Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 5.8s

Alternatives: 11

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	t_1 = fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))))) - t_0), t_0)
	return Float32(Float32(-s) * log(Float32(Float32((t_1 ^ Float32(-2.0)) - Float32(1.0)) / Float32(Float32(1.0) + Float32(Float32(1.0) / t_1)))))
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 98.3%

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

4. Taylor expanded in s around 0

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

6. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

Alternative 3: 97.7% accurate, 1.4× speedup

Math

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

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))))

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));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * 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))))
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)));
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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 97.7%

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

Alternative 4: 37.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((u * (single(0.5) - (single(1.0) / (single(2.0) + (single(pi) / s))))) + (single(1.0) / (single(1.0) + exp((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. Taylor expanded in s around inf

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

4. Applied rewrites 37.7%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 37.7

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

7. Applied rewrites 37.7%

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

Alternative 5: 25.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(fma(Float32(fma(Float32(Float32(pi) * Float32(0.5)), u, Float32(Float32(-0.25) * Float32(pi))) / 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. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 25.0%

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

Alternative 6: 14.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\log 1\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \left(u \cdot \frac{\mathsf{fma}\left(-2, \pi, \frac{\pi}{u}\right)}{s}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 9.999999682655225e-21)
   (log 1.0)
   (* (- s) (* u (/ (fma -2.0 PI (/ PI u)) s)))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 9.999999682655225e-21f) {
		tmp = logf(1.0f);
	} else {
		tmp = -s * (u * (fmaf(-2.0f, ((float) M_PI), (((float) M_PI) / u)) / s));
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-21))
		tmp = log(Float32(1.0));
	else
		tmp = Float32(Float32(-s) * Float32(u * Float32(fma(Float32(-2.0), Float32(pi), Float32(Float32(pi) / u)) / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-21

   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 s around inf

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{\mathsf{PI}\left(\right)}{s}}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

      2. metadata-eval N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-PI.f32 -0.0

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

   7. Applied rewrites -0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \color{blue}{\left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      2. lift-log.f32 N/A

         \[\leadsto \left(-s\right) \cdot \color{blue}{\log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      3. log-pow-rev N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

      4. lower-log.f32 N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

   9. Applied rewrites -0.0%

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

   10. Taylor expanded in s around 0

       \[\leadsto \log \color{blue}{1} \]
   11. Step-by-step derivation

   12. Applied rewrites 13.2%

       \[\leadsto \log \color{blue}{1} \]

   if 9.99999968e-21 < s

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 15.0%

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

   5. Taylor expanded in u around inf

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lower-*.f32 15.0

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

   7. Applied rewrites 15.0%

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

   8. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lift-PI.f32 15.0

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

   10. Applied rewrites 15.0%

       \[\leadsto \left(-s\right) \cdot \left(u \cdot \frac{\mathsf{fma}\left(-2, \pi, \frac{\pi}{u}\right)}{s}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 14.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\log 1\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \frac{u \cdot \mathsf{fma}\left(-2, \pi, \frac{\pi}{u}\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 9.999999682655225e-21)
   (log 1.0)
   (* (- s) (/ (* u (fma -2.0 PI (/ PI u))) s))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 9.999999682655225e-21f) {
		tmp = logf(1.0f);
	} else {
		tmp = -s * ((u * fmaf(-2.0f, ((float) M_PI), (((float) M_PI) / u))) / s);
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-21))
		tmp = log(Float32(1.0));
	else
		tmp = Float32(Float32(-s) * Float32(Float32(u * fma(Float32(-2.0), Float32(pi), Float32(Float32(pi) / u))) / s));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-21

   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 s around inf

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{\mathsf{PI}\left(\right)}{s}}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

      2. metadata-eval N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-PI.f32 -0.0

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

   7. Applied rewrites -0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \color{blue}{\left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      2. lift-log.f32 N/A

         \[\leadsto \left(-s\right) \cdot \color{blue}{\log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      3. log-pow-rev N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

      4. lower-log.f32 N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

   9. Applied rewrites -0.0%

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

   10. Taylor expanded in s around 0

       \[\leadsto \log \color{blue}{1} \]
   11. Step-by-step derivation

   12. Applied rewrites 13.2%

       \[\leadsto \log \color{blue}{1} \]

   if 9.99999968e-21 < s

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 15.0%

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

   5. Taylor expanded in u around inf

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lower-*.f32 15.0

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

   7. Applied rewrites 15.0%

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

   8. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lift-PI.f32 15.0

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

   10. Applied rewrites 15.0%

       \[\leadsto \left(-s\right) \cdot \frac{u \cdot \mathsf{fma}\left(-2, \pi, \frac{\pi}{u}\right)}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 14.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\log 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, u, -0.25 \cdot \pi\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 9.999999682655225e-21)
   (log 1.0)
   (* (fma (* PI 0.5) u (* -0.25 PI)) 4.0)))

C

float code(float u, float s) {
	float tmp;
	if (s <= 9.999999682655225e-21f) {
		tmp = logf(1.0f);
	} else {
		tmp = fmaf((((float) M_PI) * 0.5f), u, (-0.25f * ((float) M_PI))) * 4.0f;
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-21))
		tmp = log(Float32(1.0));
	else
		tmp = Float32(fma(Float32(Float32(pi) * Float32(0.5)), u, Float32(Float32(-0.25) * Float32(pi))) * Float32(4.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-21

   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 s around inf

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{\mathsf{PI}\left(\right)}{s}}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

      2. metadata-eval N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-PI.f32 -0.0

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

   7. Applied rewrites -0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \color{blue}{\left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      2. lift-log.f32 N/A

         \[\leadsto \left(-s\right) \cdot \color{blue}{\log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      3. log-pow-rev N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

      4. lower-log.f32 N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

   9. Applied rewrites -0.0%

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

   10. Taylor expanded in s around 0

       \[\leadsto \log \color{blue}{1} \]
   11. Step-by-step derivation

   12. Applied rewrites 13.2%

       \[\leadsto \log \color{blue}{1} \]

   if 9.99999968e-21 < s

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi \cdot 0.5, u, -0.25 \cdot \pi\right) \cdot 4} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 14.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\left(-s\right) \cdot \frac{\pi}{s}\\ \mathbf{else}:\\ \;\;\;\;\log 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
           1.0)))
        -1.999999936531045e-19)
     (* (- s) (/ PI s))
     (log 1.0))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float tmp;
	if ((-s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f))) <= -1.999999936531045e-19f) {
		tmp = -s * (((float) M_PI) / s);
	} else {
		tmp = logf(1.0f);
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	tmp = Float32(0.0)
	if (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)))) <= Float32(-1.999999936531045e-19))
		tmp = Float32(Float32(-s) * Float32(Float32(pi) / s));
	else
		tmp = log(Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = single(0.0);
	if ((-s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)))) <= single(-1.999999936531045e-19))
		tmp = -s * (single(pi) / s);
	else
		tmp = log(single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.99999994e-19

   1. Initial program 99.1%

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

   2. Taylor expanded in u around 0

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

      1. lift-/.f32 N/A

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

      2. lift-PI.f32 15.1

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

   4. Applied rewrites 15.1%

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

   if -1.99999994e-19 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 36.3%

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

   5. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{\mathsf{PI}\left(\right)}{s}}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

      2. metadata-eval N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-PI.f32 0.0

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

   7. Applied rewrites 0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \color{blue}{\left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      2. lift-log.f32 N/A

         \[\leadsto \left(-s\right) \cdot \color{blue}{\log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      3. log-pow-rev N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

      4. lower-log.f32 N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

   9. Applied rewrites 0.0%

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

   10. Taylor expanded in s around 0

       \[\leadsto \log \color{blue}{1} \]
   11. Step-by-step derivation

   12. Applied rewrites 13.0%

       \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 14.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;-\pi\\ \mathbf{else}:\\ \;\;\;\;\log 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
           1.0)))
        -1.999999936531045e-19)
     (- PI)
     (log 1.0))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float tmp;
	if ((-s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f))) <= -1.999999936531045e-19f) {
		tmp = -((float) M_PI);
	} else {
		tmp = logf(1.0f);
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	tmp = Float32(0.0)
	if (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)))) <= Float32(-1.999999936531045e-19))
		tmp = Float32(-Float32(pi));
	else
		tmp = log(Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = single(0.0);
	if ((-s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)))) <= single(-1.999999936531045e-19))
		tmp = -single(pi);
	else
		tmp = log(single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.99999994e-19

   1. Initial program 99.1%

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

   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 15.1

         \[\leadsto -\pi \]

   4. Applied rewrites 15.1%

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

   if -1.99999994e-19 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.9%

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

   2. Taylor expanded in s around inf

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

   4. Applied rewrites 36.3%

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

   5. Taylor expanded in s around inf

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{\mathsf{PI}\left(\right)}{s}}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

      2. metadata-eval N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-PI.f32 0.0

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

   7. Applied rewrites 0.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \color{blue}{\left(0.5 - 0.25 \cdot \frac{\pi}{s}\right)}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      2. lift-log.f32 N/A

         \[\leadsto \left(-s\right) \cdot \color{blue}{\log \left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]

      3. log-pow-rev N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

      4. lower-log.f32 N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{1}{u \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{\pi}{s}\right)\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)}^{\left(-s\right)}\right)} \]

   9. Applied rewrites 0.0%

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

   10. Taylor expanded in s around 0

       \[\leadsto \log \color{blue}{1} \]
   11. Step-by-step derivation

   12. Applied rewrites 13.0%

       \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 11.3% 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 11.3

      \[\leadsto -\pi \]

4. Applied rewrites 11.3%

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

Reproduce

herbie shell --seed 2025106 
(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))))