HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 19.6s

Alternatives: 13

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

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

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 99.5%

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

2. Taylor expanded in v around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-def 99.5%

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

4. Simplified 99.5%

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

5. Final simplification 99.5%

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

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. Final simplification 99.5%

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

Alternative 3: 91.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{u \cdot u}{v}, -4 \cdot \left(\frac{u}{v} \cdot \frac{u}{v}\right) - u \cdot \left(-2 + \left(\frac{-2}{v} + \frac{-1.3333333333333333}{v \cdot v}\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    (fma
     -2.0
     (/ (* u u) v)
     (-
      (* -4.0 (* (/ u v) (/ u v)))
      (* u (+ -2.0 (+ (/ -2.0 v) (/ -1.3333333333333333 (* v v)))))))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, ((u * u) / v), ((-4.0f * ((u / v) * (u / v))) - (u * (-2.0f + ((-2.0f / v) + (-1.3333333333333333f / (v * v))))))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(u * u) / v), Float32(Float32(Float32(-4.0) * Float32(Float32(u / v) * Float32(u / v))) - Float32(u * Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) / v) + Float32(Float32(-1.3333333333333333) / Float32(v * v))))))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.6%

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

   2. Taylor expanded in u around 0 69.0%

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

      1. associate--l+ 68.7%

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

      2. unpow2 68.7%

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

      3. associate-*r* 68.7%

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

      4. mul-1-neg 68.7%

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

      5. fma-neg 68.7%

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

   4. Simplified 68.7%

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

   5. Taylor expanded in v around inf 63.5%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{{u}^{2}}{v} + \left(-0.5 \cdot \frac{-8 \cdot {u}^{2} - -16 \cdot {u}^{2}}{{v}^{2}} + \left(1.3333333333333333 \cdot \frac{u}{{v}^{2}} + \left(2 \cdot u + 2 \cdot \frac{u}{v}\right)\right)\right)\right) - 1} \]
   6. Step-by-step derivation

      1. sub-neg 63.5%

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{{u}^{2}}{v} + \left(-0.5 \cdot \frac{-8 \cdot {u}^{2} - -16 \cdot {u}^{2}}{{v}^{2}} + \left(1.3333333333333333 \cdot \frac{u}{{v}^{2}} + \left(2 \cdot u + 2 \cdot \frac{u}{v}\right)\right)\right)\right) + \left(-1\right)} \]

   7. Simplified 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{u \cdot u}{v}, \mathsf{fma}\left(-0.5, \frac{\left(u \cdot u\right) \cdot 8}{v \cdot v}, \mathsf{fma}\left(1.3333333333333333, \frac{u}{v \cdot v}, 2 \cdot \left(u + \frac{u}{v}\right)\right)\right)\right) + -1} \]

   8. Taylor expanded in u around -inf 63.5%

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

      1. mul-1-neg 63.5%

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

      2. unsub-neg 63.5%

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

      3. unpow2 63.5%

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

      4. unpow2 63.5%

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

      5. times-frac 63.5%

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

      6. sub-neg 63.5%

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

      7. distribute-lft-in 63.5%

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

      8. metadata-eval 63.5%

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

      9. metadata-eval 63.5%

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

      10. distribute-lft-neg-in 63.5%

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

      11. associate-*r/ 63.5%

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

      12. metadata-eval 63.5%

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

      13. distribute-neg-frac 63.5%

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

      14. metadata-eval 63.5%

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

      15. associate-+l+ 63.5%

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

   10. Simplified 63.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{u \cdot u}{v}, -4 \cdot \left(\frac{u}{v} \cdot \frac{u}{v}\right) - u \cdot \left(-2 + \left(\frac{-2}{v} + \frac{-1.3333333333333333}{v \cdot v}\right)\right)\right) + -1\\ \end{array} \]

Alternative 4: 91.3% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (fma
     -2.0
     (/ (* u u) v)
     (* u (+ (+ (/ 1.3333333333333333 (* v v)) 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 + fmaf(-2.0f, ((u * u) / v), (u * (((1.3333333333333333f / (v * v)) + 2.0f) + (2.0f / v))));
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.6%

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

   2. Taylor expanded in u around 0 69.0%

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

      1. associate--l+ 68.7%

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

      2. unpow2 68.7%

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

      3. associate-*r* 68.7%

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

      4. mul-1-neg 68.7%

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

      5. fma-neg 68.7%

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

   4. Simplified 68.7%

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

   5. Taylor expanded in v around inf 63.5%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{{u}^{2}}{v} + \left(-0.5 \cdot \frac{-8 \cdot {u}^{2} - -16 \cdot {u}^{2}}{{v}^{2}} + \left(1.3333333333333333 \cdot \frac{u}{{v}^{2}} + \left(2 \cdot u + 2 \cdot \frac{u}{v}\right)\right)\right)\right) - 1} \]
   6. Step-by-step derivation

      1. sub-neg 63.5%

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{{u}^{2}}{v} + \left(-0.5 \cdot \frac{-8 \cdot {u}^{2} - -16 \cdot {u}^{2}}{{v}^{2}} + \left(1.3333333333333333 \cdot \frac{u}{{v}^{2}} + \left(2 \cdot u + 2 \cdot \frac{u}{v}\right)\right)\right)\right) + \left(-1\right)} \]

   7. Simplified 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{u \cdot u}{v}, \mathsf{fma}\left(-0.5, \frac{\left(u \cdot u\right) \cdot 8}{v \cdot v}, \mathsf{fma}\left(1.3333333333333333, \frac{u}{v \cdot v}, 2 \cdot \left(u + \frac{u}{v}\right)\right)\right)\right) + -1} \]

   8. Taylor expanded in u around 0 61.4%

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

      1. +-commutative 61.4%

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

      2. distribute-lft-in 61.4%

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

      3. metadata-eval 61.4%

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

      4. associate-+r+ 61.4%

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

      5. associate-*r/ 61.4%

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

      6. metadata-eval 61.4%

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

      7. unpow2 61.4%

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

      8. associate-*r/ 61.4%

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

      9. metadata-eval 61.4%

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

   10. Simplified 61.4%

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

4. Final simplification 91.1%

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

Alternative 5: 90.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = ((-2.0f * (u * u)) / v) + fmaf(2.0f, (u + (u / v)), -1.0f);
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.6%

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

   2. Taylor expanded in u around 0 69.0%

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

      1. associate--l+ 68.7%

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

      2. unpow2 68.7%

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

      3. associate-*r* 68.7%

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

      4. mul-1-neg 68.7%

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

      5. fma-neg 68.7%

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

   4. Simplified 68.7%

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

   5. Taylor expanded in v around inf 57.6%

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

      1. associate--l+ 57.6%

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

      2. associate-*r/ 57.6%

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

      3. unpow2 57.6%

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

      4. distribute-lft-out 57.6%

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

      5. fma-neg 57.6%

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

      6. metadata-eval 57.6%

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

   7. Simplified 57.6%

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

4. Final simplification 90.8%

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

Alternative 6: 91.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+ 1.0 (- (* u (* v (expm1 (/ 2.0 v)))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * expm1f((2.0f / v)))) - 2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))) - Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.300000012 < v

   1. Initial program 93.0%

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

   2. Taylor expanded in u around 0 67.4%

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

      1. sub-neg 67.4%

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

      2. rec-exp 67.4%

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

      3. metadata-eval 67.4%

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

   4. Applied egg-rr 67.4%

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

      1. metadata-eval 67.4%

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

      2. sub-neg 67.4%

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

      3. distribute-neg-frac 67.4%

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

      4. metadata-eval 67.4%

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

      5. expm1-def 67.4%

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

   6. Simplified 67.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) - 2\right)\\ \end{array} \]

Alternative 7: 91.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + e^{\frac{2}{v}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+ -1.0 (* u (* v (+ -1.0 (exp (/ 2.0 v))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (v * (-1.0f + expf((2.0f / v)))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (v * ((-1.0e0) + exp((2.0e0 / v)))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(v * Float32(Float32(-1.0) + exp(Float32(Float32(2.0) / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (v * (single(-1.0) + exp((single(2.0) / v)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.300000012 < v

   1. Initial program 93.0%

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

   2. Taylor expanded in u around 0 79.1%

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

      1. associate--l+ 78.7%

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

      2. unpow2 78.7%

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

      3. associate-*r* 78.7%

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

      4. mul-1-neg 78.7%

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

      5. fma-neg 78.7%

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

   4. Simplified 78.7%

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

   5. Taylor expanded in u around 0 67.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + e^{\frac{2}{v}}\right)\right)\\ \end{array} \]

Alternative 8: 90.6% accurate, 8.5× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * (-1.0f + (1.0f / (1.0f + ((2.0f / (v * v)) - (2.0f / v))))))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (v * ((-1.0e0) + (1.0e0 / (1.0e0 + ((2.0e0 / (v * v)) - (2.0e0 / v))))))) - 2.0e0)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.6%

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

   2. Taylor expanded in u around 0 56.7%

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

   3. Taylor expanded in v around inf 56.1%

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

      1. associate--l+ 56.1%

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

      2. associate-*r/ 56.1%

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

      3. metadata-eval 56.1%

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

      4. unpow2 56.1%

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

      5. associate-*r/ 56.1%

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

      6. metadata-eval 56.1%

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

   5. Simplified 56.1%

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

4. Final simplification 90.7%

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

Alternative 9: 90.8% accurate, 11.1× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * ((2.0f / v) + (2.0f / (v * v))))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (v * ((2.0e0 / v) + (2.0e0 / (v * v))))) - 2.0e0)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.6%

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

   2. Taylor expanded in u around 0 56.7%

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

   3. Taylor expanded in v around inf 55.1%

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

      1. associate-*r/ 55.1%

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

      2. metadata-eval 55.1%

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

      3. associate-*r/ 55.1%

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

      4. metadata-eval 55.1%

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

      5. unpow2 55.1%

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

   5. Simplified 55.1%

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

4. Final simplification 90.7%

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

Alternative 10: 90.8% accurate, 16.2× speedup

Math

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

FPCore

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

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))) - 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 + ((2.0e0 * (u + (u / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.6%

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

   2. Taylor expanded in u around 0 56.7%

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

   3. Taylor expanded in v around inf 55.1%

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

      1. distribute-lft-out 55.1%

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

   5. Simplified 55.1%

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

4. Final simplification 90.7%

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

Alternative 11: 90.8% accurate, 19.1× 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 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.6%

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

   2. Taylor expanded in u around 0 56.3%

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

   3. Taylor expanded in v around inf 55.0%

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

      1. sub-neg 55.0%

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

      2. distribute-lft-out 55.0%

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

      3. metadata-eval 55.0%

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

   5. Simplified 55.0%

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

4. Final simplification 90.7%

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

Alternative 12: 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. Taylor expanded in u around 0 5.7%

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

3. Final simplification 5.7%

   \[\leadsto -1 \]

Alternative 13: 87.2% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in v around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-def 99.5%

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

4. Simplified 99.5%

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

5. Taylor expanded in v around 0 87.1%

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

6. Final simplification 87.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023271 
(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))))))))