Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 11.1s

Alternatives: 10

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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / Float32(-s))) + Float32(1.0))) - Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0)))) * u) + Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(pi) / s)))))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((((single(1.0) / (exp((single(pi) / -s)) + single(1.0))) - (single(1.0) / (exp((single(pi) / s)) + single(1.0)))) * u) + (single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (single(pi) / s)))))) - single(1.0)));
end

Derivation

1. Initial program 99.0%

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

   1. lift-exp.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. *-lft-identity N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-PI.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-exp.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. *-lft-identity N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-PI.f32 99.0

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 99.0%

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

Alternative 3: 97.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in u around inf

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

6. Applied rewrites 97.4%

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

Alternative 4: 37.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u around inf

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

4. Applied rewrites 97.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 37.2%

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

Alternative 5: 37.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{1 + 1} - \frac{1}{2 + \frac{\pi}{s}}\right) \cdot u} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (- (/ 1.0 (* (- (/ 1.0 (+ 1.0 1.0)) (/ 1.0 (+ 2.0 (/ PI s)))) u)) 1.0))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u around inf

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

4. Applied rewrites 97.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 37.2%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.1%

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

Alternative 6: 37.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u around inf

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

4. Applied rewrites 97.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 37.2%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 37.1%

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

11. Taylor expanded in s around 0

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

    1. lift-/.f32 N/A

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

    2. lift-PI.f32 37.1

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

13. Applied rewrites 37.1%

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

Alternative 7: 25.0% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (fma (/ (* PI (fma 0.5 u -0.25)) s) -4.0 1.0))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

   1. lift-exp.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. *-lft-identity N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-PI.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-exp.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. *-lft-identity N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-PI.f32 99.0

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 25.0%

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

Alternative 8: 25.0% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (fma (* PI (fma 0.5 u -0.25)) (/ -4.0 s) 1.0))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

   1. lift-exp.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. *-lft-identity N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-PI.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-exp.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. *-lft-identity N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-PI.f32 99.0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 25.0%

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

Alternative 9: 11.8% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

   1. lift-exp.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. *-lft-identity N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-PI.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-exp.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. *-lft-identity N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f32 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-PI.f32 99.0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 11.8%

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

Alternative 10: 11.6% accurate, 46.3× 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 99.0%

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

2. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f32 N/A

      \[\leadsto -\mathsf{PI}\left(\right) \]

   3. lift-PI.f32 11.6

      \[\leadsto -\pi \]

4. Applied rewrites 11.6%

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

Reproduce

herbie shell --seed 2025130 
(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))))