HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 5.0s

Alternatives: 18

Speedup: 0.8×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

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

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

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

Fortran

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

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

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

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

Fortran

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.8× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\mathsf{fma}\left(\frac{-1}{-\frac{-1}{\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right)}}, -v, 1\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma
 (/ -1.0 (- (/ -1.0 (log (fma (exp (/ -2.0 v)) (- 1.0 u) u)))))
 (- v)
 1.0))

C

float code(float u, float v) {
	return fmaf((-1.0f / -(-1.0f / logf(fmaf(expf((-2.0f / v)), (1.0f - u), u)))), -v, 1.0f);
}

Julia

function code(u, v)
	return fma(Float32(Float32(-1.0) / Float32(-Float32(Float32(-1.0) / log(fma(exp(Float32(Float32(-2.0) / v)), Float32(Float32(1.0) - u), u))))), Float32(-v), Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

   \[\leadsto 1 + \log \left({\left(\frac{1}{\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)}\right)}^{\left(-v\right)}\right) \]
4. Step-by-step derivation

5. Applied rewrites 99.4%

   \[\leadsto \left(-v\right) \cdot \log \left(\frac{1}{\left|\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right|}\right) - -1 \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(-\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), -v, 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\frac{-1}{-\frac{-1}{\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right)}}, -v, 1\right) \]
10. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\mathsf{fma}\left(-\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), -v, 1\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma (- (log (fma (- 1.0 u) (exp (/ -2.0 v)) u))) (- v) 1.0))

C

float code(float u, float v) {
	return fmaf(-logf(fmaf((1.0f - u), expf((-2.0f / v)), u)), -v, 1.0f);
}

Julia

function code(u, v)
	return fma(Float32(-log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u))), Float32(-v), Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

   \[\leadsto 1 + \log \left({\left(\frac{1}{\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)}\right)}^{\left(-v\right)}\right) \]
4. Step-by-step derivation

5. Applied rewrites 99.4%

   \[\leadsto \left(-v\right) \cdot \log \left(\frac{1}{\left|\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right|}\right) - -1 \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(-\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), -v, 1\right) \]
8. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\mathsf{fma}\left(\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right), v, 1\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v 1.0))

C

float code(float u, float v) {
	return fmaf(logf(((expf((-2.0f / v)) * (1.0f - u)) + u)), v, 1.0f);
}

Julia

function code(u, v)
	return fma(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)), v, Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 95.9%

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

6. Applied rewrites 95.9%

   \[\leadsto \mathsf{fma}\left(\log \left(e^{\frac{-2}{v}} + u\right), v, 1\right) \]

7. Taylor expanded in v around 0

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

9. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right), v, 1\right) \]
10. Add Preprocessing

Alternative 4: 95.9% accurate, 1.2× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\mathsf{fma}\left(\log \left(e^{\frac{-2}{v}} + u\right), v, 1\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma (log (+ (exp (/ -2.0 v)) u)) v 1.0))

C

float code(float u, float v) {
	return fmaf(logf((expf((-2.0f / v)) + u)), v, 1.0f);
}

Julia

function code(u, v)
	return fma(log(Float32(exp(Float32(Float32(-2.0) / v)) + u)), v, Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 95.9%

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

6. Applied rewrites 95.9%

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

Alternative 5: 94.1% accurate, 1.3× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right) \cdot v - -1 \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (- (* (log (* (expm1 (/ -2.0 v)) (- u))) v) -1.0))

C

float code(float u, float v) {
	return (logf((expm1f((-2.0f / v)) * -u)) * v) - -1.0f;
}

Julia

function code(u, v)
	return Float32(Float32(log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u))) * v) - Float32(-1.0))
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around -inf

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

4. Applied rewrites 94.1%

   \[\leadsto 1 + v \cdot \log \left(-1 \cdot \left(u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 94.1%

   \[\leadsto \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right) \cdot v - -1 \]
7. Add Preprocessing

Alternative 6: 94.1% accurate, 1.3× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\mathsf{fma}\left(\log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right), v, 1\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma (log (* (expm1 (/ -2.0 v)) (- u))) v 1.0))

C

float code(float u, float v) {
	return fmaf(logf((expm1f((-2.0f / v)) * -u)), v, 1.0f);
}

Julia

function code(u, v)
	return fma(log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u))), v, Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around -inf

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

4. Applied rewrites 94.1%

   \[\leadsto 1 + v \cdot \log \left(-1 \cdot \left(u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 94.1%

   \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right), v, 1\right) \]
7. Add Preprocessing

Alternative 7: 91.2% accurate, 0.6× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{expm1}\left(\frac{2}{v}\right) \cdot v, u, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
  (fma (* (expm1 (/ 2.0 v)) v) u -1.0)
  (+ (/ 1.0 (* 2.0 (/ 0.5 0.5))) 0.5)))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf((expm1f((2.0f / v)) * v), u, -1.0f);
	} else {
		tmp = (1.0f / (2.0f * (0.5f / 0.5f))) + 0.5f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = fma(Float32(expm1(Float32(Float32(2.0) / v)) * v), u, Float32(-1.0));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(Float32(0.5) / Float32(0.5)))) + Float32(0.5));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 10.8%

      \[\leadsto \frac{u \cdot \left(v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)\right)}{e^{\frac{-2}{v}}} - 1 \]
   5. Step-by-step derivation

   6. Applied rewrites 10.8%

      \[\leadsto \mathsf{fma}\left(v \cdot \left(-1 + e^{\frac{2}{v}}\right), u, -1\right) \]

   8. Applied rewrites 10.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{expm1}\left(\frac{2}{v}\right) \cdot v, u, -1\right) \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right) + 0.5 \]
   4. Step-by-step derivation

      1. pow1 N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{1}\right)}^{-1}} + \frac{1}{2} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{\left(\mathsf{neg}\left(-1\right)\right)}\right)}^{-1}} + \frac{1}{2} \]

      3. pow-neg N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      4. remove-sound-/ N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      5. lower-/.f32 N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      6. remove-sound-pow N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      7. lower-pow.f32 99.3%

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}}\right)}^{-1}} + 0.5 \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}} + 0.5 \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\mathsf{fma}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), v, 0.5\right)}} + 0.5 \]

   8. Taylor expanded in v around 0

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\frac{1}{2}}} + 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 86.7%

       \[\leadsto \frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.0% accurate, 0.6× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\left(-\frac{\frac{-1.3333333333333333}{v} - 2}{v}\right) + 2, u, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
  (fma (+ (- (/ (- (/ -1.3333333333333333 v) 2.0) v)) 2.0) u -1.0)
  (+ (/ 1.0 (* 2.0 (/ 0.5 0.5))) 0.5)))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf((-(((-1.3333333333333333f / v) - 2.0f) / v) + 2.0f), u, -1.0f);
	} else {
		tmp = (1.0f / (2.0f * (0.5f / 0.5f))) + 0.5f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = fma(Float32(Float32(-Float32(Float32(Float32(Float32(-1.3333333333333333) / v) - Float32(2.0)) / v)) + Float32(2.0)), u, Float32(-1.0));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(Float32(0.5) / Float32(0.5)))) + Float32(0.5));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

      \[\leadsto u \cdot \left(u \cdot \left(\frac{-1}{2} \cdot \frac{v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}}{e^{\frac{-4}{v}}} + \frac{1}{3} \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{3}\right)}{e^{\frac{-6}{v}}}\right) + \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   4. Applied rewrites 6.0%

      \[\leadsto u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(-0.5, \frac{v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}}{e^{\frac{-4}{v}}}, 0.3333333333333333 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{3}\right)}{e^{\frac{-6}{v}}}\right), \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 12.8%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + u \cdot \left(2.6666666666666665 \cdot u - 4\right)}{v}, 2 \cdot u\right) - 2}{v}\right) - 1 \]

   8. Taylor expanded in u around 0

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\frac{\frac{-4}{3}}{v} - 2}{v}\right) - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 12.6%

       \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\frac{-1.3333333333333333}{v} - 2}{v}\right) - 1 \]

   12. Applied rewrites 12.6%

       \[\leadsto \mathsf{fma}\left(\left(-\frac{\frac{-1.3333333333333333}{v} - 2}{v}\right) + 2, u, -1\right) \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right) + 0.5 \]
   4. Step-by-step derivation

      1. pow1 N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{1}\right)}^{-1}} + \frac{1}{2} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{\left(\mathsf{neg}\left(-1\right)\right)}\right)}^{-1}} + \frac{1}{2} \]

      3. pow-neg N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      4. remove-sound-/ N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      5. lower-/.f32 N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      6. remove-sound-pow N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      7. lower-pow.f32 99.3%

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}}\right)}^{-1}} + 0.5 \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}} + 0.5 \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\mathsf{fma}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), v, 0.5\right)}} + 0.5 \]

   8. Taylor expanded in v around 0

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\frac{1}{2}}} + 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 86.7%

       \[\leadsto \frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.8% accurate, 0.7× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 - \left(u + u\right)}{v} - -2, u, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
  (fma (- (/ (- 2.0 (+ u u)) v) -2.0) u -1.0)
  (+ (/ 1.0 (* 2.0 (/ 0.5 0.5))) 0.5)))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf((((2.0f - (u + u)) / v) - -2.0f), u, -1.0f);
	} else {
		tmp = (1.0f / (2.0f * (0.5f / 0.5f))) + 0.5f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = fma(Float32(Float32(Float32(Float32(2.0) - Float32(u + u)) / v) - Float32(-2.0)), u, Float32(-1.0));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(Float32(0.5) / Float32(0.5)))) + Float32(0.5));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

      \[\leadsto u \cdot \left(\frac{-1}{2} \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}} + \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 5.7%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{2 \cdot u - 2}{v}\right) - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 14.6%

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

   9. Applied rewrites 14.6%

      \[\leadsto \mathsf{fma}\left(\left(-\frac{\mathsf{fma}\left(2, u, -2\right)}{v}\right) - -2, u, -1\right) \]

   11. Applied rewrites 14.6%

       \[\leadsto \mathsf{fma}\left(\frac{2 - \left(u + u\right)}{v} - -2, u, -1\right) \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right) + 0.5 \]
   4. Step-by-step derivation

      1. pow1 N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{1}\right)}^{-1}} + \frac{1}{2} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{\left(\mathsf{neg}\left(-1\right)\right)}\right)}^{-1}} + \frac{1}{2} \]

      3. pow-neg N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      4. remove-sound-/ N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      5. lower-/.f32 N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      6. remove-sound-pow N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      7. lower-pow.f32 99.3%

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}}\right)}^{-1}} + 0.5 \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}} + 0.5 \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\mathsf{fma}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), v, 0.5\right)}} + 0.5 \]

   8. Taylor expanded in v around 0

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\frac{1}{2}}} + 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 86.7%

       \[\leadsto \frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 90.8% accurate, 0.7× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{v} - -2, u, -2\right) - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
  (- (fma (- (/ 2.0 v) -2.0) u -2.0) -1.0)
  (+ (/ 1.0 (* 2.0 (/ 0.5 0.5))) 0.5)))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf(((2.0f / v) - -2.0f), u, -2.0f) - -1.0f;
	} else {
		tmp = (1.0f / (2.0f * (0.5f / 0.5f))) + 0.5f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(fma(Float32(Float32(Float32(2.0) / v) - Float32(-2.0)), u, Float32(-2.0)) - Float32(-1.0));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(Float32(0.5) / Float32(0.5)))) + Float32(0.5));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 14.6%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, 0.5 \cdot \frac{\mathsf{fma}\left(-4, {\left(1 - u\right)}^{2}, 4 \cdot \left(1 - u\right)\right)}{v}\right) \]

   5. Taylor expanded in u around 0

      \[\leadsto 1 + \left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) - 2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 14.4%

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

   9. Applied rewrites 14.4%

      \[\leadsto \mathsf{fma}\left(\frac{2}{v} - -2, u, -2\right) - -1 \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right) + 0.5 \]
   4. Step-by-step derivation

      1. pow1 N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{1}\right)}^{-1}} + \frac{1}{2} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{\left(\mathsf{neg}\left(-1\right)\right)}\right)}^{-1}} + \frac{1}{2} \]

      3. pow-neg N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      4. remove-sound-/ N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      5. lower-/.f32 N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      6. remove-sound-pow N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      7. lower-pow.f32 99.3%

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}}\right)}^{-1}} + 0.5 \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}} + 0.5 \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\mathsf{fma}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), v, 0.5\right)}} + 0.5 \]

   8. Taylor expanded in v around 0

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\frac{1}{2}}} + 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 86.7%

       \[\leadsto \frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 90.8% accurate, 0.7× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{v} + 2, u, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
  (fma (+ (/ 2.0 v) 2.0) u -1.0)
  (+ (/ 1.0 (* 2.0 (/ 0.5 0.5))) 0.5)))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf(((2.0f / v) + 2.0f), u, -1.0f);
	} else {
		tmp = (1.0f / (2.0f * (0.5f / 0.5f))) + 0.5f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = fma(Float32(Float32(Float32(2.0) / v) + Float32(2.0)), u, Float32(-1.0));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(Float32(0.5) / Float32(0.5)))) + Float32(0.5));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

      \[\leadsto u \cdot \left(\frac{-1}{2} \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}} + \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 5.7%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{2 \cdot u - 2}{v}\right) - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 14.6%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 14.4%

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

   12. Applied rewrites 14.4%

       \[\leadsto \mathsf{fma}\left(\frac{2}{v} + 2, u, -1\right) \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right) + 0.5 \]
   4. Step-by-step derivation

      1. pow1 N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{1}\right)}^{-1}} + \frac{1}{2} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{{\left({\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{\left(\mathsf{neg}\left(-1\right)\right)}\right)}^{-1}} + \frac{1}{2} \]

      3. pow-neg N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      4. remove-sound-/ N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      5. lower-/.f32 N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      6. remove-sound-pow N/A

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, \frac{1}{2}\right)\right)}^{-1}}\right)}^{-1}} + \frac{1}{2} \]

      7. lower-pow.f32 99.3%

         \[\leadsto \frac{1}{{\left(\frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}}\right)}^{-1}} + 0.5 \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{1}{{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), v, 0.5\right)\right)}^{-1}} + 0.5 \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\mathsf{fma}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), v, 0.5\right)}} + 0.5 \]

   8. Taylor expanded in v around 0

      \[\leadsto \frac{1}{2 \cdot \frac{0.5}{\frac{1}{2}}} + 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 86.7%

       \[\leadsto \frac{1}{2 \cdot \frac{0.5}{0.5}} + 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 87.9% accurate, 1.5× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\mathsf{fma}\left(-v, \log \left(\mathsf{fma}\left(\frac{1 - u}{v}, 2, 1\right)\right), 1\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma (- v) (log (fma (/ (- 1.0 u) v) 2.0 1.0)) 1.0))

C

float code(float u, float v) {
	return fmaf(-v, logf(fmaf(((1.0f - u) / v), 2.0f, 1.0f)), 1.0f);
}

Julia

function code(u, v)
	return fma(Float32(-v), log(fma(Float32(Float32(Float32(1.0) - u) / v), Float32(2.0), Float32(1.0))), Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

   \[\leadsto 1 + \log \left({\left(\frac{1}{\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)}\right)}^{\left(-v\right)}\right) \]
4. Step-by-step derivation

5. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(-v, \log \left(\frac{1}{\left|\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right|}\right), 1\right) \]

6. Taylor expanded in v around inf

   \[\leadsto \mathsf{fma}\left(-v, \log \left(1 + 2 \cdot \frac{1 - u}{v}\right), 1\right) \]
7. Step-by-step derivation

8. Applied rewrites 87.9%

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

10. Applied rewrites 87.9%

    \[\leadsto \mathsf{fma}\left(-v, \log \left(\mathsf{fma}\left(\frac{1 - u}{v}, 2, 1\right)\right), 1\right) \]
11. Add Preprocessing

Alternative 13: 87.7% accurate, 1.8× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\mathsf{fma}\left(-v, \log \left(1 + \frac{2}{v}\right), 1\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma (- v) (log (+ 1.0 (/ 2.0 v))) 1.0))

C

float code(float u, float v) {
	return fmaf(-v, logf((1.0f + (2.0f / v))), 1.0f);
}

Julia

function code(u, v)
	return fma(Float32(-v), log(Float32(Float32(1.0) + Float32(Float32(2.0) / v))), Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

   \[\leadsto 1 + \log \left({\left(\frac{1}{\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)}\right)}^{\left(-v\right)}\right) \]
4. Step-by-step derivation

5. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(-v, \log \left(\frac{1}{\left|\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right|}\right), 1\right) \]

6. Taylor expanded in v around inf

   \[\leadsto \mathsf{fma}\left(-v, \log \left(1 + 2 \cdot \frac{1 - u}{v}\right), 1\right) \]
7. Step-by-step derivation

8. Applied rewrites 87.9%

   \[\leadsto \mathsf{fma}\left(-v, \log \left(1 + 2 \cdot \frac{1 - u}{v}\right), 1\right) \]

9. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(-v, \log \left(1 + \frac{2}{v}\right), 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 87.7%

    \[\leadsto \mathsf{fma}\left(-v, \log \left(1 + \frac{2}{v}\right), 1\right) \]
12. Add Preprocessing

Alternative 14: 50.4% accurate, 0.7× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{v} + 2, u, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u + u\right) - -1\\ \end{array} \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
  (fma (+ (/ 2.0 v) 2.0) u -1.0)
  (- (+ u u) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf(((2.0f / v) + 2.0f), u, -1.0f);
	} else {
		tmp = (u + u) - -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = fma(Float32(Float32(Float32(2.0) / v) + Float32(2.0)), u, Float32(-1.0));
	else
		tmp = Float32(Float32(u + u) - Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 99.5%

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

   2. Taylor expanded in u around 0

      \[\leadsto u \cdot \left(\frac{-1}{2} \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}} + \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 5.7%

      \[\leadsto u \cdot \mathsf{fma}\left(-0.5, \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{e^{\frac{-4}{v}}}, \frac{v \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}{e^{\frac{-2}{v}}}\right) - 1 \]

   5. Taylor expanded in v around -inf

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{2 \cdot u - 2}{v}\right) - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 14.6%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 14.4%

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

   12. Applied rewrites 14.4%

       \[\leadsto \mathsf{fma}\left(\frac{2}{v} + 2, u, -1\right) \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 8.2%

      \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]

   5. Taylor expanded in u around inf

      \[\leadsto 1 + 2 \cdot u \]
   6. Step-by-step derivation

   7. Applied rewrites 46.4%

      \[\leadsto 1 + 2 \cdot u \]
   8. Step-by-step derivation

   9. Applied rewrites 46.4%

      \[\leadsto \left(u + u\right) - -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 46.4% accurate, 5.7× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\left(u + u\right) - -1 \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (- (+ u u) -1.0))

C

float code(float u, float v) {
	return (u + u) - -1.0f;
}

Fortran

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (u + u) - (-1.0e0)
end function

Julia

function code(u, v)
	return Float32(Float32(u + u) - Float32(-1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (u + u) - single(-1.0);
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in v around inf

   \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]
3. Step-by-step derivation

4. Applied rewrites 8.2%

   \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]

5. Taylor expanded in u around inf

   \[\leadsto 1 + 2 \cdot u \]
6. Step-by-step derivation

7. Applied rewrites 46.4%

   \[\leadsto 1 + 2 \cdot u \]
8. Step-by-step derivation

9. Applied rewrites 46.4%

   \[\leadsto \left(u + u\right) - -1 \]
10. Add Preprocessing

Alternative 16: 19.9% accurate, 5.7× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[0 + \left(u + u\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (+ 0.0 (+ u u)))

C

float code(float u, float v) {
	return 0.0f + (u + u);
}

Fortran

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 0.0e0 + (u + u)
end function

Julia

function code(u, v)
	return Float32(Float32(0.0) + Float32(u + u))
end

MATLAB

function tmp = code(u, v)
	tmp = single(0.0) + (u + u);
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in v around inf

   \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]
3. Step-by-step derivation

4. Applied rewrites 8.2%

   \[\leadsto 1 + -2 \cdot \left(1 - u\right) \]

5. Taylor expanded in u around inf

   \[\leadsto 1 + 2 \cdot u \]
6. Step-by-step derivation

7. Applied rewrites 46.4%

   \[\leadsto 1 + 2 \cdot u \]

8. Taylor expanded in undef-var around zero

   \[\leadsto 0 + 2 \cdot u \]
9. Step-by-step derivation

10. Applied rewrites 19.9%

    \[\leadsto 0 + 2 \cdot u \]
11. Step-by-step derivation

12. Applied rewrites 19.9%

    \[\leadsto 0 + \left(u + u\right) \]
13. Add Preprocessing

Alternative 17: 8.2% accurate, 5.9× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\mathsf{fma}\left(2, u, -1\right) \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  (fma 2.0 u -1.0))

C

float code(float u, float v) {
	return fmaf(2.0f, u, -1.0f);
}

Julia

function code(u, v)
	return fma(Float32(2.0), u, Float32(-1.0))
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 10.8%

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

5. Taylor expanded in v around inf

   \[\leadsto 2 \cdot u - 1 \]
6. Step-by-step derivation

7. Applied rewrites 8.2%

   \[\leadsto 2 \cdot u - 1 \]

9. Applied rewrites 8.2%

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

Alternative 18: 6.1% accurate, 35.6× speedup

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[-1 \]

FPCore

(FPCore (u v)
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0))
     (and (<= 0.0 v) (<= v 109.746574)))
  -1.0)

C

float code(float u, float v) {
	return -1.0f;
}

Fortran

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = -1.0e0
end function

Julia

function code(u, v)
	return Float32(-1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(-1.0);
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

   \[\leadsto -1 \]
3. Step-by-step derivation

4. Applied rewrites 6.1%

   \[\leadsto -1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))