Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 96.4%

Time: 6.7s

Alternatives: 4

Speedup: 11.4×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.0007999999797903001:\\ \;\;\;\;s \cdot \left(\left(u \cdot u\right) \cdot 8 - -4 \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.0007999999797903001)
   (* s (- (* (* u u) 8.0) (* -4.0 u)))
   (* s (log (/ 1.0 (- 1.0 (* 4.0 u)))))))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.0007999999797903001f) {
		tmp = s * (((u * u) * 8.0f) - (-4.0f * u));
	} else {
		tmp = s * logf((1.0f / (1.0f - (4.0f * u))));
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if (u <= 0.0007999999797903001e0) then
        tmp = s * (((u * u) * 8.0e0) - ((-4.0e0) * u))
    else
        tmp = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.0007999999797903001))
		tmp = Float32(s * Float32(Float32(Float32(u * u) * Float32(8.0)) - Float32(Float32(-4.0) * u)));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if (u <= single(0.0007999999797903001))
		tmp = s * (((u * u) * single(8.0)) - (single(-4.0) * u));
	else
		tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u < 7.9999998e-4

   1. Initial program 49.5%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]
   4. Step-by-step derivation

      1. lower-*.f32 86.1

         \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

   5. Applied rewrites 86.1%

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + 8 \cdot u\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + 8 \cdot u\right) \cdot u\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + 8 \cdot u\right) \cdot u\right)} \]

      3. +-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left(8 \cdot u + 4\right)} \cdot u\right) \]

      4. lower-fma.f32 86.1

         \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8, u, 4\right)} \cdot u\right) \]

   8. Applied rewrites 86.1%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(8, u, 4\right) \cdot u\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.2%

       \[\leadsto s \cdot \left(\left(u \cdot u\right) \cdot 8 - \color{blue}{-4 \cdot u}\right) \]

   if 7.9999998e-4 < u

   1. Initial program 90.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(u \cdot u\right) \cdot 8 - -4 \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (- (* (* u u) 8.0) (* -4.0 u))))

C

float code(float s, float u) {
	return s * (((u * u) * 8.0f) - (-4.0f * u));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (((u * u) * 8.0e0) - ((-4.0e0) * u))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(u * u) * Float32(8.0)) - Float32(Float32(-4.0) * u)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (((u * u) * single(8.0)) - (single(-4.0) * u));
end

Derivation

1. Initial program 60.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]
4. Step-by-step derivation

   1. lower-*.f32 75.0

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

5. Applied rewrites 75.0%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + 8 \cdot u\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + 8 \cdot u\right) \cdot u\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + 8 \cdot u\right) \cdot u\right)} \]

   3. +-commutative N/A

      \[\leadsto s \cdot \left(\color{blue}{\left(8 \cdot u + 4\right)} \cdot u\right) \]

   4. lower-fma.f32 75.0

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8, u, 4\right)} \cdot u\right) \]

8. Applied rewrites 75.0%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(8, u, 4\right) \cdot u\right)} \]
9. Step-by-step derivation

10. Applied rewrites 88.0%

    \[\leadsto s \cdot \left(\left(u \cdot u\right) \cdot 8 - \color{blue}{-4 \cdot u}\right) \]
11. Add Preprocessing

Alternative 3: 86.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(4 - -8 \cdot u\right) \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* (- 4.0 (* -8.0 u)) u)))

C

float code(float s, float u) {
	return s * ((4.0f - (-8.0f * u)) * u);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((4.0e0 - ((-8.0e0) * u)) * u)
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(4.0) - Float32(Float32(-8.0) * u)) * u))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((single(4.0) - (single(-8.0) * u)) * u);
end

Derivation

1. Initial program 60.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]
4. Step-by-step derivation

   1. lower-*.f32 75.0

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

5. Applied rewrites 75.0%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + 8 \cdot u\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + 8 \cdot u\right) \cdot u\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + 8 \cdot u\right) \cdot u\right)} \]

   3. +-commutative N/A

      \[\leadsto s \cdot \left(\color{blue}{\left(8 \cdot u + 4\right)} \cdot u\right) \]

   4. lower-fma.f32 75.0

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8, u, 4\right)} \cdot u\right) \]

8. Applied rewrites 75.0%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(8, u, 4\right) \cdot u\right)} \]
9. Step-by-step derivation

10. Applied rewrites 87.8%

    \[\leadsto s \cdot \left(\left(4 - -8 \cdot u\right) \cdot u\right) \]
11. Add Preprocessing

Alternative 4: 73.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(4 \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* 4.0 u)))

C

float code(float s, float u) {
	return s * (4.0f * u);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (4.0e0 * u)
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(4.0) * u))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (single(4.0) * u);
end

Derivation

1. Initial program 60.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]
4. Step-by-step derivation

   1. lower-*.f32 75.0

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

5. Applied rewrites 75.0%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))