Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 14.1s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (exp (/ PI s)))))
   (*
    (- s)
    (log
     (+
      -1.0
      (/
       1.0
       (fma
        u
        (+ (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ -1.0 t_0))
        (/ 1.0 t_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 / fmaf(u, ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) + (-1.0f / t_0)), (1.0f / t_0)))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) + Float32(Float32(-1.0) / t_0)), Float32(Float32(1.0) / t_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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 37.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (exp (/ PI s)))))
   (* (- s) (log (+ -1.0 (/ 1.0 (fma u (+ (/ -1.0 t_0) 0.5) (/ 1.0 t_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 / fmaf(u, ((-1.0f / t_0) + 0.5f), (1.0f / t_0)))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(u, Float32(Float32(Float32(-1.0) / t_0) + Float32(0.5)), Float32(Float32(1.0) / t_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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in s around inf

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

8. Simplified 38.0%

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

9. Taylor expanded in s around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

    3. distribute-rgt-neg-in N/A

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

    4. mul-1-neg N/A

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

    5. lower-*.f32 N/A

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

11. Simplified 38.0%

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

12. Final simplification 38.0%

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

Alternative 3: 37.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{\mathsf{fma}\left(u, 0.5 + \frac{-1}{1 + \left(1 + \frac{\pi + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, \pi \cdot \left(\pi \cdot 0.5\right)\right)}{s}}{s}\right)}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (fma
      u
      (+
       0.5
       (/
        -1.0
        (+
         1.0
         (+
          1.0
          (/
           (+
            PI
            (/
             (fma 0.16666666666666666 (/ (* PI (* PI PI)) s) (* PI (* PI 0.5)))
             s))
           s)))))
      (/ 1.0 (+ 1.0 (exp (/ PI s))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + (1.0f / fmaf(u, (0.5f + (-1.0f / (1.0f + (1.0f + ((((float) M_PI) + (fmaf(0.16666666666666666f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) / s), (((float) M_PI) * (((float) M_PI) * 0.5f))) / s)) / s))))), (1.0f / (1.0f + expf((((float) M_PI) / s))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(u, Float32(Float32(0.5) + Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(pi) + Float32(fma(Float32(0.16666666666666666), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) / s), Float32(Float32(pi) * Float32(Float32(pi) * Float32(0.5)))) / s)) / s))))), Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in s around inf

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

8. Simplified 38.0%

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

9. Taylor expanded in s around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

    3. distribute-rgt-neg-in N/A

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

    4. mul-1-neg N/A

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

    5. lower-*.f32 N/A

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

11. Simplified 38.0%

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

12. Taylor expanded in s around -inf

    \[\leadsto \log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{2} + \frac{-1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}, \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} + -1\right) \cdot \left(\mathsf{neg}\left(s\right)\right) \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{2} + \frac{-1}{1 + \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)}\right)}, \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} + -1\right) \cdot \left(\mathsf{neg}\left(s\right)\right) \]

    2. unsub-neg N/A

       \[\leadsto \log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{2} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}, \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} + -1\right) \cdot \left(\mathsf{neg}\left(s\right)\right) \]

    3. lower--.f32 N/A

       \[\leadsto \log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{2} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}, \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} + -1\right) \cdot \left(\mathsf{neg}\left(s\right)\right) \]

    4. lower-/.f32 N/A

       \[\leadsto \log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{2} + \frac{-1}{1 + \left(1 - \color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}}\right)}, \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} + -1\right) \cdot \left(\mathsf{neg}\left(s\right)\right) \]

14. Simplified 38.0%

    \[\leadsto \log \left(\frac{1}{\mathsf{fma}\left(u, 0.5 + \frac{-1}{1 + \color{blue}{\left(1 - \frac{\left(-\pi\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, \pi \cdot \left(\pi \cdot 0.5\right)\right)}{s}}{s}\right)}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + -1\right) \cdot \left(-s\right) \]

15. Final simplification 38.0%

    \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{1}{\mathsf{fma}\left(u, 0.5 + \frac{-1}{1 + \left(1 + \frac{\pi + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, \pi \cdot \left(\pi \cdot 0.5\right)\right)}{s}}{s}\right)}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}\right) \]
16. Add Preprocessing

Alternative 4: 37.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(u, Float32(Float32(0.5) + Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(fma(Float32(-0.5), Float32(Float32(Float32(pi) * Float32(pi)) / s), Float32(-Float32(pi))) / s))))), Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in s around inf

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

8. Simplified 38.0%

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

9. Taylor expanded in s around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

    3. distribute-rgt-neg-in N/A

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

    4. mul-1-neg N/A

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

    5. lower-*.f32 N/A

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

11. Simplified 38.0%

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

12. Taylor expanded in s around -inf

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 N/A

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

    5. +-commutative N/A

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

    6. lower-fma.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lower-PI.f32 N/A

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

    11. lower-PI.f32 N/A

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

    12. mul-1-neg N/A

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

    13. lower-neg.f32 N/A

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

    14. lower-PI.f32 38.0

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

14. Simplified 38.0%

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

15. Final simplification 38.0%

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

Alternative 5: 37.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(u, Float32(Float32(0.5) + Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(pi) / s))))), Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in s around inf

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

8. Simplified 38.0%

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

9. Taylor expanded in s around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

    3. distribute-rgt-neg-in N/A

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

    4. mul-1-neg N/A

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

    5. lower-*.f32 N/A

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

11. Simplified 38.0%

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

12. Taylor expanded in s around inf

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-PI.f32 38.0

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

14. Simplified 38.0%

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

15. Final simplification 38.0%

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

Alternative 6: 37.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(1.0) / (u * ((single(-1.0) / (single(1.0) + exp((single(pi) / s)))) + single(0.5))))));
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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in s around inf

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

8. Simplified 38.0%

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

9. Taylor expanded in u around inf

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. lower-+.f32 N/A

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

11. Simplified 37.7%

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

12. Final simplification 37.7%

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

Alternative 7: 36.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(fma(Float32(-0.5), Float32(Float32(Float32(pi) * Float32(pi)) / s), Float32(-Float32(pi))) / s)))
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(u, Float32(Float32(Float32(-1.0) / t_0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(1.0)))), Float32(Float32(1.0) / t_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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in s around inf

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

8. Simplified 38.0%

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

9. Taylor expanded in s around -inf

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 N/A

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

    5. +-commutative N/A

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

    6. lower-fma.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lower-PI.f32 N/A

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

    11. lower-PI.f32 N/A

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

    12. mul-1-neg N/A

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

    13. lower-neg.f32 N/A

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

    14. lower-PI.f32 37.5

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

11. Simplified 37.5%

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

12. Taylor expanded in s around -inf

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 N/A

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

    5. +-commutative N/A

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

    6. lower-fma.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lower-PI.f32 N/A

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

    11. lower-PI.f32 N/A

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

    12. mul-1-neg N/A

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

    13. lower-neg.f32 N/A

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

    14. lower-PI.f32 37.5

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

14. Simplified 37.5%

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

15. Final simplification 37.5%

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

Alternative 8: 36.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(u, Float32(Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(pi) / s)))) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(1.0)))), Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(fma(Float32(-0.5), Float32(Float32(Float32(pi) * Float32(pi)) / s), Float32(-Float32(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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in s around inf

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

8. Simplified 38.0%

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

9. Taylor expanded in s around -inf

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 N/A

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

    4. lower-/.f32 N/A

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

    5. +-commutative N/A

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

    6. lower-fma.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lower-PI.f32 N/A

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

    11. lower-PI.f32 N/A

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

    12. mul-1-neg N/A

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

    13. lower-neg.f32 N/A

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

    14. lower-PI.f32 37.5

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

11. Simplified 37.5%

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

12. Taylor expanded in s around inf

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-PI.f32 37.5

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

14. Simplified 37.5%

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

15. Final simplification 37.5%

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

Alternative 9: 25.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(1.0) - Float32(fma(Float32(pi), u, Float32(-Float32(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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in s around inf

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

8. Simplified 38.0%

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

9. Taylor expanded in s around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

    3. distribute-rgt-neg-in N/A

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

    4. mul-1-neg N/A

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

    5. lower-*.f32 N/A

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

11. Simplified 38.0%

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

12. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. associate-*r/ N/A

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

    3. distribute-lft-out-- N/A

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

    4. associate-*r* N/A

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

    5. metadata-eval N/A

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

    6. associate-*r/ N/A

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

    7. mul-1-neg N/A

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

    8. distribute-neg-frac2 N/A

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

    9. mul-1-neg N/A

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

    10. lower-/.f32 N/A

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

    11. sub-neg N/A

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

    12. *-commutative N/A

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

    13. mul-1-neg N/A

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

    14. lower-fma.f32 N/A

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

    15. lower-PI.f32 N/A

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

    16. mul-1-neg N/A

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

    17. lower-neg.f32 N/A

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

    18. lower-PI.f32 N/A

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

    19. mul-1-neg N/A

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

    20. lower-neg.f32 25.2

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

14. Simplified 25.2%

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

15. Final simplification 25.2%

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

Alternative 10: 11.6% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(u * Float32(fma(Float32(pi), Float32(u * Float32(-2.0)), Float32(pi)) / Float32(-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. Add Preprocessing

3. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

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

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

   6. metadata-eval N/A

      \[\leadsto 4 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot u + \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto 4 \cdot \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u\right)} + \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   8. *-commutative N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u\right) + \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}\right) \]

   9. distribute-lft-out N/A

      \[\leadsto 4 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto 4 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right)} \]

   11. lower-PI.f32 N/A

       \[\leadsto 4 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right) \]

   12. lower-fma.f32 11.3

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

5. Simplified 11.3%

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

6. Taylor expanded in u around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f32 11.3

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

8. Simplified 11.3%

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

9. Taylor expanded in u around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. associate-*l* N/A

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

    7. lower-fma.f32 N/A

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

    8. lower-PI.f32 N/A

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

    9. lower-*.f32 N/A

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

    10. lower-PI.f32 11.3

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

11. Simplified 11.3%

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

12. Final simplification 11.3%

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

Alternative 11: 11.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(4.0) * fma(u, Float32(Float32(pi) * Float32(0.5)), Float32(Float32(pi) * Float32(-0.25))))
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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. cancel-sign-sub-inv N/A

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

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

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

   4. metadata-eval N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right) \]

   5. *-commutative N/A

      \[\leadsto 4 \cdot \left(u \cdot \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\frac{-1}{4}} \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto 4 \cdot \color{blue}{\mathsf{fma}\left(u, \frac{1}{2} \cdot \mathsf{PI}\left(\right), \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto 4 \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto 4 \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   10. lower-PI.f32 N/A

       \[\leadsto 4 \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   11. *-commutative N/A

       \[\leadsto 4 \cdot \mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot \frac{1}{2}, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}\right) \]

   12. lower-*.f32 N/A

       \[\leadsto 4 \cdot \mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot \frac{1}{2}, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}\right) \]

   13. lower-PI.f32 11.3

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

8. Simplified 11.3%

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

Alternative 12: 11.6% accurate, 42.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

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

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

   6. metadata-eval N/A

      \[\leadsto 4 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot u + \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto 4 \cdot \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u\right)} + \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   8. *-commutative N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u\right) + \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}\right) \]

   9. distribute-lft-out N/A

      \[\leadsto 4 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto 4 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right)} \]

   11. lower-PI.f32 N/A

       \[\leadsto 4 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right) \]

   12. lower-fma.f32 11.3

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

5. Simplified 11.3%

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

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-fma.f32 11.3

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

8. Simplified 11.3%

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

Alternative 13: 11.4% accurate, 170.0× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   3. lower-PI.f32 11.3

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

5. Simplified 11.3%

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

Reproduce

herbie shell --seed 2024215 
(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))))