Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 10.6s

Alternatives: 14

Speedup: N/A×

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: 98.9% 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: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 25.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\log s - \left(-2 \cdot {u}^{2} + \left(u \cdot -2 + -2.6666666666666665 \cdot {u}^{3}\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log s)
   (+
    (* -2.0 (pow u 2.0))
    (+ (* u -2.0) (* -2.6666666666666665 (pow u 3.0)))))))

C

float code(float u, float s) {
	return s * (logf(s) - ((-2.0f * powf(u, 2.0f)) + ((u * -2.0f) + (-2.6666666666666665f * powf(u, 3.0f)))));
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * (log(s) - (((-2.0e0) * (u ** 2.0e0)) + ((u * (-2.0e0)) + ((-2.6666666666666665e0) * (u ** 3.0e0)))))
end function

Julia

function code(u, s)
	return Float32(s * Float32(log(s) - Float32(Float32(Float32(-2.0) * (u ^ Float32(2.0))) + Float32(Float32(u * Float32(-2.0)) + Float32(Float32(-2.6666666666666665) * (u ^ Float32(3.0)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (log(s) - ((single(-2.0) * (u ^ single(2.0))) + ((u * single(-2.0)) + (single(-2.6666666666666665) * (u ^ single(3.0))))));
end

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 24.5%

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

5. Taylor expanded in s around 0 24.5%

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

   1. mul-1-neg 24.5%

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

   2. *-commutative 24.5%

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

   3. distribute-rgt-neg-in 24.5%

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

7. Simplified 24.5%

   \[\leadsto \color{blue}{\left(\log \left(-4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)\right) - \log s\right) \cdot \left(-s\right)} \]

8. Taylor expanded in u around 0 25.1%

   \[\leadsto \left(\color{blue}{\left(-2 \cdot {u}^{2} + \left(\log \pi + \left(-2 \cdot u + -2.6666666666666665 \cdot {u}^{3}\right)\right)\right)} - \log s\right) \cdot \left(-s\right) \]

9. Taylor expanded in u around inf 25.1%

   \[\leadsto \left(\left(-2 \cdot {u}^{2} + \color{blue}{\left(-2 \cdot u + -2.6666666666666665 \cdot {u}^{3}\right)}\right) - \log s\right) \cdot \left(-s\right) \]

10. Final simplification 25.1%

    \[\leadsto s \cdot \left(\log s - \left(-2 \cdot {u}^{2} + \left(u \cdot -2 + -2.6666666666666665 \cdot {u}^{3}\right)\right)\right) \]

Alternative 3: 25.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\log s - \left(-2 \cdot {u}^{2} + -2.6666666666666665 \cdot {u}^{3}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (- (log s) (+ (* -2.0 (pow u 2.0)) (* -2.6666666666666665 (pow u 3.0))))))

C

float code(float u, float s) {
	return s * (logf(s) - ((-2.0f * powf(u, 2.0f)) + (-2.6666666666666665f * powf(u, 3.0f))));
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * (log(s) - (((-2.0e0) * (u ** 2.0e0)) + ((-2.6666666666666665e0) * (u ** 3.0e0))))
end function

Julia

function code(u, s)
	return Float32(s * Float32(log(s) - Float32(Float32(Float32(-2.0) * (u ^ Float32(2.0))) + Float32(Float32(-2.6666666666666665) * (u ^ Float32(3.0))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (log(s) - ((single(-2.0) * (u ^ single(2.0))) + (single(-2.6666666666666665) * (u ^ single(3.0)))));
end

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 24.5%

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

5. Taylor expanded in s around 0 24.5%

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

   1. mul-1-neg 24.5%

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

   2. *-commutative 24.5%

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

   3. distribute-rgt-neg-in 24.5%

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

7. Simplified 24.5%

   \[\leadsto \color{blue}{\left(\log \left(-4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)\right) - \log s\right) \cdot \left(-s\right)} \]

8. Taylor expanded in u around 0 25.1%

   \[\leadsto \left(\color{blue}{\left(-2 \cdot {u}^{2} + \left(\log \pi + \left(-2 \cdot u + -2.6666666666666665 \cdot {u}^{3}\right)\right)\right)} - \log s\right) \cdot \left(-s\right) \]

9. Taylor expanded in u around inf 25.1%

   \[\leadsto \left(\left(-2 \cdot {u}^{2} + \color{blue}{-2.6666666666666665 \cdot {u}^{3}}\right) - \log s\right) \cdot \left(-s\right) \]

10. Final simplification 25.1%

    \[\leadsto s \cdot \left(\log s - \left(-2 \cdot {u}^{2} + -2.6666666666666665 \cdot {u}^{3}\right)\right) \]

Alternative 4: 25.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 24.5%

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

5. Taylor expanded in s around 0 24.5%

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

   1. mul-1-neg 24.5%

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

   2. *-commutative 24.5%

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

   3. distribute-rgt-neg-in 24.5%

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

7. Simplified 24.5%

   \[\leadsto \color{blue}{\left(\log \left(-4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)\right) - \log s\right) \cdot \left(-s\right)} \]

8. Taylor expanded in u around 0 24.9%

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

   1. *-commutative 24.9%

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

10. Simplified 24.9%

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

11. Final simplification 24.9%

    \[\leadsto s \cdot \left(\log s - \left(u \cdot -2 + \log \pi\right)\right) \]

Alternative 5: 25.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 24.5%

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

5. Taylor expanded in u around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. *-commutative 24.7%

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

   3. distribute-rgt-neg-in 24.7%

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

   4. log1p-def 24.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \cdot \left(-s\right) \]

7. Simplified 24.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]

8. Taylor expanded in s around 0 24.9%

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

   1. mul-1-neg 24.9%

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

   2. log-rec 24.8%

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

   3. +-commutative 24.8%

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

   4. log-rec 24.9%

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

   5. sub-neg 24.9%

      \[\leadsto \color{blue}{\left(\log \pi - \log s\right)} \cdot \left(-s\right) \]

10. Simplified 24.9%

    \[\leadsto \color{blue}{\left(\log \pi - \log s\right)} \cdot \left(-s\right) \]

11. Final simplification 24.9%

    \[\leadsto s \cdot \left(\log s - \log \pi\right) \]

Alternative 6: 25.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 2\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- (* u (* s 2.0)) (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return (u * (s * 2.0f)) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(u * Float32(s * Float32(2.0))) - Float32(s * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 24.5%

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

5. Taylor expanded in u around 0 24.7%

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

   1. +-commutative 24.7%

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

   2. mul-1-neg 24.7%

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

   3. unsub-neg 24.7%

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

7. Simplified 24.7%

   \[\leadsto \color{blue}{\frac{4}{\frac{1 + \frac{\pi}{s}}{\left(s \cdot u\right) \cdot \frac{\pi}{\frac{s}{0.5}}}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

8. Taylor expanded in s around 0 24.7%

   \[\leadsto \color{blue}{2 \cdot \left(s \cdot u\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
9. Step-by-step derivation

   1. associate-*r* 24.7%

      \[\leadsto \color{blue}{\left(2 \cdot s\right) \cdot u} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

   2. *-commutative 24.7%

      \[\leadsto \color{blue}{\left(s \cdot 2\right)} \cdot u - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

10. Simplified 24.7%

    \[\leadsto \color{blue}{\left(s \cdot 2\right) \cdot u} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

11. Final simplification 24.7%

    \[\leadsto u \cdot \left(s \cdot 2\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

Alternative 7: 25.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(\pi \cdot \frac{1}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (* PI (/ 1.0 s)))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) * (1.0f / s)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) * Float32(Float32(1.0) / s))))
end

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 24.5%

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

5. Taylor expanded in u around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. *-commutative 24.7%

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

   3. distribute-rgt-neg-in 24.7%

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

   4. log1p-def 24.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \cdot \left(-s\right) \]

7. Simplified 24.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]
8. Step-by-step derivation

   1. div-inv 24.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\pi \cdot \frac{1}{s}}\right) \cdot \left(-s\right) \]

9. Applied egg-rr 24.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\pi \cdot \frac{1}{s}}\right) \cdot \left(-s\right) \]

10. Final simplification 24.7%

    \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\pi \cdot \frac{1}{s}\right) \]

Alternative 8: 25.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 24.5%

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

5. Taylor expanded in u around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. *-commutative 24.7%

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

   3. distribute-rgt-neg-in 24.7%

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

   4. log1p-def 24.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \cdot \left(-s\right) \]

7. Simplified 24.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]

8. Taylor expanded in s around 0 24.9%

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

   1. mul-1-neg 24.9%

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

   2. log-rec 24.8%

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

   3. +-commutative 24.8%

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

   4. log-rec 24.9%

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

   5. sub-neg 24.9%

      \[\leadsto \color{blue}{\left(\log \pi - \log s\right)} \cdot \left(-s\right) \]

   6. log-div 24.7%

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

10. Simplified 24.7%

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

11. Final simplification 24.7%

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

Alternative 9: 25.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 24.5%

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

5. Taylor expanded in u around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. *-commutative 24.7%

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

   3. distribute-rgt-neg-in 24.7%

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

   4. log1p-def 24.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \cdot \left(-s\right) \]

7. Simplified 24.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]

8. Final simplification 24.7%

   \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

Alternative 10: 14.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(-\frac{\pi}{s}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.000000031374395e-22)
   (* 2.6666666666666665 (* s (pow u 3.0)))
   (* s (- (/ PI s)))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 1.000000031374395e-22f) {
		tmp = 2.6666666666666665f * (s * powf(u, 3.0f));
	} else {
		tmp = s * -(((float) M_PI) / s);
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(2.6666666666666665) * Float32(s * (u ^ Float32(3.0))));
	else
		tmp = Float32(s * Float32(-Float32(Float32(pi) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	tmp = single(0.0);
	if (s <= single(1.000000031374395e-22))
		tmp = single(2.6666666666666665) * (s * (u ^ single(3.0)));
	else
		tmp = s * -(single(pi) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 1.00000003e-22

   1. Initial program 98.6%

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

      1. distribute-lft-neg-out 98.6%

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

      2. distribute-rgt-neg-in 98.6%

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

      3. sub-neg 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in s around inf 21.9%

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

   5. Taylor expanded in s around 0 22.3%

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

      1. mul-1-neg 22.3%

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

      2. *-commutative 22.3%

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

      3. distribute-rgt-neg-in 22.3%

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

   7. Simplified 22.3%

      \[\leadsto \color{blue}{\left(\log \left(-4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)\right) - \log s\right) \cdot \left(-s\right)} \]

   8. Taylor expanded in u around 0 23.0%

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {u}^{2} + \left(\log \pi + \left(-2 \cdot u + -2.6666666666666665 \cdot {u}^{3}\right)\right)\right)} - \log s\right) \cdot \left(-s\right) \]

   9. Taylor expanded in u around inf 14.7%

      \[\leadsto \color{blue}{2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 14.7%

          \[\leadsto 2.6666666666666665 \cdot \color{blue}{\left({u}^{3} \cdot s\right)} \]

   11. Simplified 14.7%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \left({u}^{3} \cdot s\right)} \]

   if 1.00000003e-22 < s

   1. Initial program 98.7%

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

      1. distribute-lft-neg-out 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. sub-neg 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in s around inf 26.6%

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

   5. Taylor expanded in u around 0 26.5%

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

      1. mul-1-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. distribute-rgt-neg-in 26.5%

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

      4. log1p-def 26.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \cdot \left(-s\right) \]

   7. Simplified 26.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]

   8. Taylor expanded in s around inf 13.2%

      \[\leadsto \color{blue}{\frac{\pi}{s}} \cdot \left(-s\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(-\frac{\pi}{s}\right)\\ \end{array} \]

Alternative 11: 14.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-1 + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.000000031374395e-22)
   (* 2.6666666666666665 (* s (pow u 3.0)))
   (* PI (+ -1.0 (* u 2.0)))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 1.000000031374395e-22f) {
		tmp = 2.6666666666666665f * (s * powf(u, 3.0f));
	} else {
		tmp = ((float) M_PI) * (-1.0f + (u * 2.0f));
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(2.6666666666666665) * Float32(s * (u ^ Float32(3.0))));
	else
		tmp = Float32(Float32(pi) * Float32(Float32(-1.0) + Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	tmp = single(0.0);
	if (s <= single(1.000000031374395e-22))
		tmp = single(2.6666666666666665) * (s * (u ^ single(3.0)));
	else
		tmp = single(pi) * (single(-1.0) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 1.00000003e-22

   1. Initial program 98.6%

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

      1. distribute-lft-neg-out 98.6%

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

      2. distribute-rgt-neg-in 98.6%

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

      3. sub-neg 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in s around inf 21.9%

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

   5. Taylor expanded in s around 0 22.3%

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

      1. mul-1-neg 22.3%

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

      2. *-commutative 22.3%

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

      3. distribute-rgt-neg-in 22.3%

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

   7. Simplified 22.3%

      \[\leadsto \color{blue}{\left(\log \left(-4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)\right) - \log s\right) \cdot \left(-s\right)} \]

   8. Taylor expanded in u around 0 23.0%

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {u}^{2} + \left(\log \pi + \left(-2 \cdot u + -2.6666666666666665 \cdot {u}^{3}\right)\right)\right)} - \log s\right) \cdot \left(-s\right) \]

   9. Taylor expanded in u around inf 14.7%

      \[\leadsto \color{blue}{2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 14.7%

          \[\leadsto 2.6666666666666665 \cdot \color{blue}{\left({u}^{3} \cdot s\right)} \]

   11. Simplified 14.7%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \left({u}^{3} \cdot s\right)} \]

   if 1.00000003e-22 < s

   1. Initial program 98.7%

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

      1. distribute-lft-neg-out 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. sub-neg 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in u around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. unsub-neg 1.4%

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

   6. Simplified 1.4%

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

   7. Taylor expanded in s around -inf 13.7%

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

      1. associate-*r/ 13.7%

         \[\leadsto \frac{s}{\color{blue}{\frac{-0.25 \cdot s}{\left(-0.25 \cdot \pi - 0.25 \cdot \pi\right) \cdot u}}} - \pi \]

      2. *-commutative 13.7%

         \[\leadsto \frac{s}{\frac{-0.25 \cdot s}{\color{blue}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}}} - \pi \]

      3. distribute-rgt-out-- 13.7%

         \[\leadsto \frac{s}{\frac{-0.25 \cdot s}{u \cdot \color{blue}{\left(\pi \cdot \left(-0.25 - 0.25\right)\right)}}} - \pi \]

      4. metadata-eval 13.7%

         \[\leadsto \frac{s}{\frac{-0.25 \cdot s}{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right)}} - \pi \]

   9. Simplified 13.7%

      \[\leadsto \frac{s}{\color{blue}{\frac{-0.25 \cdot s}{u \cdot \left(\pi \cdot -0.5\right)}}} - \pi \]

   10. Taylor expanded in s around 0 13.7%

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

       1. sub-neg 13.7%

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

       2. associate-*r* 13.7%

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

       3. neg-mul-1 13.7%

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

       4. distribute-rgt-out 13.7%

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

   12. Simplified 13.7%

       \[\leadsto \color{blue}{\pi \cdot \left(2 \cdot u + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-1 + u \cdot 2\right)\\ \end{array} \]

Alternative 12: 14.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;{u}^{3} \cdot \left(s \cdot 2.6666666666666665\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-1 + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.000000031374395e-22)
   (* (pow u 3.0) (* s 2.6666666666666665))
   (* PI (+ -1.0 (* u 2.0)))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 1.000000031374395e-22f) {
		tmp = powf(u, 3.0f) * (s * 2.6666666666666665f);
	} else {
		tmp = ((float) M_PI) * (-1.0f + (u * 2.0f));
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(1.000000031374395e-22))
		tmp = Float32((u ^ Float32(3.0)) * Float32(s * Float32(2.6666666666666665)));
	else
		tmp = Float32(Float32(pi) * Float32(Float32(-1.0) + Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	tmp = single(0.0);
	if (s <= single(1.000000031374395e-22))
		tmp = (u ^ single(3.0)) * (s * single(2.6666666666666665));
	else
		tmp = single(pi) * (single(-1.0) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 1.00000003e-22

   1. Initial program 98.6%

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

      1. distribute-lft-neg-out 98.6%

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

      2. distribute-rgt-neg-in 98.6%

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

      3. sub-neg 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in s around inf 21.9%

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

   5. Taylor expanded in s around 0 22.3%

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

      1. mul-1-neg 22.3%

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

      2. *-commutative 22.3%

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

      3. distribute-rgt-neg-in 22.3%

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

   7. Simplified 22.3%

      \[\leadsto \color{blue}{\left(\log \left(-4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)\right) - \log s\right) \cdot \left(-s\right)} \]

   8. Taylor expanded in u around 0 23.0%

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {u}^{2} + \left(\log \pi + \left(-2 \cdot u + -2.6666666666666665 \cdot {u}^{3}\right)\right)\right)} - \log s\right) \cdot \left(-s\right) \]

   9. Taylor expanded in u around inf 14.7%

      \[\leadsto \color{blue}{2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 14.7%

          \[\leadsto \color{blue}{\left(s \cdot {u}^{3}\right) \cdot 2.6666666666666665} \]

       2. *-commutative 14.7%

          \[\leadsto \color{blue}{\left({u}^{3} \cdot s\right)} \cdot 2.6666666666666665 \]

       3. associate-*l* 14.7%

          \[\leadsto \color{blue}{{u}^{3} \cdot \left(s \cdot 2.6666666666666665\right)} \]

   11. Simplified 14.7%

       \[\leadsto \color{blue}{{u}^{3} \cdot \left(s \cdot 2.6666666666666665\right)} \]

   if 1.00000003e-22 < s

   1. Initial program 98.7%

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

      1. distribute-lft-neg-out 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. sub-neg 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in u around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. unsub-neg 1.4%

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

   6. Simplified 1.4%

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

   7. Taylor expanded in s around -inf 13.7%

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

      1. associate-*r/ 13.7%

         \[\leadsto \frac{s}{\color{blue}{\frac{-0.25 \cdot s}{\left(-0.25 \cdot \pi - 0.25 \cdot \pi\right) \cdot u}}} - \pi \]

      2. *-commutative 13.7%

         \[\leadsto \frac{s}{\frac{-0.25 \cdot s}{\color{blue}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}}} - \pi \]

      3. distribute-rgt-out-- 13.7%

         \[\leadsto \frac{s}{\frac{-0.25 \cdot s}{u \cdot \color{blue}{\left(\pi \cdot \left(-0.25 - 0.25\right)\right)}}} - \pi \]

      4. metadata-eval 13.7%

         \[\leadsto \frac{s}{\frac{-0.25 \cdot s}{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right)}} - \pi \]

   9. Simplified 13.7%

      \[\leadsto \frac{s}{\color{blue}{\frac{-0.25 \cdot s}{u \cdot \left(\pi \cdot -0.5\right)}}} - \pi \]

   10. Taylor expanded in s around 0 13.7%

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

       1. sub-neg 13.7%

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

       2. associate-*r* 13.7%

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

       3. neg-mul-1 13.7%

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

       4. distribute-rgt-out 13.7%

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

   12. Simplified 13.7%

       \[\leadsto \color{blue}{\pi \cdot \left(2 \cdot u + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;{u}^{3} \cdot \left(s \cdot 2.6666666666666665\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-1 + u \cdot 2\right)\\ \end{array} \]

Alternative 13: 11.3% accurate, 7.2× 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 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in u around 0 10.5%

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

   1. mul-1-neg 10.5%

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

6. Simplified 10.5%

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

7. Final simplification 10.5%

   \[\leadsto -\pi \]

Alternative 14: 10.4% accurate, 243.3× speedup

Math

\[\begin{array}{l} \\ s \cdot 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s 0.0))

C

float code(float u, float s) {
	return s * 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * 0.0e0
end function

Julia

function code(u, s)
	return Float32(s * Float32(0.0))
end

MATLAB

function tmp = code(u, s)
	tmp = s * single(0.0);
end

Derivation

1. Initial program 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around inf 10.7%

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

5. Final simplification 10.7%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023188 
(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))))