Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 6.6s

Alternatives: 21

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

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
  (*
   (- s)
   (log
    (-
     (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
     1.0)))))

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (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: 97.6% accurate, 1.2× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (* -1.0 s)
 (log
  (/
   (fma
    -1.0
    u
    (/
     1.0
     (-
      (/ 1.0 (+ 1.0 (exp (* -1.0 (/ PI s)))))
      (/ 1.0 (+ 1.0 (exp (/ PI s)))))))
   u))))

C

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

Julia

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

Derivation

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. Step-by-step derivation

3. Applied rewrites 98.6%

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

4. Taylor expanded in u around inf

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

6. Applied rewrites 97.2%

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

7. Taylor expanded in u around 0

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

9. Applied rewrites 97.3%

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

11. Applied rewrites 97.6%

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

Alternative 2: 97.5% accurate, 1.2× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (- s)
 (-
  (log
   (fabs
    (/
     1.0
     (-
      (/
       1.0
       (*
        (-
         (/ -1.0 (- -1.0 (exp (/ (- PI) s))))
         (/ -1.0 (- -1.0 (exp (/ PI s)))))
        u))
      1.0)))))))

C

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

Julia

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

MATLAB

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

Derivation

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 u around inf

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

4. Applied rewrites 97.5%

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

6. Applied rewrites 97.5%

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

8. Applied rewrites 97.5%

   \[\leadsto \left(-s\right) \cdot \left(0 - \log \left(\left|\frac{1}{\frac{1}{\left(\frac{-1}{-1 - e^{\frac{-\pi}{s}}} - \frac{-1}{-1 - e^{\frac{\pi}{s}}}\right) \cdot u} - 1}\right|\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 97.5%

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

Alternative 3: 97.5% accurate, 1.4× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-3.1415927410125732}{s}} + 1} - \frac{1}{e^{\frac{\pi}{s}} + 1}\right) \cdot u} - 1\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (- s)
 (log
  (-
   (/
    1.0
    (*
     (-
      (/ 1.0 (+ (exp (/ -3.1415927410125732 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((-3.1415927410125732f / 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(-3.1415927410125732) / 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(-3.1415927410125732) / s)) + single(1.0))) - (single(1.0) / (exp((single(pi) / s)) + single(1.0)))) * u)) - single(1.0)));
end

Derivation

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 u around inf

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

4. Applied rewrites 97.5%

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

6. Applied rewrites 97.5%

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

7. Evaluated real constant 97.5%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{-3.1415927410125732}{s}} + 1} - \frac{1}{e^{\frac{\pi}{s}} + 1}\right) \cdot u} - 1\right) \]
8. Add Preprocessing

Alternative 4: 94.2% accurate, 1.7× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{-1}{-1 - e^{\frac{-3.1415927410125732}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right) \cdot u} - 1\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (- s)
 (log
  (-
   (/
    1.0
    (*
     (-
      (/ -1.0 (- -1.0 (exp (/ -3.1415927410125732 s))))
      (/ 1.0 (+ 2.0 (/ PI s))))
     u))
   1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / (((-1.0f / (-1.0f - expf((-3.1415927410125732f / s)))) - (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(Float32(-1.0) - exp(Float32(Float32(-3.1415927410125732) / s)))) - 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) / (single(-1.0) - exp((single(-3.1415927410125732) / s)))) - (single(1.0) / (single(2.0) + (single(pi) / s)))) * u)) - single(1.0)));
end

Derivation

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 u around inf

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 94.2%

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

9. Applied rewrites 94.2%

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

10. Evaluated real constant 94.2%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{-1}{-1 - e^{\frac{-3.1415927410125732}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right) \cdot u} - 1\right) \]
11. Add Preprocessing

Alternative 5: 85.5% accurate, 2.2× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (- s)
 (log
  (fma
   (/ 1.0 (fma (* u (/ (* -0.5 PI) s)) -2.0 1.0))
   (+ 2.0 (/ PI s))
   -1.0))))

C

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

Julia

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

Derivation

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

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

4. Applied rewrites 86.0%

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

6. Applied rewrites 85.5%

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

7. Taylor expanded in s around -inf

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

9. Applied rewrites 85.4%

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

11. Applied rewrites 85.5%

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

Alternative 6: 85.5% accurate, 2.3× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (- s)
 (log
  (-
   (/ (+ 2.0 (/ PI s)) (fma (* u (/ (* -0.5 PI) s)) -2.0 1.0))
   1.0))))

C

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

Julia

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

Derivation

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

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

4. Applied rewrites 86.0%

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

6. Applied rewrites 85.5%

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

7. Taylor expanded in s around -inf

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

9. Applied rewrites 85.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{2 + \frac{\pi}{s}}{1 + -2 \cdot \frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}{s}} - 1\right) \]

11. Applied rewrites 85.5%

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

Alternative 7: 85.4% accurate, 2.3× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{1}{1 + -2 \cdot \frac{u \cdot -1.5707963705062866}{s}}, 2 + \frac{\pi}{s}, -1\right)\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (- s)
 (log
  (fma
   (/ 1.0 (+ 1.0 (* -2.0 (/ (* u -1.5707963705062866) s))))
   (+ 2.0 (/ PI s))
   -1.0))))

C

float code(float u, float s) {
	return -s * logf(fmaf((1.0f / (1.0f + (-2.0f * ((u * -1.5707963705062866f) / s)))), (2.0f + (((float) M_PI) / s)), -1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(fma(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(u * Float32(-1.5707963705062866)) / s)))), Float32(Float32(2.0) + Float32(Float32(pi) / s)), Float32(-1.0))))
end

Derivation

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

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

4. Applied rewrites 86.0%

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

6. Applied rewrites 85.5%

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

7. Taylor expanded in s around -inf

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

9. Applied rewrites 85.4%

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

10. Evaluated real constant 85.4%

    \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{1}{1 + -2 \cdot \frac{u \cdot -1.5707963705062866}{s}}, 2 + \frac{\pi}{s}, -1\right)\right) \]
11. Add Preprocessing

Alternative 8: 85.4% accurate, 2.5× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\left(-s\right) \cdot \log \left(\frac{2 + \frac{\pi}{s}}{1 + -2 \cdot \frac{u \cdot -1.5707963705062866}{s}} - 1\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (- s)
 (log
  (-
   (/
    (+ 2.0 (/ PI s))
    (+ 1.0 (* -2.0 (/ (* u -1.5707963705062866) s))))
   1.0))))

C

float code(float u, float s) {
	return -s * logf((((2.0f + (((float) M_PI) / s)) / (1.0f + (-2.0f * ((u * -1.5707963705062866f) / s)))) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(pi) / s)) / Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(u * Float32(-1.5707963705062866)) / s)))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((((single(2.0) + (single(pi) / s)) / (single(1.0) + (single(-2.0) * ((u * single(-1.5707963705062866)) / s)))) - single(1.0)));
end

Derivation

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

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

4. Applied rewrites 86.0%

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

6. Applied rewrites 85.5%

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

7. Taylor expanded in s around -inf

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

9. Applied rewrites 85.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{2 + \frac{\pi}{s}}{1 + -2 \cdot \frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}{s}} - 1\right) \]

10. Evaluated real constant 85.4%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{2 + \frac{\pi}{s}}{1 + -2 \cdot \frac{u \cdot -1.5707963705062866}{s}} - 1\right) \]
11. Add Preprocessing

Alternative 9: 25.1% accurate, 2.8× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (*
 (- s)
 (log (fabs (fma (/ (fma (* 0.5 PI) u (* -0.25 PI)) s) -4.0 1.0)))))

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(abs(fma(Float32(fma(Float32(Float32(0.5) * Float32(pi)), u, Float32(Float32(-0.25) * Float32(pi))) / s), Float32(-4.0), Float32(1.0)))))
end

Derivation

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

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

4. Applied rewrites 24.9%

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

6. Applied rewrites 25.1%

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

Alternative 10: 25.1% accurate, 3.2× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* (/ 1.0 (/ -1.0 s)) (log (+ 1.0 (/ 1.0 (/ s PI))))))

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

3. Applied rewrites 98.6%

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

4. Taylor expanded in s around inf

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

6. Applied rewrites 24.9%

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

8. Applied rewrites 24.9%

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

9. Taylor expanded in u around 0

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

11. Applied rewrites 25.1%

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

Alternative 11: 25.1% accurate, 3.8× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* (/ 1.0 (/ -1.0 s)) (log (+ 1.0 (/ PI s)))))

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

3. Applied rewrites 98.6%

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

4. Taylor expanded in s around inf

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

6. Applied rewrites 24.9%

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

7. Taylor expanded in u around 0

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

9. Applied rewrites 25.1%

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

Alternative 12: 25.1% accurate, 4.3× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (/ (log (- (/ PI s) -1.0)) (/ -1.0 s)))

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

3. Applied rewrites 98.6%

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

4. Taylor expanded in s around inf

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

6. Applied rewrites 24.9%

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

7. Taylor expanded in u around 0

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

9. Applied rewrites 25.1%

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

11. Applied rewrites 25.1%

    \[\leadsto \frac{\log \left(\frac{\pi}{s} - -1\right)}{\frac{-1}{s}} \]
12. Add Preprocessing

Alternative 13: 25.1% accurate, 5.2× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- 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 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

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

4. Applied rewrites 24.9%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 25.1%

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

Alternative 14: 25.1% accurate, 5.2× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (/ (+ s PI) s))))

C

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

Julia

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

MATLAB

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

Derivation

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

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

4. Applied rewrites 24.9%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 25.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 25.1%

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

Alternative 15: 14.3% accurate, 6.2× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\frac{s \cdot \frac{s}{u}}{-0.5 \cdot \pi} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (/ (* s (/ s u)) (* -0.5 PI)))

C

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

Julia

function code(u, s)
	return Float32(Float32(s * Float32(s / u)) / Float32(Float32(-0.5) * Float32(pi)))
end

MATLAB

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

Derivation

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 u around inf

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 14.3%

   \[\leadsto \frac{{s}^{2}}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)} \]
8. Step-by-step derivation

9. Applied rewrites 14.3%

   \[\leadsto \frac{\frac{s \cdot s}{u}}{-0.5 \cdot \pi} \]
10. Step-by-step derivation

11. Applied rewrites 14.3%

    \[\leadsto \frac{s \cdot \frac{s}{u}}{-0.5 \cdot \pi} \]
12. Add Preprocessing

Alternative 16: 14.3% accurate, 6.5× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[s \cdot \frac{s}{\left(-0.5 \cdot \pi\right) \cdot u} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (* s (/ s (* (* -0.5 PI) u))))

C

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

Julia

function code(u, s)
	return Float32(s * Float32(s / Float32(Float32(Float32(-0.5) * Float32(pi)) * u)))
end

MATLAB

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

Derivation

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 u around inf

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 14.3%

   \[\leadsto \frac{{s}^{2}}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)} \]
8. Step-by-step derivation

9. Applied rewrites 14.3%

   \[\leadsto \left(s \cdot s\right) \cdot \frac{1}{\left(-0.5 \cdot \pi\right) \cdot u} \]
10. Step-by-step derivation

11. Applied rewrites 14.3%

    \[\leadsto s \cdot \frac{s}{\left(-0.5 \cdot \pi\right) \cdot u} \]
12. Add Preprocessing

Alternative 17: 14.3% accurate, 7.8× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\frac{\frac{s \cdot s}{u}}{-1.5707963705062866} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (/ (/ (* s s) u) -1.5707963705062866))

C

float code(float u, float s) {
	return ((s * s) / u) / -1.5707963705062866f;
}

Fortran

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = ((s * s) / u) / (-1.5707963705062866e0)
end function

Julia

function code(u, s)
	return Float32(Float32(Float32(s * s) / u) / Float32(-1.5707963705062866))
end

MATLAB

function tmp = code(u, s)
	tmp = ((s * s) / u) / single(-1.5707963705062866);
end

Derivation

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 u around inf

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 14.3%

   \[\leadsto \frac{{s}^{2}}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)} \]
8. Step-by-step derivation

9. Applied rewrites 14.3%

   \[\leadsto \frac{\frac{s \cdot s}{u}}{-0.5 \cdot \pi} \]

10. Evaluated real constant 14.3%

    \[\leadsto \frac{\frac{s \cdot s}{u}}{-1.5707963705062866} \]
11. Add Preprocessing

Alternative 18: 14.3% accurate, 8.4× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\frac{s \cdot s}{-1.5707963705062866 \cdot u} \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (/ (* s s) (* -1.5707963705062866 u)))

C

float code(float u, float s) {
	return (s * s) / (-1.5707963705062866f * u);
}

Fortran

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = (s * s) / ((-1.5707963705062866e0) * u)
end function

Julia

function code(u, s)
	return Float32(Float32(s * s) / Float32(Float32(-1.5707963705062866) * u))
end

MATLAB

function tmp = code(u, s)
	tmp = (s * s) / (single(-1.5707963705062866) * u);
end

Derivation

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 u around inf

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 14.3%

   \[\leadsto \frac{{s}^{2}}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)} \]
8. Step-by-step derivation

9. Applied rewrites 14.3%

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

10. Evaluated real constant 14.3%

    \[\leadsto \frac{s \cdot s}{-1.5707963705062866 \cdot u} \]
11. Add Preprocessing

Alternative 19: 11.7% accurate, 12.9× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\mathsf{fma}\left(6.2831854820251465, u, -\pi\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  (fma 6.2831854820251465 u (- PI)))

C

float code(float u, float s) {
	return fmaf(6.2831854820251465f, u, -((float) M_PI));
}

Julia

function code(u, s)
	return fma(Float32(6.2831854820251465), u, Float32(-Float32(pi)))
end

Derivation

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

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

4. Applied rewrites 11.7%

   \[\leadsto -4 \cdot \left(u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right) - -0.25 \cdot \pi\right) \]

5. Evaluated real constant 11.7%

   \[\leadsto -4 \cdot \left(u \cdot -1.5707963705062866 - -0.25 \cdot \pi\right) \]

6. Taylor expanded in u around 0

   \[\leadsto -1 \cdot \pi + \frac{13176795}{2097152} \cdot u \]
7. Step-by-step derivation

8. Applied rewrites 11.7%

   \[\leadsto \mathsf{fma}\left(-1, \pi, 6.2831854820251465 \cdot u\right) \]
9. Step-by-step derivation

10. Applied rewrites 11.7%

    \[\leadsto \mathsf{fma}\left(6.2831854820251465, u, -\pi\right) \]
11. Add Preprocessing

Alternative 20: 11.4% accurate, 90.2× speedup

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[-3.1415927410125732 \]

FPCore

(FPCore (u s)
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0))
     (and (<= 0.0 s) (<= s 1.0651631)))
  -3.1415927410125732)

C

float code(float u, float s) {
	return -3.1415927410125732f;
}

Fortran

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -3.1415927410125732e0
end function

Julia

function code(u, s)
	return Float32(-3.1415927410125732)
end

MATLAB

function tmp = code(u, s)
	tmp = single(-3.1415927410125732);
end

Derivation

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 u around 0

   \[\leadsto -1 \cdot \pi \]
3. Step-by-step derivation

4. Applied rewrites 11.4%

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

5. Evaluated real constant 11.4%

   \[\leadsto -3.1415927410125732 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2026084 
(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))))