HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 15.6s

Alternatives: 14

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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(u * Float32(Float32(1.0) + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(Float32(1.0) / u) + Float32(-1.0)))))), 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-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in u around inf 99.5%

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

   1. +-commutative 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. associate-/l* 99.5%

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

   4. *-commutative 99.5%

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

   5. distribute-lft-out 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)))), 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-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.5%

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

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%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 4: 91.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / (Float32(exp(1)) ^ Float32(Float32(-2.0) / v)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (v * (single(-1.0) + (single(1.0) / (single(2.71828182845904523536) ^ (single(-2.0) / v))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 67.0%

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

      1. *-un-lft-identity 67.0%

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

      2. exp-prod 67.0%

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

   5. Applied egg-rr 67.0%

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

      1. exp-1-e 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 89.5%

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

Alternative 5: 91.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (v * (-1.0f + (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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(v * Float32(Float32(-1.0) + 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(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 67.0%

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

4. Final simplification 89.5%

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

Alternative 6: 91.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 67.0%

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

      1. sub-neg 67.0%

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

      2. rec-exp 67.0%

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

      3. metadata-eval 67.0%

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

   5. Applied egg-rr 67.0%

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

      1. metadata-eval 67.0%

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

      2. sub-neg 67.0%

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

      3. expm1-undefine 67.0%

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

      4. distribute-neg-frac 67.0%

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

      5. metadata-eval 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 89.5%

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

Alternative 7: 90.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 - \left(\frac{u \cdot -2 - \frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v}}{v} - u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10599999874830246)
   1.0
   (-
    -1.0
    (-
     (/
      (-
       (* u -2.0)
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v))
      v)
     (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10599999874830246f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f - ((((u * -2.0f) - (((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v)) / v) - (u * 2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10599999874830246e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) - ((((u * (-2.0e0)) - (((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v)) / v) - (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10599999874830246))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) - Float32(Float32(Float32(Float32(u * Float32(-2.0)) - Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v)) / v) - Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10599999874830246))
		tmp = single(1.0);
	else
		tmp = single(-1.0) - ((((u * single(-2.0)) - (((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v)) / v) - (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.105999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 91.8%

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

   if 0.105999999 < v

   1. Initial program 93.9%

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

   3. Taylor expanded in u around 0 60.2%

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

   4. Taylor expanded in v around -inf 60.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 - \left(\frac{u \cdot -2 - \frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v}}{v} - u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(2 + \frac{2 - \frac{0.6666666666666666 \cdot \frac{-1}{v} - 1.3333333333333333}{v}}{v}\right) + \frac{-1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10599999874830246)
   1.0
   (*
    u
    (+
     (+
      2.0
      (/
       (- 2.0 (/ (- (* 0.6666666666666666 (/ -1.0 v)) 1.3333333333333333) v))
       v))
     (/ -1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10599999874830246f) {
		tmp = 1.0f;
	} else {
		tmp = u * ((2.0f + ((2.0f - (((0.6666666666666666f * (-1.0f / v)) - 1.3333333333333333f) / v)) / v)) + (-1.0f / u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10599999874830246e0) then
        tmp = 1.0e0
    else
        tmp = u * ((2.0e0 + ((2.0e0 - (((0.6666666666666666e0 * ((-1.0e0) / v)) - 1.3333333333333333e0) / v)) / v)) + ((-1.0e0) / u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10599999874830246))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(Float32(2.0) + Float32(Float32(Float32(2.0) - Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(Float32(-1.0) / v)) - Float32(1.3333333333333333)) / v)) / v)) + Float32(Float32(-1.0) / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10599999874830246))
		tmp = single(1.0);
	else
		tmp = u * ((single(2.0) + ((single(2.0) - (((single(0.6666666666666666) * (single(-1.0) / v)) - single(1.3333333333333333)) / v)) / v)) + (single(-1.0) / u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.105999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 91.8%

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

   if 0.105999999 < v

   1. Initial program 93.9%

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

   3. Taylor expanded in u around 0 60.9%

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

   4. Taylor expanded in u around inf 60.1%

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

      1. rec-exp 60.1%

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

      2. expm1-undefine 60.1%

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

      3. distribute-neg-frac 60.1%

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

      4. metadata-eval 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in v around -inf 60.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(2 + \frac{2 - \frac{0.6666666666666666 \cdot \frac{-1}{v} - 1.3333333333333333}{v}}{v}\right) + \frac{-1}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10599999874830246)
   1.0
   (+
    -1.0
    (- (* u 2.0) (/ (+ (* u -2.0) (* (/ u v) -1.3333333333333333)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10599999874830246f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - (((u * -2.0f) + ((u / v) * -1.3333333333333333f)) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10599999874830246e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - (((u * (-2.0e0)) + ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10599999874830246))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10599999874830246))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - (((u * single(-2.0)) + ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.105999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 91.8%

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

   if 0.105999999 < v

   1. Initial program 93.9%

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

   3. Taylor expanded in u around 0 60.2%

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

   4. Taylor expanded in v around -inf 58.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.7% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (u * (-2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(u * Float32(-2.0))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (u * single(-2.0))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in v around inf 59.2%

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

      1. fma-define 59.2%

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

      2. associate-*r/ 59.2%

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

      3. *-commutative 59.2%

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

      4. associate-/l* 59.2%

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

      5. *-commutative 59.2%

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

      6. unpow2 59.2%

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

      7. associate-*l* 59.2%

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

      8. *-commutative 59.2%

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

      9. distribute-lft-out 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in u around 0 59.5%

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

   7. Taylor expanded in v around 0 59.5%

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

      1. *-commutative 59.5%

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

   9. Simplified 59.5%

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

4. Final simplification 89.0%

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

Alternative 11: 90.5% accurate, 13.3× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 66.3%

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

   4. Taylor expanded in v around inf 56.4%

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

      1. distribute-lft-out 56.4%

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

   6. Simplified 56.4%

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

4. Final simplification 88.8%

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

Alternative 12: 90.5% accurate, 15.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0 66.3%

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

   4. Taylor expanded in v around inf 56.4%

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

      1. sub-neg 56.4%

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

      2. distribute-lft-out 56.4%

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

      3. metadata-eval 56.4%

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

   6. Simplified 56.4%

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

4. Final simplification 88.8%

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

Alternative 13: 5.9% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0 5.8%

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

4. Final simplification 5.8%

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

Alternative 14: 86.7% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in u around inf 99.5%

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

   1. +-commutative 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. associate-/l* 99.5%

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

   4. *-commutative 99.5%

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

   5. distribute-lft-out 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in v around 0 85.5%

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

9. Final simplification 85.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))))))