HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 10.4s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

Alternative 2: 96.1% 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 99.5%

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

3. Taylor expanded in u around 0 97.3%

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

Alternative 3: 94.5% 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 99.5%

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

3. Taylor expanded in u around 0 97.3%

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

4. Taylor expanded in u around inf 95.9%

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

   1. mul-1-neg 95.9%

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

   2. distribute-rgt-neg-in 95.9%

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

   3. log-rec 95.9%

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

   4. remove-double-neg 95.9%

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

6. Simplified 95.9%

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

Alternative 4: 90.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.07999999821186066)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/ (+ 2.0 (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.07999999821186066f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((1.3333333333333333f + (0.6666666666666666f / 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 <= 0.07999999821186066e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 + (0.6666666666666666e0 / v)) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.07999999821186066))
		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) / v)) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.07999999821186066))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) + (single(0.6666666666666666) / v)) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0799999982

   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. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

      2. fma-define 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 91.4%

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

   if 0.0799999982 < v

   1. Initial program 92.0%

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

      1. +-commutative 92.0%

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

      2. fma-define 91.7%

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

      3. +-commutative 91.7%

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

      4. fma-define 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in u around 0 69.8%

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

   6. Taylor expanded in v around -inf 72.6%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)} - 1 \]

   7. Taylor expanded in v around 0 72.6%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.6% accurate, 9.7× speedup

Math

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

FPCore

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

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. count-2 99.9%

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

      2. fma-define 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-define 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in v around 0 91.2%

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

   if 0.100000001 < v

   1. Initial program 91.6%

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

      1. +-commutative 91.6%

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

      2. fma-define 91.3%

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

      3. +-commutative 91.3%

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

      4. fma-define 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in u around 0 74.0%

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

   6. Taylor expanded in v around -inf 75.3%

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

4. Final simplification 90.4%

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

Alternative 6: 90.6% 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. 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. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. count-2 99.9%

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

      2. fma-define 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-define 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in v around 0 91.2%

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

   if 0.100000001 < v

   1. Initial program 91.6%

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

      1. +-commutative 91.6%

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

      2. fma-define 91.3%

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

      3. +-commutative 91.3%

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

      4. fma-define 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in u around 0 74.0%

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

   6. Taylor expanded in v around -inf 75.9%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)} - 1 \]

   7. Taylor expanded in v around inf 75.3%

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

      1. associate-*r/ 75.3%

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

      2. metadata-eval 75.3%

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

   9. Simplified 75.3%

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

4. Final simplification 90.4%

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

Alternative 7: 90.4% accurate, 15.2× speedup

Math

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

FPCore

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

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. count-2 99.9%

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

      2. fma-define 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-define 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in v around 0 91.2%

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

   if 0.100000001 < v

   1. Initial program 91.6%

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

      1. +-commutative 91.6%

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

      2. fma-define 91.3%

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

      3. +-commutative 91.3%

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

      4. fma-define 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in u around 0 74.0%

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

   6. Taylor expanded in v around inf 73.3%

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

      1. associate-*r/ 73.3%

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

      2. metadata-eval 73.3%

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

   8. Simplified 73.3%

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

4. Final simplification 90.3%

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

Alternative 8: 86.7% 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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. +-commutative 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. log-prod 99.5%

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

   5. +-commutative 99.5%

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

   6. fma-undefine 99.5%

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

   7. +-commutative 99.5%

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

   8. fma-undefine 99.5%

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

6. Applied egg-rr 99.5%

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

   1. count-2 99.5%

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

   2. fma-define 99.5%

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

   3. *-commutative 99.5%

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

   4. fma-define 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in v around 0 86.8%

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

Alternative 9: 5.9% 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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u around 0 5.5%

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

Reproduce

herbie shell --seed 2024170 
(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))))))))