HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.9s

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: 99.5% 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 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 3: 90.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.5%

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

   if 0.200000003 < v

   1. Initial program 95.2%

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

   3. Taylor expanded in u around 0 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*l* 70.0%

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

      3. *-commutative 70.0%

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

      4. fma-neg 70.0%

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

      5. rec-exp 70.0%

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

      6. expm1-define 70.2%

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

      7. distribute-neg-frac 70.2%

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

      8. metadata-eval 70.2%

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

      9. metadata-eval 70.2%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in v around 0 70.7%

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

4. Final simplification 91.6%

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

Alternative 4: 90.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.5%

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

   if 0.200000003 < v

   1. Initial program 95.2%

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

   3. Taylor expanded in u around 0 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*l* 70.0%

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

      3. *-commutative 70.0%

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

      4. fma-neg 70.0%

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

      5. rec-exp 70.0%

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

      6. expm1-define 70.2%

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

      7. distribute-neg-frac 70.2%

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

      8. metadata-eval 70.2%

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

      9. metadata-eval 70.2%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in v around 0 70.7%

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

      1. pow1 70.7%

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

      2. expm1-define 70.7%

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

   8. Applied egg-rr 70.7%

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

      1. unpow1 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*r* 70.7%

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

   10. Simplified 70.7%

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

4. Final simplification 91.6%

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

Alternative 5: 90.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (+
     (/
      (-
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v)
       (* u -2.0))
      v)
     (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) + (u * 2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) + (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v) + Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.5%

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

   if 0.200000003 < v

   1. Initial program 95.2%

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

   3. Taylor expanded in u around 0 70.0%

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

   4. Taylor expanded in v around -inf 67.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 - 0.6666666666666666 \cdot \frac{-1}{v}}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+ 2.0 (/ (- 1.3333333333333333 (* 0.6666666666666666 (/ -1.0 v))) v))
       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 + ((1.3333333333333333f - (0.6666666666666666f * (-1.0f / v))) / v)) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 - (0.6666666666666666e0 * ((-1.0e0) / v))) / v)) / 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(Float32(Float32(1.3333333333333333) - Float32(Float32(0.6666666666666666) * Float32(Float32(-1.0) / v))) / v)) / 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) + ((single(1.3333333333333333) - (single(0.6666666666666666) * (single(-1.0) / v))) / v)) / 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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.5%

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

   if 0.200000003 < v

   1. Initial program 95.2%

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

   3. Taylor expanded in u around 0 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*l* 70.0%

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

      3. *-commutative 70.0%

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

      4. fma-neg 70.0%

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

      5. rec-exp 70.0%

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

      6. expm1-define 70.2%

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

      7. distribute-neg-frac 70.2%

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

      8. metadata-eval 70.2%

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

      9. metadata-eval 70.2%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in v around 0 70.7%

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

   7. Taylor expanded in v around -inf 67.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 - 0.6666666666666666 \cdot \frac{-1}{v}}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\frac{-1}{u} - \frac{1.3333333333333333 \cdot \frac{-1}{v} - 2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (*
    u
    (+ 2.0 (- (/ -1.0 u) (/ (- (* 1.3333333333333333 (/ -1.0 v)) 2.0) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = u * (2.0f + ((-1.0f / u) - (((1.3333333333333333f * (-1.0f / v)) - 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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = u * (2.0e0 + (((-1.0e0) / u) - (((1.3333333333333333e0 * ((-1.0e0) / v)) - 2.0e0) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(-1.0) / u) - Float32(Float32(Float32(Float32(1.3333333333333333) * Float32(Float32(-1.0) / v)) - Float32(2.0)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = u * (single(2.0) + ((single(-1.0) / u) - (((single(1.3333333333333333) * (single(-1.0) / v)) - single(2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. Step-by-step derivation

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.8%

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

   if 0.119999997 < v

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0 67.6%

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

   4. Taylor expanded in v around -inf 63.4%

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

   5. Taylor expanded in u around -inf 63.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\frac{-1}{u} - \frac{1.3333333333333333 \cdot \frac{-1}{v} - 2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.6% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(\frac{2 + \frac{1.3333333333333333}{v}}{v} + \frac{-1}{u}\right) - -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (* u (- (+ (/ (+ 2.0 (/ 1.3333333333333333 v)) v) (/ -1.0 u)) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = u * ((((2.0f + (1.3333333333333333f / v)) / v) + (-1.0f / u)) - -2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = u * ((((2.0e0 + (1.3333333333333333e0 / v)) / v) + ((-1.0e0) / u)) - (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) + Float32(Float32(-1.0) / u)) - Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = u * ((((single(2.0) + (single(1.3333333333333333) / v)) / v) + (single(-1.0) / u)) - single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. Step-by-step derivation

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.8%

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

   if 0.119999997 < v

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0 67.6%

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

   4. Taylor expanded in v around -inf 63.4%

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

   5. Taylor expanded in u around -inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. distribute-rgt-neg-in 63.7%

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

      3. sub-neg 63.7%

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

      4. +-commutative 63.7%

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

      5. mul-1-neg 63.7%

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

      6. unsub-neg 63.7%

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

      7. associate-*r/ 63.7%

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

      8. metadata-eval 63.7%

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

      9. metadata-eval 63.7%

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

   7. Simplified 63.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(\frac{2 + \frac{1.3333333333333333}{v}}{v} + \frac{-1}{u}\right) - -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.3% accurate, 11.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (u * ((2.0f / v) - (-2.0f + (2.0f / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (u * ((2.0e0 / v) - ((-2.0e0) + (2.0e0 / u))))
    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(Float32(2.0) / v) - Float32(Float32(-2.0) + Float32(Float32(2.0) / u)))));
	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) / v) - (single(-2.0) + (single(2.0) / u))));
	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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.5%

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

   if 0.200000003 < v

   1. Initial program 95.2%

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

   3. Taylor expanded in u around 0 70.0%

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

   4. Taylor expanded in u around -inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. distribute-rgt-neg-in 69.8%

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

      3. fma-define 69.8%

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

      4. rec-exp 69.8%

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

      5. distribute-neg-frac 69.8%

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

      6. metadata-eval 69.8%

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

      7. fma-define 69.8%

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

      8. +-commutative 69.8%

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

      9. mul-1-neg 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in v around inf 60.2%

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

      1. associate--r+ 60.2%

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

      2. sub-neg 60.2%

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

      3. associate-*r/ 60.2%

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

      4. metadata-eval 60.2%

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

      5. metadata-eval 60.2%

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

      6. associate-*r/ 60.2%

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

      7. metadata-eval 60.2%

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

   9. Simplified 60.2%

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

4. Final simplification 90.8%

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

Alternative 10: 90.3% accurate, 13.3× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ 1.0 (+ -2.0 (* 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 + (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) + (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(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) + (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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.5%

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

   if 0.200000003 < v

   1. Initial program 95.2%

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

   3. Taylor expanded in u around 0 70.0%

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

   4. Taylor expanded in v around inf 60.1%

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

      1. sub-neg 60.1%

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

      2. distribute-lft-out 60.1%

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

      3. metadata-eval 60.1%

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

   6. Simplified 60.1%

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

4. Final simplification 90.7%

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

Alternative 11: 90.3% 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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.5%

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

   if 0.200000003 < v

   1. Initial program 95.2%

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

   3. Taylor expanded in u around 0 70.0%

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

   4. Taylor expanded in v around inf 60.1%

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

      1. sub-neg 60.1%

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

      2. distribute-lft-out 60.1%

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

      3. metadata-eval 60.1%

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

   6. Simplified 60.1%

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

4. Final simplification 90.7%

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

Alternative 12: 89.7% accurate, 21.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 93.5%

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

   if 0.200000003 < v

   1. Initial program 95.2%

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

   3. Taylor expanded in u around 0 70.0%

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

   4. Taylor expanded in v around inf 52.5%

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

4. Final simplification 90.1%

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

Alternative 13: 86.8% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in v around 0 86.6%

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

Alternative 14: 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 99.6%

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

3. Taylor expanded in u around 0 6.3%

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

Reproduce

herbie shell --seed 2024103 
(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))))))))