Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 99.4%

Time: 3.8s

Alternatives: 6

Speedup: 2.8×

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

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

FPCore

(FPCore (s u)
  :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))))))

C

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

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    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

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

Math

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

FPCore

(FPCore (s u)
  :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))))))

C

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

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    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: 99.4% accurate, 1.1× speedup

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

Math

\[s \cdot \left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (- (log1p (* -4.0 u)))))

C

float code(float s, float u) {
	return s * -log1pf((-4.0f * u));
}

Julia

function code(s, u)
	return Float32(s * Float32(-log1p(Float32(Float32(-4.0) * u))))
end

Derivation

1. Initial program 61.4%

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

3. Applied rewrites 63.9%

   \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \]

5. Applied rewrites 99.4%

   \[\leadsto s \cdot \left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \]
6. Add Preprocessing

Alternative 2: 98.4% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9819999933242798:\\ \;\;\;\;s \cdot \left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{\mathsf{fma}\left(-0.3333333333333333, u, -0.5\right) + \frac{0.25}{u}}\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (if (<= (- 1.0 (* 4.0 u)) 0.9819999933242798)
  (* s (- (log (fma -4.0 u 1.0))))
  (/ s (+ (fma -0.3333333333333333 u -0.5) (/ 0.25 u)))))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9819999933242798f) {
		tmp = s * -logf(fmaf(-4.0f, u, 1.0f));
	} else {
		tmp = s / (fmaf(-0.3333333333333333f, u, -0.5f) + (0.25f / u));
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9819999933242798))
		tmp = Float32(s * Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))));
	else
		tmp = Float32(s / Float32(fma(Float32(-0.3333333333333333), u, Float32(-0.5)) + Float32(Float32(0.25) / u)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.981999993

   1. Initial program 61.4%

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

   3. Applied rewrites 63.9%

      \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \]

   if 0.981999993 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 61.4%

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

   3. Applied rewrites 62.1%

      \[\leadsto \frac{1}{{\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right)}^{-1}} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.1%

      \[\leadsto \frac{1}{\frac{\frac{-1}{\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)}}{s}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \frac{1}{\frac{\frac{\frac{1}{4} + u \cdot \left(\frac{-1}{3} \cdot u - \frac{1}{2}\right)}{u}}{s}} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.9%

      \[\leadsto \frac{1}{\frac{\frac{0.25 + u \cdot \left(-0.3333333333333333 \cdot u - 0.5\right)}{u}}{s}} \]

   10. Applied rewrites 92.6%

       \[\leadsto \frac{s}{\mathsf{fma}\left(-0.3333333333333333, u, -0.5\right) + \frac{0.25}{u}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.3% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9957000017166138:\\ \;\;\;\;s \cdot \left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{\frac{0.25}{u} - 0.5}\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (if (<= (- 1.0 (* 4.0 u)) 0.9957000017166138)
  (* s (- (log (fma -4.0 u 1.0))))
  (/ s (- (/ 0.25 u) 0.5))))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9957000017166138f) {
		tmp = s * -logf(fmaf(-4.0f, u, 1.0f));
	} else {
		tmp = s / ((0.25f / u) - 0.5f);
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9957000017166138))
		tmp = Float32(s * Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))));
	else
		tmp = Float32(s / Float32(Float32(Float32(0.25) / u) - Float32(0.5)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.995700002

   1. Initial program 61.4%

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

   3. Applied rewrites 63.9%

      \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \]

   if 0.995700002 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 61.4%

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

   3. Applied rewrites 62.1%

      \[\leadsto \frac{1}{{\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right)}^{-1}} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.1%

      \[\leadsto \frac{1}{\frac{\frac{-1}{\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)}}{s}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \frac{1}{\frac{\frac{\frac{1}{4} + \frac{-1}{2} \cdot u}{u}}{s}} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.0%

      \[\leadsto \frac{1}{\frac{\frac{0.25 + -0.5 \cdot u}{u}}{s}} \]

   10. Applied rewrites 88.7%

       \[\leadsto \frac{s}{\frac{0.25}{u} - 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.7% accurate, 1.7× speedup

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

Math

\[\frac{s}{\frac{0.25}{u} - 0.5} \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (/ s (- (/ 0.25 u) 0.5)))

C

float code(float s, float u) {
	return s / ((0.25f / u) - 0.5f);
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s / ((0.25e0 / u) - 0.5e0)
end function

Julia

function code(s, u)
	return Float32(s / Float32(Float32(Float32(0.25) / u) - Float32(0.5)))
end

MATLAB

function tmp = code(s, u)
	tmp = s / ((single(0.25) / u) - single(0.5));
end

Derivation

1. Initial program 61.4%

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

3. Applied rewrites 62.1%

   \[\leadsto \frac{1}{{\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right)}^{-1}} \]
4. Step-by-step derivation

5. Applied rewrites 62.1%

   \[\leadsto \frac{1}{\frac{\frac{-1}{\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)}}{s}} \]

6. Taylor expanded in u around 0

   \[\leadsto \frac{1}{\frac{\frac{\frac{1}{4} + \frac{-1}{2} \cdot u}{u}}{s}} \]
7. Step-by-step derivation

8. Applied rewrites 85.0%

   \[\leadsto \frac{1}{\frac{\frac{0.25 + -0.5 \cdot u}{u}}{s}} \]

10. Applied rewrites 88.7%

    \[\leadsto \frac{s}{\frac{0.25}{u} - 0.5} \]
11. Add Preprocessing

Alternative 5: 73.7% accurate, 2.8× speedup

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

Math

\[s \cdot \left(u \cdot 4\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (* u 4.0)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

2. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(u \cdot \left(4 + 8 \cdot u\right)\right) \]
3. Step-by-step derivation

4. Applied rewrites 86.6%

   \[\leadsto s \cdot \left(u \cdot \left(4 + 8 \cdot u\right)\right) \]

5. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(u \cdot 4\right) \]
6. Step-by-step derivation

7. Applied rewrites 73.7%

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

Alternative 6: 73.5% accurate, 2.8× speedup

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

Math

\[4 \cdot \left(s \cdot u\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0))
     (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* 4.0 (* s u)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

2. Taylor expanded in u around 0

   \[\leadsto 4 \cdot \left(s \cdot u\right) \]
3. Step-by-step derivation

4. Applied rewrites 73.5%

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

Reproduce

herbie shell --seed 2026086 
(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))))))