HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 14.3s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 3: 90.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \left(\frac{u \cdot 1.3333333333333333}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (+
      (/ (* u 1.3333333333333333) (pow v 2.0))
      (*
       0.5
       (/ (+ (* -4.0 (+ 1.0 (* u (+ u -2.0)))) (* (- 1.0 u) 4.0)) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((u * 1.3333333333333333f) / powf(v, 2.0f)) + (0.5f * (((-4.0f * (1.0f + (u * (u + -2.0f)))) + ((1.0f - u) * 4.0f)) / 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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (((u * 1.3333333333333333e0) / (v ** 2.0e0)) + (0.5e0 * ((((-4.0e0) * (1.0e0 + (u * (u + (-2.0e0))))) + ((1.0e0 - u) * 4.0e0)) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(u * Float32(1.3333333333333333)) / (v ^ Float32(2.0))) + Float32(Float32(0.5) * Float32(Float32(Float32(Float32(-4.0) * Float32(Float32(1.0) + Float32(u * Float32(u + Float32(-2.0))))) + Float32(Float32(Float32(1.0) - u) * Float32(4.0))) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((u * single(1.3333333333333333)) / (v ^ single(2.0))) + (single(0.5) * (((single(-4.0) * (single(1.0) + (u * (u + single(-2.0))))) + ((single(1.0) - u) * single(4.0))) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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-def 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-def 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-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 94.0%

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

   if 0.0500000007 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around inf 82.0%

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

   4. Taylor expanded in u around 0 82.0%

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

   5. Taylor expanded in u around 0 82.0%

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

      1. +-commutative 72.7%

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

      2. unpow2 72.7%

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

      3. distribute-rgt-out 72.7%

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

   7. Simplified 82.0%

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

   8. Taylor expanded in u around 0 78.8%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(\color{blue}{1.3333333333333333 \cdot \frac{u}{{v}^{2}}} + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 78.8%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(\color{blue}{\frac{u}{{v}^{2}} \cdot 1.3333333333333333} + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

      2. associate-*l/ 78.8%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(\color{blue}{\frac{u \cdot 1.3333333333333333}{{v}^{2}}} + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

   10. Simplified 78.8%

       \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(\color{blue}{\frac{u \cdot 1.3333333333333333}{{v}^{2}}} + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \left(\frac{u \cdot 1.3333333333333333}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5) 1.0 (+ (* u (* v (+ (/ 1.0 (exp (/ -2.0 v))) -1.0))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (v * ((1.0f / expf((-2.0f / v))) + -1.0f))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.5e0) then
        tmp = 1.0e0
    else
        tmp = (u * (v * ((1.0e0 / exp(((-2.0e0) / v))) + (-1.0e0)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(v * Float32(Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))) + Float32(-1.0)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.5))
		tmp = single(1.0);
	else
		tmp = (u * (v * ((single(1.0) / exp((single(-2.0) / v))) + single(-1.0)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   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-def 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-def 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-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.7%

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

   if 0.5 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in u around 0 82.7%

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

4. Final simplification 93.0%

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

Alternative 5: 90.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5) 1.0 (+ (* (* v u) (expm1 (/ 2.0 v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f;
	} else {
		tmp = ((v * u) * expm1f((2.0f / v))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(v * u) * expm1(Float32(Float32(2.0) / v))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   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-def 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-def 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-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.7%

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

   if 0.5 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in u around 0 82.1%

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

      1. sub-neg 82.1%

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

      2. rec-exp 82.1%

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

      3. metadata-eval 82.1%

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

   5. Applied egg-rr 82.1%

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

      1. metadata-eval 82.1%

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

      2. sub-neg 82.1%

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

      3. distribute-neg-frac 82.1%

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

      4. metadata-eval 82.1%

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

      5. metadata-eval 82.1%

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

      6. associate-*r/ 82.1%

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

      7. expm1-def 82.1%

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

      8. associate-*r/ 82.1%

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

      9. metadata-eval 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in u around 0 82.6%

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

      1. expm1-log1p-u 82.6%

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

      2. expm1-udef 82.1%

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

      3. expm1-def 82.1%

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

   10. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right)\right)} - 1\right)} - 1 \]
   11. Step-by-step derivation

       1. expm1-def 82.6%

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

       2. expm1-log1p 82.6%

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

       3. associate-*r* 82.6%

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

   12. Simplified 82.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (* 0.5 (/ (+ (* -4.0 (+ 1.0 (* u (+ u -2.0)))) (* (- 1.0 u) 4.0)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (0.5f * (((-4.0f * (1.0f + (u * (u + -2.0f)))) + ((1.0f - u) * 4.0f)) / 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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (0.5e0 * ((((-4.0e0) * (1.0e0 + (u * (u + (-2.0e0))))) + ((1.0e0 - u) * 4.0e0)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(0.5) * Float32(Float32(Float32(Float32(-4.0) * Float32(Float32(1.0) + Float32(u * Float32(u + Float32(-2.0))))) + Float32(Float32(Float32(1.0) - u) * Float32(4.0))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (single(0.5) * (((single(-4.0) * (single(1.0) + (u * (u + single(-2.0))))) + ((single(1.0) - u) * single(4.0))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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-def 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-def 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-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 94.0%

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

   if 0.0500000007 < v

   1. Initial program 91.7%

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

      1. +-commutative 91.7%

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

      2. fma-def 91.9%

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

      3. +-commutative 91.9%

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

      4. fma-def 91.3%

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

   3. Simplified 91.3%

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

      1. fma-udef 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in v around inf 72.7%

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

   8. Taylor expanded in u around 0 72.7%

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

      1. +-commutative 72.7%

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

      2. unpow2 72.7%

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

      3. distribute-rgt-out 72.7%

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

   10. Simplified 72.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806) 1.0 (+ (* 2.0 (+ u (/ u v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = (2.0f * (u + (u / v))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (2.0e0 * (u + (u / v))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(u + Float32(u / v))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = (single(2.0) * (u + (u / v))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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-def 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-def 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-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 94.0%

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

   if 0.0500000007 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in u around 0 77.5%

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

   4. Taylor expanded in v around inf 71.2%

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

      1. sub-neg 71.2%

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

      2. distribute-lft-out 71.2%

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

      3. metadata-eval 71.2%

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

   6. Simplified 71.2%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806) 1.0 (+ 1.0 (* (- 1.0 u) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((1.0f - u) * -2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((1.0e0 - u) * (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(1.0) - u) * single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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-def 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-def 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-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 94.0%

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

   if 0.0500000007 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in v around inf 60.3%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.5% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot 2 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806) 1.0 (+ (* u 2.0) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = (u * 2.0f) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (u * 2.0e0) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(2.0)) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = (u * single(2.0)) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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-def 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-def 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-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 94.0%

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

   if 0.0500000007 < v

   1. Initial program 91.7%

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

   3. Taylor expanded in u around 0 77.5%

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

   4. Taylor expanded in v around inf 60.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot 2 + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 5.8% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(u, v)
	return Float32(-1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(-1.0);
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0 6.0%

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

4. Final simplification 6.0%

   \[\leadsto -1 \]
5. Add Preprocessing

Alternative 11: 86.8% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

function code(u, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.5%

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

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around 0 88.2%

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

8. Final simplification 88.2%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))))))))