Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.9% → 89.0%

Time: 6.1s

Alternatives: 3

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: 60.9% 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: 89.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.00022000000171829015:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - e^{\log \left(u \cdot 4\right)}}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.00022000000171829015)
   (* (* u 4.0) s)
   (* (log (/ 1.0 (- 1.0 (exp (log (* u 4.0)))))) s)))

C

float code(float s, float u) {
	float tmp;
	if ((u * 4.0f) <= 0.00022000000171829015f) {
		tmp = (u * 4.0f) * s;
	} else {
		tmp = logf((1.0f / (1.0f - expf(logf((u * 4.0f)))))) * s;
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((u * 4.0e0) <= 0.00022000000171829015e0) then
        tmp = (u * 4.0e0) * s
    else
        tmp = log((1.0e0 / (1.0e0 - exp(log((u * 4.0e0)))))) * s
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(u * Float32(4.0)) <= Float32(0.00022000000171829015))
		tmp = Float32(Float32(u * Float32(4.0)) * s);
	else
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - exp(log(Float32(u * Float32(4.0))))))) * s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((u * single(4.0)) <= single(0.00022000000171829015))
		tmp = (u * single(4.0)) * s;
	else
		tmp = log((single(1.0) / (single(1.0) - exp(log((u * single(4.0))))))) * s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 2.20000002e-4

   1. Initial program 43.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

   if 2.20000002e-4 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 86.1%

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

      1. rem-exp-log N/A

         \[\leadsto s \cdot \log \left(\frac{1}{1 - \color{blue}{e^{\log \left(4 \cdot u\right)}}}\right) \]

      2. lower-exp.f32 N/A

         \[\leadsto s \cdot \log \left(\frac{1}{1 - \color{blue}{e^{\log \left(4 \cdot u\right)}}}\right) \]

      3. lower-log.f32 86.2

         \[\leadsto s \cdot \log \left(\frac{1}{1 - e^{\color{blue}{\log \left(4 \cdot u\right)}}}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto s \cdot \log \left(\frac{1}{1 - e^{\log \color{blue}{\left(4 \cdot u\right)}}}\right) \]

      5. *-commutative N/A

         \[\leadsto s \cdot \log \left(\frac{1}{1 - e^{\log \color{blue}{\left(u \cdot 4\right)}}}\right) \]

      6. lower-*.f32 86.2

         \[\leadsto s \cdot \log \left(\frac{1}{1 - e^{\log \color{blue}{\left(u \cdot 4\right)}}}\right) \]

   4. Applied rewrites 86.2%

      \[\leadsto s \cdot \log \left(\frac{1}{1 - \color{blue}{e^{\log \left(u \cdot 4\right)}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.00022000000171829015:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - e^{\log \left(u \cdot 4\right)}}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.00022000000171829015:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.00022000000171829015)
   (* (* u 4.0) s)
   (* (log (/ 1.0 (- 1.0 (* u 4.0)))) s)))

C

float code(float s, float u) {
	float tmp;
	if ((u * 4.0f) <= 0.00022000000171829015f) {
		tmp = (u * 4.0f) * s;
	} else {
		tmp = logf((1.0f / (1.0f - (u * 4.0f)))) * s;
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((u * 4.0e0) <= 0.00022000000171829015e0) then
        tmp = (u * 4.0e0) * s
    else
        tmp = log((1.0e0 / (1.0e0 - (u * 4.0e0)))) * s
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(u * Float32(4.0)) <= Float32(0.00022000000171829015))
		tmp = Float32(Float32(u * Float32(4.0)) * s);
	else
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(u * Float32(4.0))))) * s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((u * single(4.0)) <= single(0.00022000000171829015))
		tmp = (u * single(4.0)) * s;
	else
		tmp = log((single(1.0) / (single(1.0) - (u * single(4.0))))) * s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 2.20000002e-4

   1. Initial program 43.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

   if 2.20000002e-4 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 86.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.00022000000171829015:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 72.8

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

5. Applied rewrites 72.8%

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

6. Final simplification 72.8%

   \[\leadsto \left(u \cdot 4\right) \cdot s \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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))))))