HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.3s

Alternatives: 11

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u)), Float32(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 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v))))), Float32(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 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 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 4: 96.2% 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 97.2%

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

Alternative 5: 94.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log u \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (+ 1.0 (* v (log u))))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log(u));
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-exp-log 99.6%

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

   2. *-commutative 99.6%

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

   3. log-prod 99.6%

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

   4. add-log-exp 99.6%

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

   5. sub-neg 99.6%

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

   6. log1p-define 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in u around inf 95.3%

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

   1. mul-1-neg 95.3%

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

   2. distribute-rgt-neg-in 95.3%

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

   3. log-rec 95.3%

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

   4. remove-double-neg 95.3%

      \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

7. Simplified 95.3%

   \[\leadsto 1 + \color{blue}{v \cdot \log u} \]
8. Add Preprocessing

Alternative 6: 91.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot \left(2 + \frac{-1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    (/
     (-
      (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v)
      (* u -2.0))
     v)
    (* u (+ 2.0 (/ -1.0 u))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) + (u * (2.0e0 + ((-1.0e0) / u)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v) + 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (u * (single(2.0) + (single(-1.0) / u)));
	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. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)}\right) \]

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.8%

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

   if 0.100000001 < v

   1. Initial program 94.1%

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

      1. +-commutative 94.1%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in u around 0 63.6%

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

      1. *-commutative 63.6%

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

      2. associate-*l* 63.6%

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

      3. *-commutative 63.6%

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

      4. fmm-def 63.6%

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

      5. rec-exp 63.6%

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

      6. expm1-define 63.6%

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

      7. distribute-neg-frac 63.6%

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

      8. metadata-eval 63.6%

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

      9. metadata-eval 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in u around inf 62.4%

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

   9. Taylor expanded in v around -inf 64.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-2 \cdot u + -1 \cdot \frac{0.6666666666666666 \cdot \frac{u}{v} + 1.3333333333333333 \cdot u}{v}}{v} + u \cdot \left(2 - \frac{1}{u}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot \left(2 + \frac{-1}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+ 2.0 (/ (+ 1.3333333333333333 (* 0.6666666666666666 (/ 1.0 v))) 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 + (0.6666666666666666f * (1.0f / v))) / 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 + (0.6666666666666666e0 * (1.0e0 / v))) / 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(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) * Float32(Float32(1.0) / v))) / 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(0.6666666666666666) * (single(1.0) / v))) / 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. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)}\right) \]

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.8%

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

   if 0.100000001 < v

   1. Initial program 94.1%

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

      1. +-commutative 94.1%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in u around 0 63.6%

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

   6. Taylor expanded in v around -inf 64.1%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 11.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (* u (+ 2.0 (/ (+ 2.0 (/ 1.3333333333333333 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 / 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 / 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) / 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) / 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. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)}\right) \]

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.8%

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

   if 0.100000001 < v

   1. Initial program 94.1%

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

      1. +-commutative 94.1%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in u around 0 63.6%

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

   6. Taylor expanded in v around -inf 63.2%

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

   7. Taylor expanded in u around 0 63.2%

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

      1. associate-/l* 63.2%

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

      2. associate-*r/ 63.2%

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

      3. metadata-eval 63.2%

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

   9. Simplified 63.2%

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

       1. fma-define 63.2%

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

       2. *-commutative 63.2%

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

   11. Applied egg-rr 63.2%

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

       1. fma-undefine 63.2%

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

       2. distribute-lft-out 63.2%

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

   13. Simplified 63.2%

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

4. Final simplification 90.8%

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

Alternative 9: 90.6% accurate, 15.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + (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 * (2.0e0 + (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(Float32(2.0) + 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 * (single(2.0) + (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. expm1-log1p-u 99.6%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 99.6%

         \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)}\right) \]

      3. expm1-undefine 99.6%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.200000003 < v

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. fma-define 93.1%

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

      3. +-commutative 93.1%

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

      4. fma-define 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in u around 0 74.5%

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

   6. Taylor expanded in v around inf 65.6%

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

      1. associate-*r/ 65.6%

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

      2. metadata-eval 65.6%

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

   8. Simplified 65.6%

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

4. Final simplification 90.4%

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

Alternative 10: 87.1% 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. expm1-log1p-u 94.8%

      \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

   2. log1p-define 94.8%

      \[\leadsto 1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)}\right) \]

   3. expm1-undefine 94.8%

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

   4. add-exp-log 99.6%

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

   5. +-commutative 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-undefine 99.6%

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

   8. fma-undefine 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0 87.2%

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

Alternative 11: 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 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in u around 0 5.7%

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

Reproduce

herbie shell --seed 2024181 
(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))))))))