Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 18.2s

Alternatives: 10

Speedup: 0.2×

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: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := t\_0 - -1\\ t_2 := \frac{u}{e^{\frac{\pi}{-s}} - -1}\\ t_3 := \frac{u}{-1 - t\_0}\\ t_4 := \frac{{t\_2}^{3} - {\left(\frac{u}{t\_1}\right)}^{3}}{\mathsf{fma}\left(t\_3 - t\_2, t\_3, {t\_2}^{2}\right)} - \frac{-1}{t\_1}\\ s \cdot \left(-\log \left(\frac{1 - t\_4}{t\_4}\right)\right) \end{array} \]

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 (/ u (- -1.0 t_0)))
        (t_4
         (-
          (/
           (- (pow t_2 3.0) (pow (/ u t_1) 3.0))
           (fma (- t_3 t_2) t_3 (pow t_2 2.0)))
          (/ -1.0 t_1))))
   (* s (- (log (/ (- 1.0 t_4) t_4))))))

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 = u / (-1.0f - t_0);
	float t_4 = ((powf(t_2, 3.0f) - powf((u / t_1), 3.0f)) / fmaf((t_3 - t_2), t_3, powf(t_2, 2.0f))) - (-1.0f / t_1);
	return s * -logf(((1.0f - t_4) / t_4));
}

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(pi) / Float32(-s))) - Float32(-1.0)))
	t_3 = Float32(u / Float32(Float32(-1.0) - t_0))
	t_4 = Float32(Float32(Float32((t_2 ^ Float32(3.0)) - (Float32(u / t_1) ^ Float32(3.0))) / fma(Float32(t_3 - t_2), t_3, (t_2 ^ Float32(2.0)))) - Float32(Float32(-1.0) / t_1))
	return Float32(s * Float32(-log(Float32(Float32(Float32(1.0) - t_4) / t_4))))
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. flip3-+ N/A

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

3. Applied rewrites 98.8%

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

5. Applied rewrites 98.8%

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

7. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 - \left(\frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(\frac{u}{-1 - e^{\frac{\pi}{s}}} - \frac{u}{e^{\frac{\pi}{-s}} - -1}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)} - \frac{-1}{e^{\frac{\pi}{s}} - -1}\right)\right) \cdot \frac{-1}{\frac{-1}{e^{\frac{\pi}{s}} - -1} - \frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(\frac{u}{-1 - e^{\frac{\pi}{s}}} - \frac{u}{e^{\frac{\pi}{-s}} - -1}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)}}\right)} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

9. Applied rewrites 98.9%

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

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := t\_0 - -1\\ t_2 := \frac{-1}{t\_1}\\ t_3 := \frac{u}{e^{\frac{\pi}{-s}} - -1}\\ t_4 := \frac{{t\_3}^{3} - {\left(\frac{u}{t\_1}\right)}^{3}}{\mathsf{fma}\left(-0.5 \cdot u - 0.5 \cdot u, \frac{u}{-1 - t\_0}, {t\_3}^{2}\right)}\\ \left(-s\right) \cdot \log \left(\left(1 - \left(t\_4 - t\_2\right)\right) \cdot \frac{-1}{t\_2 - t\_4}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s)))
        (t_1 (- t_0 -1.0))
        (t_2 (/ -1.0 t_1))
        (t_3 (/ u (- (exp (/ PI (- s))) -1.0)))
        (t_4
         (/
          (- (pow t_3 3.0) (pow (/ u t_1) 3.0))
          (fma (- (* -0.5 u) (* 0.5 u)) (/ u (- -1.0 t_0)) (pow t_3 2.0)))))
   (* (- s) (log (* (- 1.0 (- t_4 t_2)) (/ -1.0 (- t_2 t_4)))))))

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 = -1.0f / t_1;
	float t_3 = u / (expf((((float) M_PI) / -s)) - -1.0f);
	float t_4 = (powf(t_3, 3.0f) - powf((u / t_1), 3.0f)) / fmaf(((-0.5f * u) - (0.5f * u)), (u / (-1.0f - t_0)), powf(t_3, 2.0f));
	return -s * logf(((1.0f - (t_4 - t_2)) * (-1.0f / (t_2 - t_4))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(t_0 - Float32(-1.0))
	t_2 = Float32(Float32(-1.0) / t_1)
	t_3 = Float32(u / Float32(exp(Float32(Float32(pi) / Float32(-s))) - Float32(-1.0)))
	t_4 = Float32(Float32((t_3 ^ Float32(3.0)) - (Float32(u / t_1) ^ Float32(3.0))) / fma(Float32(Float32(Float32(-0.5) * u) - Float32(Float32(0.5) * u)), Float32(u / Float32(Float32(-1.0) - t_0)), (t_3 ^ Float32(2.0))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) - Float32(t_4 - t_2)) * Float32(Float32(-1.0) / Float32(t_2 - t_4)))))
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. flip3-+ N/A

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

3. Applied rewrites 98.8%

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

5. Applied rewrites 98.8%

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\left(1 - \left(\frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot u - \frac{1}{2} \cdot u}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)} - \frac{-1}{e^{\frac{\pi}{s}} - -1}\right)\right) \cdot \frac{-1}{\frac{-1}{e^{\frac{\pi}{s}} - -1} - \frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(\frac{u}{-1 - e^{\frac{\pi}{s}}} - \frac{u}{e^{\frac{\pi}{-s}} - -1}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower--.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\left(1 - \left(\frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(-0.5 \cdot u - 0.5 \cdot \color{blue}{u}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)} - \frac{-1}{e^{\frac{\pi}{s}} - -1}\right)\right) \cdot \frac{-1}{\frac{-1}{e^{\frac{\pi}{s}} - -1} - \frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(\frac{u}{-1 - e^{\frac{\pi}{s}}} - \frac{u}{e^{\frac{\pi}{-s}} - -1}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)}}\right) \]

10. Applied rewrites 98.9%

    \[\leadsto \left(-s\right) \cdot \log \left(\left(1 - \left(\frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(\color{blue}{-0.5 \cdot u - 0.5 \cdot u}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)} - \frac{-1}{e^{\frac{\pi}{s}} - -1}\right)\right) \cdot \frac{-1}{\frac{-1}{e^{\frac{\pi}{s}} - -1} - \frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(\frac{u}{-1 - e^{\frac{\pi}{s}}} - \frac{u}{e^{\frac{\pi}{-s}} - -1}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)}}\right) \]

11. Taylor expanded in s around inf

    \[\leadsto \left(-s\right) \cdot \log \left(\left(1 - \left(\frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(-0.5 \cdot u - 0.5 \cdot u, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)} - \frac{-1}{e^{\frac{\pi}{s}} - -1}\right)\right) \cdot \frac{-1}{\frac{-1}{e^{\frac{\pi}{s}} - -1} - \frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot u - \frac{1}{2} \cdot u}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower--.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-*.f32 98.9%

       \[\leadsto \left(-s\right) \cdot \log \left(\left(1 - \left(\frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(-0.5 \cdot u - 0.5 \cdot u, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)} - \frac{-1}{e^{\frac{\pi}{s}} - -1}\right)\right) \cdot \frac{-1}{\frac{-1}{e^{\frac{\pi}{s}} - -1} - \frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(-0.5 \cdot u - 0.5 \cdot \color{blue}{u}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)}}\right) \]

13. Applied rewrites 98.9%

    \[\leadsto \left(-s\right) \cdot \log \left(\left(1 - \left(\frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(-0.5 \cdot u - 0.5 \cdot u, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)} - \frac{-1}{e^{\frac{\pi}{s}} - -1}\right)\right) \cdot \frac{-1}{\frac{-1}{e^{\frac{\pi}{s}} - -1} - \frac{{\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{3} - {\left(\frac{u}{e^{\frac{\pi}{s}} - -1}\right)}^{3}}{\mathsf{fma}\left(\color{blue}{-0.5 \cdot u - 0.5 \cdot u}, \frac{u}{-1 - e^{\frac{\pi}{s}}}, {\left(\frac{u}{e^{\frac{\pi}{-s}} - -1}\right)}^{2}\right)}}\right) \]
14. Add Preprocessing

Alternative 3: 97.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (*
      u
      (-
       (/ 1.0 (+ 1.0 (exp (* -1.0 (/ PI s)))))
       (/ 1.0 (+ 1.0 (exp (/ 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 + expf((((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) + exp(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) + 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 97.5%

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

Alternative 4: 85.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 2.0 (/ 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 / (2.0f + (((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(2.0) + 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(2.0) + (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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 94.8%

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

4. Applied rewrites 94.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 85.9%

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

7. Applied rewrites 85.9%

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

Alternative 5: 24.8% 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 24.8%

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

Alternative 6: 14.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 14.5%

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

7. Applied rewrites 14.5%

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

Alternative 7: 14.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * (single(1.0) / ((u * ((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 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 14.5%

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

7. Applied rewrites 14.5%

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

Alternative 8: 14.5% 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 14.5%

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

7. Applied rewrites 14.5%

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

Alternative 9: 11.5% 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 11.5%

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

4. Applied rewrites 11.5%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
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 11.5%

      \[\leadsto -\pi \]

6. Applied rewrites 11.5%

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

7. Evaluated real constant 11.5%

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

Reproduce

herbie shell --seed 2025187 
(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))))