HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 12.4s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (+ 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)
    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

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (+ 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)
    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: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\log \mathsf{E}\left(\right) \cdot \frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (* (log (E)) (/ -2.0 v)))))))))

Derivation

1. Initial program 99.4%

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

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. pow-to-exp N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. exp-1-e N/A

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

   9. lower-E.f32 99.4

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

4. Applied rewrites 99.4%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\log \mathsf{E}\left(\right) \cdot \frac{-2}{v}}}\right) \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (+ 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)
    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

Derivation

1. Initial program 99.4%

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

Alternative 3: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 - \frac{-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  1.0
  (*
   v
   (log
    (+
     u
     (*
      (- 1.0 u)
      (/
       1.0
       (- 1.0 (/ (+ -2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v)) v)))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * (1.0f / (1.0f - ((-2.0f + ((-2.0f + (-1.3333333333333333f / v)) / v)) / v)))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * (1.0e0 / (1.0e0 - (((-2.0e0) + (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v)) / v)))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)) / v))))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(1.0) / (single(1.0) - ((single(-2.0) + ((single(-2.0) + (single(-1.3333333333333333) / v)) / v)) / v)))))));
end

Derivation

1. Initial program 99.4%

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

   1. frac-2neg N/A

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

   2. distribute-frac-neg2 N/A

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

   3. exp-neg N/A

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

   4. lower-/.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}\right)\right)}}\right) \]

   2. unsub-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]

   3. lower--.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]

   4. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 - \color{blue}{\frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]

7. Applied rewrites 96.0%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}}{v}}}\right) \]
8. Add Preprocessing

Alternative 4: 90.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(u \cdot \left(\frac{-2}{v} + \frac{\left(\frac{-1}{u} - \frac{-2}{v}\right) - -2}{u}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (* u (* u (+ (/ -2.0 v) (/ (- (- (/ -1.0 u) (/ -2.0 v)) -2.0) u))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = u * (u * ((-2.0f / v) + ((((-1.0f / u) - (-2.0f / v)) - -2.0f) / u)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = u * (u * (((-2.0e0) / v) + (((((-1.0e0) / u) - ((-2.0e0) / v)) - (-2.0e0)) / u)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(u * Float32(Float32(Float32(-2.0) / v) + Float32(Float32(Float32(Float32(Float32(-1.0) / u) - Float32(Float32(-2.0) / v)) - Float32(-2.0)) / u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = u * (u * ((single(-2.0) / v) + ((((single(-1.0) / u) - (single(-2.0) / v)) - single(-2.0)) / u)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in u around 0

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \color{blue}{\left(4 \cdot u\right)}, \frac{\frac{-1}{2}}{v}, \mathsf{fma}\left(2, \mathsf{neg}\left(u\right), 2\right)\right)}{\mathsf{neg}\left(v\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 7.3

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in u around 0

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(u \cdot 4\right), \frac{\frac{-1}{2}}{v}, \color{blue}{2 + -2 \cdot u}\right)}{\mathsf{neg}\left(v\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f32 7.3

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

   11. Applied rewrites 7.3%

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

   12. Taylor expanded in u around -inf

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

       1. unpow2 N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. sub-neg N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f32 N/A

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

       7. lower-*.f32 N/A

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

       8. sub-neg N/A

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

       9. +-commutative N/A

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

   14. Applied rewrites 72.9%

       \[\leadsto \color{blue}{u \cdot \left(u \cdot \left(\frac{-2}{v} - \frac{-2 + \left(\frac{-2}{v} + \frac{1}{u}\right)}{u}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(u \cdot \left(\frac{-2}{v} + \frac{\left(\frac{-1}{u} - \frac{-2}{v}\right) - -2}{u}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{\left(\frac{-2 \cdot \left(u \cdot \left(u + -1\right)\right)}{v} - 2\right) - u \cdot -2}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ 1.0 (* v (/ (- (- (/ (* -2.0 (* u (+ u -1.0))) v) 2.0) (* u -2.0)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * (((((-2.0f * (u * (u + -1.0f))) / v) - 2.0f) - (u * -2.0f)) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * ((((((-2.0e0) * (u * (u + (-1.0e0)))) / v) - 2.0e0) - (u * (-2.0e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v * Float32(Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(u * Float32(u + Float32(-1.0)))) / v) - Float32(2.0)) - Float32(u * Float32(-2.0))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * (((((single(-2.0) * (u * (u + single(-1.0)))) / v) - single(2.0)) - (u * single(-2.0))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in u around 0

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \color{blue}{\left(4 \cdot u\right)}, \frac{\frac{-1}{2}}{v}, \mathsf{fma}\left(2, \mathsf{neg}\left(u\right), 2\right)\right)}{\mathsf{neg}\left(v\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 7.3

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

   8. Applied rewrites 7.3%

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \color{blue}{\left(u \cdot 4\right)}, \frac{-0.5}{v}, \mathsf{fma}\left(2, -u, 2\right)\right)}{-v} \]
   9. Step-by-step derivation

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-neg.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f32 N/A

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

   10. Applied rewrites 72.4%

       \[\leadsto 1 + v \cdot \frac{\color{blue}{\left(\frac{\left(u \cdot \left(1 - u\right)\right) \cdot -2}{v} + 2\right) + -2 \cdot u}}{-v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{\left(\frac{-2 \cdot \left(u \cdot \left(u + -1\right)\right)}{v} - 2\right) - u \cdot -2}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{\left(1 - u\right) \cdot -2}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ 1.0 (* v (/ (* (- 1.0 u) -2.0) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * (((1.0f - u) * -2.0f) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * (((1.0e0 - u) * (-2.0e0)) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v * Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * (((single(1.0) - u) * single(-2.0)) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. lower-+.f32 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f32 29.6

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in v around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 61.6

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

   8. Applied rewrites 61.6%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{-2 \cdot \left(1 - u\right)}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{\left(1 - u\right) \cdot -2}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ 1.0 (* (- 1.0 u) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((1.0f - u) * -2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((1.0e0 - u) * (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(1.0) - u) * single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. lower-+.f32 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f32 29.6

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in v around inf

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 61.5

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

   8. Applied rewrites 61.5%

      \[\leadsto 1 + \color{blue}{-2 \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot 2 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ (* u 2.0) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (u * 2.0f) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (u * 2.0e0) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(2.0)) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = (u * single(2.0)) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

      1. frac-2neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. exp-neg N/A

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

      4. lower-/.f32 N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. metadata-eval 90.2

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

   4. Applied rewrites 90.2%

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

   5. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f32 38.7

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

   7. Applied rewrites 6.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(2, u, -2\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f32 N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

         \[\leadsto 2 \cdot u + \color{blue}{-1} \]

      6. lower-+.f32 N/A

         \[\leadsto \color{blue}{2 \cdot u + -1} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{u \cdot 2} + -1 \]

      8. lower-*.f32 61.5

         \[\leadsto \color{blue}{u \cdot 2} + -1 \]

   9. Applied rewrites 61.5%

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

Alternative 9: 86.6% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

real(4) function code(u, v)
    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 99.4%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 87.3%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Alternative 10: 6.1% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 -1.0)

C

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

Fortran

real(4) function code(u, v)
    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 99.4%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.9%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))))))