HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.1s

Alternatives: 14

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: 99.5% 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 99.6%

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

Alternative 2: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((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 + exp(((-2.0e0) / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0 95.4%

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

Alternative 3: 95.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((u - ((u + -1.0f) / (1.0f - (((((1.3333333333333333f * (-1.0f / v)) - 2.0f) / v) - 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 - ((u + (-1.0e0)) / (1.0e0 - (((((1.3333333333333333e0 * ((-1.0e0) / v)) - 2.0e0) / v) - 2.0e0) / v))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-lft-neg-out 99.6%

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

   3. distribute-rgt1-in 99.6%

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

   4. +-commutative 99.6%

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

   5. sub-neg 99.6%

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

   6. rem-exp-log 99.6%

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

   7. sub-neg 99.6%

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

   8. log1p-undefine 99.6%

      \[\leadsto 1 + v \cdot \log \left(u + e^{\color{blue}{\mathsf{log1p}\left(-u\right)}} \cdot e^{\frac{-2}{v}}\right) \]

   9. exp-sum 99.6%

      \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{e^{\mathsf{log1p}\left(-u\right) + \frac{-2}{v}}}\right) \]

   10. metadata-eval 99.6%

       \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) + \frac{\color{blue}{-2}}{v}}\right) \]

   11. distribute-neg-frac 99.6%

       \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) + \color{blue}{\left(-\frac{2}{v}\right)}}\right) \]

   12. metadata-eval 99.6%

       \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) + \left(-\frac{\color{blue}{2 \cdot 1}}{v}\right)}\right) \]

   13. associate-*r/ 99.6%

       \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) + \left(-\color{blue}{2 \cdot \frac{1}{v}}\right)}\right) \]

   14. unsub-neg 99.6%

       \[\leadsto 1 + v \cdot \log \left(u + e^{\color{blue}{\mathsf{log1p}\left(-u\right) - 2 \cdot \frac{1}{v}}}\right) \]

   15. exp-diff 99.5%

       \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\frac{e^{\mathsf{log1p}\left(-u\right)}}{e^{2 \cdot \frac{1}{v}}}}\right) \]

   16. log1p-undefine 99.5%

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

   17. sub-neg 99.5%

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

   18. rem-exp-log 99.5%

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

   19. associate-*r/ 99.5%

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

   20. metadata-eval 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in v around -inf 94.6%

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

7. Final simplification 94.6%

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

Alternative 4: 94.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f + (v * logf((u + ((1.0f - u) / (1.0f + (2.0f / v))))));
	} else {
		tmp = -1.0f - (u * (v * ((-1.0f / expf((-2.0f / v))) - -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 + (v * log((u + ((1.0e0 - u) / (1.0e0 + (2.0e0 / v))))))
    else
        tmp = (-1.0e0) - (u * (v * (((-1.0e0) / exp(((-2.0e0) / v))) - (-1.0e0))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + Float32(Float32(2.0) / v)))))));
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(v * Float32(Float32(Float32(-1.0) / exp(Float32(Float32(-2.0) / v))) - 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) + (v * log((u + ((single(1.0) - u) / (single(1.0) + (single(2.0) / v))))));
	else
		tmp = single(-1.0) - (u * (v * ((single(-1.0) / exp((single(-2.0) / v))) - 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 u around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-neg-out 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. rem-exp-log 100.0%

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

      7. sub-neg 100.0%

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

      8. log1p-undefine 100.0%

         \[\leadsto 1 + v \cdot \log \left(u + e^{\color{blue}{\mathsf{log1p}\left(-u\right)}} \cdot e^{\frac{-2}{v}}\right) \]

      9. exp-sum 100.0%

         \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{e^{\mathsf{log1p}\left(-u\right) + \frac{-2}{v}}}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) + \frac{\color{blue}{-2}}{v}}\right) \]

      11. distribute-neg-frac 100.0%

          \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) + \color{blue}{\left(-\frac{2}{v}\right)}}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) + \left(-\frac{\color{blue}{2 \cdot 1}}{v}\right)}\right) \]

      13. associate-*r/ 100.0%

          \[\leadsto 1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) + \left(-\color{blue}{2 \cdot \frac{1}{v}}\right)}\right) \]

      14. unsub-neg 100.0%

          \[\leadsto 1 + v \cdot \log \left(u + e^{\color{blue}{\mathsf{log1p}\left(-u\right) - 2 \cdot \frac{1}{v}}}\right) \]

      15. exp-diff 100.0%

          \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\frac{e^{\mathsf{log1p}\left(-u\right)}}{e^{2 \cdot \frac{1}{v}}}}\right) \]

      16. log1p-undefine 100.0%

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

      17. sub-neg 100.0%

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

      18. rem-exp-log 100.0%

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

      19. associate-*r/ 100.0%

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

      20. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in v around inf 96.0%

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

      1. associate-*r/ 96.0%

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

      2. metadata-eval 96.0%

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

   8. Simplified 96.0%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 66.3%

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

4. Final simplification 93.7%

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

Alternative 5: 91.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f - (u * (v * ((-1.0f / expf((-2.0f / v))) - -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 = (-1.0e0) - (u * (v * (((-1.0e0) / exp(((-2.0e0) / v))) - (-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(-1.0) - Float32(u * Float32(v * Float32(Float32(Float32(-1.0) / exp(Float32(Float32(-2.0) / v))) - 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 = single(-1.0) - (u * (v * ((single(-1.0) / exp((single(-2.0) / v))) - 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. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

         \[\leadsto 1 + {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}}\right)}^{3} \]

   4. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

   5. Taylor expanded in v around 0 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 66.3%

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

4. Final simplification 91.7%

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

Alternative 6: 91.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (v * (-1.0f + expf((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) + (u * (v * ((-1.0e0) + exp((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(u * Float32(v * Float32(Float32(-1.0) + exp(Float32(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) + (u * (v * (single(-1.0) + exp((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. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

         \[\leadsto 1 + {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}}\right)}^{3} \]

   4. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

   5. Taylor expanded in v around 0 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 65.7%

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

      1. fmm-def 65.7%

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

      2. rec-exp 65.7%

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

      3. expm1-define 66.1%

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

      4. distribute-neg-frac 66.1%

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

      5. metadata-eval 66.1%

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

      6. associate-*r/ 66.1%

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

      7. metadata-eval 66.1%

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

      8. distribute-neg-frac 66.1%

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

      9. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in v around 0 66.3%

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

4. Final simplification 91.7%

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

Alternative 7: 91.1% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (* u (+ 2.0 (/ (+ 2.0 (* 1.3333333333333333 (/ 1.0 v))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (1.3333333333333333f * (1.0f / v))) / 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (1.3333333333333333e0 * (1.0e0 / v))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) * Float32(Float32(1.0) / v))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (single(1.3333333333333333) * (single(1.0) / v))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Step-by-step derivation

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

         \[\leadsto 1 + {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}}\right)}^{3} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

   5. Taylor expanded in v around 0 94.5%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. fma-define 94.7%

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

      3. +-commutative 94.7%

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

      4. fma-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in v around -inf 68.2%

      \[\leadsto \mathsf{fma}\left(v, \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + 0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}}, 1\right) \]

   6. Taylor expanded in u around 0 61.4%

      \[\leadsto \color{blue}{-1 \cdot \left(u \cdot \left(-1 \cdot \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v} - 2\right)\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.1% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{2 + \frac{1.3333333333333333}{v}}{v} - -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (* u (- (/ (+ 2.0 (/ 1.3333333333333333 v)) v) -2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (((2.0f + (1.3333333333333333f / v)) / v) - -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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (((2.0e0 + (1.3333333333333333e0 / v)) / v) - (-2.0e0)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) - Float32(-2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (((single(2.0) + (single(1.3333333333333333) / v)) / v) - single(-2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Step-by-step derivation

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

         \[\leadsto 1 + {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}}\right)}^{3} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

   5. Taylor expanded in v around 0 94.5%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. fma-define 94.7%

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

      3. +-commutative 94.7%

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

      4. fma-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in v around -inf 68.2%

      \[\leadsto \mathsf{fma}\left(v, \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + 0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}}, 1\right) \]

   6. Taylor expanded in u around 0 61.4%

      \[\leadsto \color{blue}{-1 \cdot \left(u \cdot \left(-1 \cdot \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v} - 2\right)\right) - 1} \]
   7. Step-by-step derivation

      1. pow1 61.4%

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

      2. fmm-def 61.4%

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

      3. un-div-inv 61.4%

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

      4. metadata-eval 61.4%

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

   8. Applied egg-rr 61.4%

      \[\leadsto -1 \cdot \color{blue}{{\left(u \cdot \mathsf{fma}\left(-1, \frac{2 + \frac{1.3333333333333333}{v}}{v}, -2\right)\right)}^{1}} - 1 \]
   9. Step-by-step derivation

      1. unpow1 61.4%

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

      2. metadata-eval 61.4%

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

      3. associate-*r/ 61.4%

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

      4. metadata-eval 61.4%

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

      5. fmm-def 61.4%

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

      6. fmm-def 61.4%

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

      7. associate-*r/ 61.4%

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

      8. metadata-eval 61.4%

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

      9. metadata-eval 61.4%

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

      10. fma-undefine 61.4%

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

      11. neg-mul-1 61.4%

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

      12. +-commutative 61.4%

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

      13. unsub-neg 61.4%

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

   10. Simplified 61.4%

       \[\leadsto -1 \cdot \color{blue}{\left(u \cdot \left(-2 - \frac{2 + \frac{1.3333333333333333}{v}}{v}\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{2 + \frac{1.3333333333333333}{v}}{v} - -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.8% accurate, 11.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((2.0f * (u / v)) - (2.0f + (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 + ((2.0e0 * (u / v)) - (2.0e0 + (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(2.0) * Float32(u / v)) - Float32(Float32(2.0) + Float32(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(2.0) * (u / v)) - (single(2.0) + (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. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

         \[\leadsto 1 + {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}}\right)}^{3} \]

   4. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

   5. Taylor expanded in v around 0 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 65.7%

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

      1. fmm-def 65.7%

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

      2. rec-exp 65.7%

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

      3. expm1-define 66.1%

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

      4. distribute-neg-frac 66.1%

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

      5. metadata-eval 66.1%

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

      6. associate-*r/ 66.1%

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

      7. metadata-eval 66.1%

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

      8. distribute-neg-frac 66.1%

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

      9. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in v around -inf 60.7%

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

      1. +-commutative 60.7%

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

      2. mul-1-neg 60.7%

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

      3. unsub-neg 60.7%

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

      4. *-commutative 60.7%

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

   10. Simplified 60.7%

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

Alternative 10: 90.8% accurate, 15.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (2.0f * (u + (u / 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) + (2.0e0 * (u + (u / 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(Float32(2.0) * Float32(u + Float32(u / 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) + (single(2.0) * (u + (u / 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. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

         \[\leadsto 1 + {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}}\right)}^{3} \]

   4. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

   5. Taylor expanded in v around 0 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 65.7%

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

      1. fmm-def 65.7%

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

      2. rec-exp 65.7%

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

      3. expm1-define 66.1%

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

      4. distribute-neg-frac 66.1%

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

      5. metadata-eval 66.1%

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

      6. associate-*r/ 66.1%

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

      7. metadata-eval 66.1%

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

      8. distribute-neg-frac 66.1%

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

      9. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in v around inf 60.7%

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

      1. sub-neg 60.7%

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

      2. distribute-lft-out 60.7%

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

      3. metadata-eval 60.7%

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

   10. Simplified 60.7%

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

4. Final simplification 91.2%

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

Alternative 11: 90.2% accurate, 17.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = u * (2.0f + (-1.0f / 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 * (2.0e0 + ((-1.0e0) / 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(Float32(2.0) + Float32(Float32(-1.0) / 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 * (single(2.0) + (single(-1.0) / 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. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

         \[\leadsto 1 + {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}}\right)}^{3} \]

   4. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

   5. Taylor expanded in v around 0 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 65.7%

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

      1. fmm-def 65.7%

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

      2. rec-exp 65.7%

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

      3. expm1-define 66.1%

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

      4. distribute-neg-frac 66.1%

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

      5. metadata-eval 66.1%

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

      6. associate-*r/ 66.1%

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

      7. metadata-eval 66.1%

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

      8. distribute-neg-frac 66.1%

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

      9. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in v around inf 52.5%

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

   9. Taylor expanded in u around inf 52.5%

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

4. Final simplification 90.6%

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

Alternative 12: 90.2% accurate, 21.2× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 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 + (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) + (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(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) + (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. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

         \[\leadsto 1 + {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}}\right)}^{3} \]

   4. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

   5. Taylor expanded in v around 0 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 65.7%

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

      1. fmm-def 65.7%

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

      2. rec-exp 65.7%

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

      3. expm1-define 66.1%

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

      4. distribute-neg-frac 66.1%

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

      5. metadata-eval 66.1%

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

      6. associate-*r/ 66.1%

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

      7. metadata-eval 66.1%

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

      8. distribute-neg-frac 66.1%

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

      9. metadata-eval 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in v around inf 52.5%

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

4. Final simplification 90.6%

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

Alternative 13: 87.2% accurate, 213.0× 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 99.6%

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

   1. add-cube-cbrt 99.4%

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

   2. pow3 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-undefine 99.4%

      \[\leadsto 1 + {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}}\right)}^{3} \]

4. Applied egg-rr 99.4%

   \[\leadsto 1 + \color{blue}{{\left(\sqrt[3]{v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

5. Taylor expanded in v around 0 86.9%

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

Alternative 14: 5.8% accurate, 213.0× 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 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u around 0 6.3%

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

Reproduce

herbie shell --seed 2024150 
(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))))))))