Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 19.8s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(log(Float32(Float32(Float32(1.0) / fma(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(Float32(1.0) / Float32(exp(Float32(Float32(Float32(1.0) / s) * Float32(pi))) + Float32(1.0))))) - Float32(1.0))) * Float32(-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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-PI.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift-PI.f32 98.9

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

6. Applied rewrites 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

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(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))) * Float32(-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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 97.9%

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

6. Final simplification 97.9%

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

Alternative 4: 25.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{2}}} - 1\right) \]
4. Step-by-step derivation

5. Applied rewrites 10.3%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{0.5}} - 1\right) \]

6. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 25.0%

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

9. Final simplification 25.0%

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

Alternative 5: 13.9% accurate, 14.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) / s) * ((-s * s) / 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 u around 0

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

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 10.7

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

5. Applied rewrites 10.7%

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

   1. --rgt-identity N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(s\right)\right) - 0\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot 0}{\left(\mathsf{neg}\left(s\right)\right) + 0}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   3. lift-neg.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot 0}{\left(\mathsf{neg}\left(s\right)\right) + 0} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   4. lift-neg.f32 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)} - 0 \cdot 0}{\left(\mathsf{neg}\left(s\right)\right) + 0} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   5. sqr-neg N/A

      \[\leadsto \frac{\color{blue}{s \cdot s} - 0 \cdot 0}{\left(\mathsf{neg}\left(s\right)\right) + 0} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{s \cdot s} - 0 \cdot 0}{\left(\mathsf{neg}\left(s\right)\right) + 0} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   7. metadata-eval N/A

      \[\leadsto \frac{s \cdot s - \color{blue}{0}}{\left(\mathsf{neg}\left(s\right)\right) + 0} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   8. --rgt-identity N/A

      \[\leadsto \frac{\color{blue}{s \cdot s}}{\left(\mathsf{neg}\left(s\right)\right) + 0} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   9. metadata-eval N/A

      \[\leadsto \frac{s \cdot s}{\left(\mathsf{neg}\left(s\right)\right) + \color{blue}{\left(\mathsf{neg}\left(0\right)\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   10. sub-neg N/A

       \[\leadsto \frac{s \cdot s}{\color{blue}{\left(\mathsf{neg}\left(s\right)\right) - 0}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   11. --rgt-identity N/A

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

   12. lower-/.f32 13.3

       \[\leadsto \color{blue}{\frac{s \cdot s}{-s}} \cdot \frac{\pi}{s} \]

7. Applied rewrites 13.3%

   \[\leadsto \color{blue}{\frac{s \cdot s}{-s}} \cdot \frac{\pi}{s} \]

8. Final simplification 13.3%

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

Alternative 6: 11.6% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(fma(Float32(2.0), Float32(pi), Float32(Float32(-Float32(pi)) / u)) * 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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 10.9

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in u around inf

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

8. Applied rewrites 10.9%

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

Alternative 7: 11.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(4.0) * fma(Float32(Float32(0.5) * Float32(pi)), u, Float32(Float32(-0.25) * Float32(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. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(0 + \left(\mathsf{neg}\left(s\right)\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. lift-neg.f32 N/A

      \[\leadsto \left(0 + \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   5. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(s\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot \left(\mathsf{neg}\left(s\right)\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(s\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot \left(\mathsf{neg}\left(s\right)\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} + {\left(\mathsf{neg}\left(s\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot \left(\mathsf{neg}\left(s\right)\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. lower-+.f32 N/A

      \[\leadsto \frac{\color{blue}{0 + {\left(\mathsf{neg}\left(s\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot \left(\mathsf{neg}\left(s\right)\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. lower-pow.f32 N/A

      \[\leadsto \frac{0 + \color{blue}{{\left(\mathsf{neg}\left(s\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot \left(\mathsf{neg}\left(s\right)\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   10. metadata-eval N/A

       \[\leadsto \frac{0 + {\left(\mathsf{neg}\left(s\right)\right)}^{3}}{\color{blue}{0} + \left(\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot \left(\mathsf{neg}\left(s\right)\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   11. lower-+.f32 N/A

       \[\leadsto \frac{0 + {\left(\mathsf{neg}\left(s\right)\right)}^{3}}{\color{blue}{0 + \left(\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot \left(\mathsf{neg}\left(s\right)\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   12. lower--.f32 N/A

       \[\leadsto \frac{0 + {\left(\mathsf{neg}\left(s\right)\right)}^{3}}{0 + \color{blue}{\left(\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right) - 0 \cdot \left(\mathsf{neg}\left(s\right)\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{0 + {\left(\mathsf{neg}\left(s\right)\right)}^{3}}{0 + \left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} - 0 \cdot \left(\mathsf{neg}\left(s\right)\right)\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   14. lower-*.f32 36.9

       \[\leadsto \frac{0 + {\left(-s\right)}^{3}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - \color{blue}{0 \cdot \left(-s\right)}\right)} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

4. Applied rewrites 36.9%

   \[\leadsto \color{blue}{\frac{0 + {\left(-s\right)}^{3}}{0 + \left(\left(-s\right) \cdot \left(-s\right) - 0 \cdot \left(-s\right)\right)}} \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

5. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 10.9

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

7. Applied rewrites 10.9%

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

8. Final simplification 10.9%

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

Alternative 8: 11.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(fma(Float32(u * Float32(pi)), Float32(0.5), Float32(Float32(-0.25) * Float32(pi))) * Float32(4.0))
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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 10.9

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

5. Applied rewrites 10.9%

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

Alternative 9: 11.5% accurate, 42.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(fma(Float32(2.0), u, Float32(-1.0)) * Float32(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 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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 10.9

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 10.9%

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

9. Final simplification 10.9%

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

Alternative 10: 11.3% 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 10.7

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

5. Applied rewrites 10.7%

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

Reproduce

herbie shell --seed 2024235 
(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))))