Float32(pi))) / Float32(s * s))))) * u)) - 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 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. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 96.7

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

10. Applied rewrites 96.7%

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

Alternative 5: 96.7% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (*
      (-
       (/ 1.0 (+ (exp (/ (- PI) s)) 1.0))
       (/ 1.0 (/ (fma 0.5 (* PI PI) (* s PI)) (* s s))))
      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 / (fmaf(0.5f, (((float) M_PI) * ((float) M_PI)), (s * ((float) M_PI))) / (s * s)))) * 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(fma(Float32(0.5), Float32(Float32(pi) * Float32(pi)), Float32(s * Float32(pi))) / Float32(s * s)))) * u)) - 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 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. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 96.7

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

10. Applied rewrites 96.7%

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

Alternative 6: 96.7% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (*
      (-
       (/ 1.0 (+ (exp (/ (- PI) s)) 1.0))
       (/ 1.0 (+ 2.0 (/ (* 0.5 (* PI PI)) (* s s)))))
      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 / (2.0f + ((0.5f * (((float) M_PI) * ((float) M_PI))) / (s * s))))) * 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(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(Float32(pi) * Float32(pi))) / Float32(s * s))))) * 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) / (single(2.0) + ((single(0.5) * (single(pi) * single(pi))) / (s * s))))) * 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. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 96.7

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

10. Applied rewrites 96.7%

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

11. Taylor expanded in s around 0

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

    1. lower-*.f32 N/A

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

    2. pow2 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-PI.f32 N/A

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

    5. lift-PI.f32 96.7

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

13. Applied rewrites 96.7%

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

Alternative 7: 94.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (* (- (/ 1.0 (+ (exp (/ (- PI) s)) 1.0)) (/ 1.0 (+ 2.0 (/ PI s)))) 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 / (2.0f + (((float) M_PI) / s)))) * 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(Float32(2.0) + Float32(Float32(pi) / s)))) * 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) / (single(2.0) + (single(pi) / s)))) * 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. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 94.5

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

7. Applied rewrites 94.5%

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

Alternative 8: 37.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((0.5f - (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(0.5) - 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(0.5) - (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. Taylor expanded in s around inf

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

7. Applied rewrites 37.2%

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

Alternative 9: 37.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((single(0.5) - (single(1.0) / (single(2.0) + ((single(pi) + (single(0.5) * ((single(pi) * single(pi)) / s))) / s)))) * 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. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.2%

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

11. Taylor expanded in s around inf

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

    1. lower-/.f32 N/A

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

    2. lower-+.f32 N/A

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

    3. lift-PI.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-/.f32 N/A

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

    6. pow2 N/A

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

    7. lift-*.f32 N/A

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

    8. lift-PI.f32 N/A

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

    9. lift-PI.f32 37.2

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

13. Applied rewrites 37.2%

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

Alternative 10: 37.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(fma(Float32(0.5), Float32(Float32(pi) * Float32(pi)), Float32(s * Float32(pi))) / Float32(s * s)))) * u)) - 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 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. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.2%

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

11. Taylor expanded in s around 0

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

    1. pow2 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-PI.f32 N/A

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

    4. lift-PI.f32 N/A

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

    5. lift-PI.f32 N/A

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

    6. lift-PI.f32 N/A

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

    7. lift-*.f32 N/A

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

    8. lower-/.f32 N/A

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

13. Applied rewrites 37.2%

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

Alternative 11: 37.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((single(0.5) - (single(1.0) / (single(2.0) + (single(pi) / s)))) * 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. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.2%

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

11. Taylor expanded in s around inf

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

    1. lift-/.f32 N/A

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

    2. lift-PI.f32 37.1

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

13. Applied rewrites 37.1%

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

Alternative 12: 25.0% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (fma (/ (fma (* PI 0.5) u (* -0.25 PI)) s) -4.0 1.0))))

C

float code(float u, float s) {
	return -s * logf(fmaf((fmaf((((float) M_PI) * 0.5f), u, (-0.25f * ((float) M_PI))) / s), -4.0f, 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(fma(Float32(fma(Float32(Float32(pi) * Float32(0.5)), u, Float32(Float32(-0.25) * Float32(pi))) / 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 25.0%

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

Alternative 13: 14.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\log 1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \mathsf{fma}\left(-1, \frac{\pi}{u}, 2 \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 9.999999682655225e-21)
   (log 1.0)
   (* u (fma -1.0 (/ PI u) (* 2.0 PI)))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 9.999999682655225e-21f) {
		tmp = logf(1.0f);
	} else {
		tmp = u * fmaf(-1.0f, (((float) M_PI) / u), (2.0f * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-21))
		tmp = log(Float32(1.0));
	else
		tmp = Float32(u * fma(Float32(-1.0), Float32(Float32(pi) / u), Float32(Float32(2.0) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-21

   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 99.0%

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

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. log-pow-rev N/A

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

   6. Applied rewrites 13.2%

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

   7. Taylor expanded in s around 0

      \[\leadsto \log \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.2%

      \[\leadsto \log \color{blue}{1} \]

   if 9.99999968e-21 < s

   1. Initial program 98.9%

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

   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 15.0%

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

   5. Taylor expanded in u around inf

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 15.0

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

   7. Applied rewrites 15.0%

      \[\leadsto u \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\pi}{u}, 2 \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 11.5% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(u * fma(Float32(-1.0), Float32(Float32(pi) / u), Float32(Float32(2.0) * 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 11.5%

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

5. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 11.5

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

7. Applied rewrites 11.5%

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

Alternative 15: 11.5% 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 11.5%

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

Alternative 16: 11.5% 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. 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 11.5%

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

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. 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. lower-*.f32 N/A

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

   5. lift-PI.f32 11.5

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

7. Applied rewrites 11.5%

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

Alternative 17: 11.3% 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 11.3

      \[\leadsto -\pi \]

4. Applied rewrites 11.3%

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

Reproduce

herbie shell --seed 2025106 
(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))))