Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 9.0s

Alternatives: 14

Speedup: 0.7×

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
 (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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := t\_0 - -1\\ t_2 := \frac{u}{e^{\frac{-\pi}{s}} - -1}\\ t_3 := \frac{{t\_2}^{2} - {\left(\frac{u}{t\_1}\right)}^{2}}{t\_2 - \frac{u}{-1 - t\_0}}\\ t_4 := \frac{-1}{t\_1}\\ \left(-s\right) \cdot \log \left(\left(1 - \left(t\_3 - t\_4\right)\right) \cdot \frac{-1}{t\_4 - t\_3}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s)))
        (t_1 (- t_0 -1.0))
        (t_2 (/ u (- (exp (/ (- PI) s)) -1.0)))
        (t_3
         (/ (- (pow t_2 2.0) (pow (/ u t_1) 2.0)) (- t_2 (/ u (- -1.0 t_0)))))
        (t_4 (/ -1.0 t_1)))
   (* (- s) (log (* (- 1.0 (- t_3 t_4)) (/ -1.0 (- t_4 t_3)))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = t_0 - -1.0f;
	float t_2 = u / (expf((-((float) M_PI) / s)) - -1.0f);
	float t_3 = (powf(t_2, 2.0f) - powf((u / t_1), 2.0f)) / (t_2 - (u / (-1.0f - t_0)));
	float t_4 = -1.0f / t_1;
	return -s * logf(((1.0f - (t_3 - t_4)) * (-1.0f / (t_4 - t_3))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	t_2 = Float32(u / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0)))
	t_3 = Float32(Float32((t_2 ^ Float32(2.0)) - (Float32(u / t_1) ^ Float32(2.0))) / Float32(t_2 - Float32(u / Float32(Float32(-1.0) - t_0))))
	t_4 = Float32(Float32(-1.0) / t_1)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) - Float32(t_3 - t_4)) * Float32(Float32(-1.0) / Float32(t_4 - t_3)))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = t_0 - single(-1.0);
	t_2 = u / (exp((-single(pi) / s)) - single(-1.0));
	t_3 = ((t_2 ^ single(2.0)) - ((u / t_1) ^ single(2.0))) / (t_2 - (u / (single(-1.0) - t_0)));
	t_4 = single(-1.0) / t_1;
	tmp = -s * log(((single(1.0) - (t_3 - t_4)) * (single(-1.0) / (t_4 - t_3))));
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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-lft-in N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f32 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f32 N/A

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

5. Applied rewrites 98.9%

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

   1. lift--.f32 N/A

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

   2. sub-to-mult N/A

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

7. Applied rewrites 99.0%

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

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{\frac{0.5 \cdot u - -0.5 \cdot u}{{\left(\frac{u}{e^{\frac{-\pi}{s}} - -1}\right)}^{2} - {\left(\frac{u}{t\_0 - -1}\right)}^{2}}} + \frac{1}{1 + t\_0}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s))))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (+
        (/
         1.0
         (/
          (- (* 0.5 u) (* -0.5 u))
          (-
           (pow (/ u (- (exp (/ (- PI) s)) -1.0)) 2.0)
           (pow (/ u (- t_0 -1.0)) 2.0))))
        (/ 1.0 (+ 1.0 t_0))))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	return -s * logf(((1.0f / ((1.0f / (((0.5f * u) - (-0.5f * u)) / (powf((u / (expf((-((float) M_PI) / s)) - -1.0f)), 2.0f) - powf((u / (t_0 - -1.0f)), 2.0f)))) + (1.0f / (1.0f + t_0)))) - 1.0f));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) * u) - Float32(Float32(-0.5) * u)) / Float32((Float32(u / Float32(exp(Float32(Float32(-Float32(pi)) / s)) - Float32(-1.0))) ^ Float32(2.0)) - (Float32(u / Float32(t_0 - Float32(-1.0))) ^ Float32(2.0))))) + Float32(Float32(1.0) / Float32(Float32(1.0) + t_0)))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	tmp = -s * log(((single(1.0) / ((single(1.0) / (((single(0.5) * u) - (single(-0.5) * u)) / (((u / (exp((-single(pi) / s)) - single(-1.0))) ^ single(2.0)) - ((u / (t_0 - single(-1.0))) ^ single(2.0))))) + (single(1.0) / (single(1.0) + t_0)))) - 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-lft-in N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f32 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f32 N/A

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in s around inf

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

   1. lower--.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 98.9%

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

8. Applied rewrites 98.9%

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

Alternative 3: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 98.9%

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

Alternative 4: 97.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- 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 -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(-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. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in u around 0

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 97.7%

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

Alternative 5: 97.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.6%

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

6. Applied rewrites 97.6%

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

Alternative 6: 94.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.4%

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

7. Applied rewrites 94.4%

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

Alternative 7: 25.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + (single(4.0) * (((u * ((single(-0.25) * single(pi)) - (single(0.25) * single(pi)))) - (single(-0.25) * 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

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

4. Applied rewrites 25.0%

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

Alternative 8: 14.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * exp((log((((single(0.5) * single(pi)) * u) / 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 u around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.4%

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

7. Applied rewrites 14.4%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. lower-unsound-exp.f32 N/A

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

   5. lower-unsound-*.f32 N/A

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

9. Applied rewrites 14.4%

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

Alternative 9: 14.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * (single(1.0) / (single(1.0) / (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

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.4%

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

7. Applied rewrites 14.4%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f32 N/A

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

   4. lower-unsound-/.f32 14.4%

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 14.4%

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-rgt-out-- N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 14.4%

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

9. Applied rewrites 14.4%

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

Alternative 10: 14.4% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.4%

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

7. Applied rewrites 14.4%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. distribute-rgt-out-- N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f32 N/A

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

   15. lower-/.f32 14.4%

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

9. Applied rewrites 14.4%

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

Alternative 11: 14.4% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (/ s (* u (- (* 0.25 PI) (* -0.25 PI))))))

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * Float32(s / Float32(u * Float32(Float32(Float32(0.25) * Float32(pi)) - Float32(Float32(-0.25) * Float32(pi))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * (s / (u * ((single(0.25) * single(pi)) - (single(-0.25) * 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

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.4%

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

7. Applied rewrites 14.4%

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

Alternative 12: 14.4% accurate, 5.8× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (/ 1.0 (/ (* u 1.5707963705062866) s))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -s * (1.0e0 / ((u * 1.5707963705062866e0) / s))
end function

Julia

function code(u, s)
	return Float32(Float32(-s) * Float32(Float32(1.0) / Float32(Float32(u * Float32(1.5707963705062866)) / s)))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * (single(1.0) / ((u * single(1.5707963705062866)) / 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 u around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.4%

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

7. Applied rewrites 14.4%

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

8. Evaluated real constant 14.4%

   \[\leadsto \left(-s\right) \cdot \frac{1}{\frac{u \cdot 1.5707963705062866}{s}} \]
9. Add Preprocessing

Alternative 13: 11.4% accurate, 87.7× speedup

Math

\[-3.1415927410125732 \]

FPCore

(FPCore (u s) :precision binary32 -3.1415927410125732)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 11.4%

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

4. Applied rewrites 11.4%

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

   1. lift-*.f32 N/A

      \[\leadsto -1 \cdot \color{blue}{\pi} \]

   2. mul-1-neg N/A

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

   3. lift-neg.f32 11.4%

      \[\leadsto -\pi \]

6. Applied rewrites 11.4%

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

7. Evaluated real constant 11.4%

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

Reproduce

herbie shell --seed 2025184 
(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))))