HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.6% → 99.6%

Time: 12.9s

Alternatives: 12

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: 99.6% 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: 99.6% 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-define 99.4%

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

3. Simplified 99.4%

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

Alternative 2: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-define 99.4%

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

5. Simplified 99.4%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

Alternative 4: 96.7% 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 99.4%

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

3. Taylor expanded in u around 0 95.4%

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

Alternative 5: 91.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) + 0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot \left(24 + u \cdot -16\right)\right)}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (+
       (* -0.5 (- (* 4.0 (+ u -1.0)) (* -4.0 (pow (- 1.0 u) 2.0))))
       (* 0.16666666666666666 (/ (* u (- 8.0 (* u (+ 24.0 (* u -16.0))))) v)))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((-0.5f * ((4.0f * (u + -1.0f)) - (-4.0f * powf((1.0f - u), 2.0f)))) + (0.16666666666666666f * ((u * (8.0f - (u * (24.0f + (u * -16.0f))))) / 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 + (((1.0e0 - u) * (-2.0e0)) + ((((-0.5e0) * ((4.0e0 * (u + (-1.0e0))) - ((-4.0e0) * ((1.0e0 - u) ** 2.0e0)))) + (0.16666666666666666e0 * ((u * (8.0e0 - (u * (24.0e0 + (u * (-16.0e0)))))) / 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(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(4.0) * Float32(u + Float32(-1.0))) - Float32(Float32(-4.0) * (Float32(Float32(1.0) - u) ^ Float32(2.0))))) + Float32(Float32(0.16666666666666666) * Float32(Float32(u * Float32(Float32(8.0) - Float32(u * Float32(Float32(24.0) + Float32(u * Float32(-16.0)))))) / 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) + (((single(1.0) - u) * single(-2.0)) + (((single(-0.5) * ((single(4.0) * (u + single(-1.0))) - (single(-4.0) * ((single(1.0) - u) ^ single(2.0))))) + (single(0.16666666666666666) * ((u * (single(8.0) - (u * (single(24.0) + (u * single(-16.0)))))) / 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. log-prod 96.9%

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

      7. *-commutative 96.9%

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

      8. exp-to-pow 96.9%

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

      9. +-commutative 96.9%

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

      10. fma-undefine 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in v around 0 93.8%

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

   if 0.100000001 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.3%

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

      3. +-commutative 93.3%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in v around -inf 66.1%

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

   6. Taylor expanded in u around 0 66.1%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{\color{blue}{u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)}}{v}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) + 0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot \left(24 + u \cdot -16\right)\right)}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (* (* v u) (expm1 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((v * u) * expm1f((2.0f / v)));
	}
	return tmp;
}

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(v * u) * expm1(Float32(Float32(2.0) / v))));
	end
	return 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. log-prod 96.9%

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

      7. *-commutative 96.9%

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

      8. exp-to-pow 96.9%

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

      9. +-commutative 96.9%

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

      10. fma-undefine 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in v around 0 93.4%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.3%

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

      3. +-commutative 93.3%

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

      4. fma-define 93.5%

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

   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. Step-by-step derivation

      1. add-log-exp 93.6%

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

      2. fma-undefine 93.4%

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

      3. fma-undefine 93.3%

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

      4. +-commutative 93.3%

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

      5. exp-sum 92.7%

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

      6. log-prod 93.0%

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

      7. *-commutative 93.0%

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

      8. exp-to-pow 92.9%

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

      9. +-commutative 92.9%

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

      10. fma-undefine 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. fma-undefine 92.9%

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

   8. Applied egg-rr 92.9%

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

   9. Taylor expanded in u around 0 66.8%

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

       1. associate-*r* 66.8%

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

       2. fmm-def 66.8%

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

       3. *-commutative 66.8%

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

       4. rec-exp 66.8%

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

       5. expm1-define 66.8%

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

       6. distribute-neg-frac 66.8%

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

       7. metadata-eval 66.8%

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

       8. metadata-eval 66.8%

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

   11. Simplified 66.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(v \cdot u, \mathsf{expm1}\left(\frac{2}{v}\right), -1\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 66.8%

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

   13. Applied egg-rr 66.8%

       \[\leadsto \color{blue}{\left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 7.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) + (u * 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(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(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) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (u * 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. log-prod 96.9%

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

      7. *-commutative 96.9%

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

      8. exp-to-pow 96.9%

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

      9. +-commutative 96.9%

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

      10. fma-undefine 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in v around 0 93.8%

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

   if 0.100000001 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.3%

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

      3. +-commutative 93.3%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

      1. add-log-exp 93.7%

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

      2. fma-undefine 93.4%

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

      3. fma-undefine 93.4%

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

      4. +-commutative 93.4%

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

      5. exp-sum 92.9%

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

      6. log-prod 93.1%

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

      7. *-commutative 93.1%

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

      8. exp-to-pow 93.1%

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

      9. +-commutative 93.1%

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

      10. fma-undefine 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. fma-undefine 93.1%

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

   8. Applied egg-rr 93.1%

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

   9. Taylor expanded in u around 0 63.9%

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

       1. associate-*r* 63.9%

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

       2. fmm-def 63.9%

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

       3. *-commutative 63.9%

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

       4. rec-exp 63.9%

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

       5. expm1-define 63.9%

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

       6. distribute-neg-frac 63.9%

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

       7. metadata-eval 63.9%

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

       8. metadata-eval 63.9%

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

   11. Simplified 63.9%

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

   12. Taylor expanded in v around -inf 62.4%

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

4. Final simplification 91.1%

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

Alternative 8: 91.0% accurate, 9.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - (((u * -2.0f) + ((u / v) * -1.3333333333333333f)) / 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) - (((u * (-2.0e0)) + ((u / v) * (-1.3333333333333333e0))) / 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(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(u / v) * Float32(-1.3333333333333333))) / 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)) - (((u * single(-2.0)) + ((u / v) * single(-1.3333333333333333))) / 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. log-prod 96.9%

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

      7. *-commutative 96.9%

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

      8. exp-to-pow 96.9%

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

      9. +-commutative 96.9%

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

      10. fma-undefine 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in v around 0 93.8%

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

   if 0.100000001 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.3%

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

      3. +-commutative 93.3%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

      1. add-log-exp 93.7%

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

      2. fma-undefine 93.4%

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

      3. fma-undefine 93.4%

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

      4. +-commutative 93.4%

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

      5. exp-sum 92.9%

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

      6. log-prod 93.1%

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

      7. *-commutative 93.1%

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

      8. exp-to-pow 93.1%

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

      9. +-commutative 93.1%

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

      10. fma-undefine 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. fma-undefine 93.1%

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

   8. Applied egg-rr 93.1%

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

   9. Taylor expanded in u around 0 63.9%

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

       1. associate-*r* 63.9%

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

       2. fmm-def 63.9%

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

       3. *-commutative 63.9%

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

       4. rec-exp 63.9%

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

       5. expm1-define 63.9%

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

       6. distribute-neg-frac 63.9%

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

       7. metadata-eval 63.9%

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

       8. metadata-eval 63.9%

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

   11. Simplified 63.9%

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

   12. Taylor expanded in v around -inf 59.8%

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

4. Final simplification 90.9%

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

Alternative 9: 90.8% accurate, 15.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		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 <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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 < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. log-prod 96.9%

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

      7. *-commutative 96.9%

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

      8. exp-to-pow 96.9%

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

      9. +-commutative 96.9%

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

      10. fma-undefine 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in v around 0 93.8%

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

   if 0.100000001 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.3%

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

      3. +-commutative 93.3%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

      1. add-log-exp 93.7%

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

      2. fma-undefine 93.4%

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

      3. fma-undefine 93.4%

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

      4. +-commutative 93.4%

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

      5. exp-sum 92.9%

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

      6. log-prod 93.1%

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

      7. *-commutative 93.1%

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

      8. exp-to-pow 93.1%

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

      9. +-commutative 93.1%

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

      10. fma-undefine 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. fma-undefine 93.1%

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

   8. Applied egg-rr 93.1%

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

   9. Taylor expanded in u around 0 63.9%

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

       1. associate-*r* 63.9%

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

       2. fmm-def 63.9%

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

       3. *-commutative 63.9%

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

       4. rec-exp 63.9%

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

       5. expm1-define 63.9%

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

       6. distribute-neg-frac 63.9%

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

       7. metadata-eval 63.9%

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

       8. metadata-eval 63.9%

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

   11. Simplified 63.9%

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

   12. Taylor expanded in v around inf 57.1%

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

       1. sub-neg 57.1%

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

       2. distribute-lft-out 57.1%

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

       3. metadata-eval 57.1%

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

   14. Simplified 57.1%

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

4. Final simplification 90.7%

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

Alternative 10: 90.3% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ -1.0 (* u 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);
	}
	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)
    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(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));
	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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. log-prod 96.9%

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

      7. *-commutative 96.9%

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

      8. exp-to-pow 96.9%

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

      9. +-commutative 96.9%

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

      10. fma-undefine 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in v around 0 93.8%

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

   if 0.100000001 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.3%

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

      3. +-commutative 93.3%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in v around inf 50.5%

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

   6. Taylor expanded in u around 0 50.5%

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

4. Final simplification 90.1%

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

Alternative 11: 89.7% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (if (<= v 0.10000000149011612) 1.0 -1.0))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = -1.0e0
    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(-1.0);
	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);
	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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. log-prod 96.9%

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

      7. *-commutative 96.9%

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

      8. exp-to-pow 96.9%

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

      9. +-commutative 96.9%

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

      10. fma-undefine 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in v around 0 93.8%

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

   if 0.100000001 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 93.3%

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

      3. +-commutative 93.3%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in u around 0 41.6%

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

Alternative 12: 5.5% 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-define 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0 6.5%

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

Reproduce

herbie shell --seed 2024172 
(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))))))))