Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.1% → 89.2%

Time: 8.8s

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: 61.1% 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: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9997000098228455:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot u\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9997000098228455) (* s (log (/ 1.0 t_0))) (* (* 4.0 u) s))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9997000098228455f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = (4.0f * u) * s;
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 - (4.0e0 * u)
    if (t_0 <= 0.9997000098228455e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = (4.0e0 * u) * s
    end if
    code = tmp
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(Float32(4.0) * u))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9997000098228455))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(Float32(Float32(4.0) * u) * s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	t_0 = single(1.0) - (single(4.0) * u);
	tmp = single(0.0);
	if (t_0 <= single(0.9997000098228455))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = (single(4.0) * u) * s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.99970001

   1. Initial program 86.8%

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

   if 0.99970001 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 44.1%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9997000098228455:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot u\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.0% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 76.4

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

5. Applied rewrites 76.4%

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

6. Final simplification 76.4%

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

Alternative 3: 73.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 76.1

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

5. Applied rewrites 76.1%

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

Reproduce

herbie shell --seed 2024216 
(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))))))