UniformSampleCone, z

Percentage Accurate: 99.9% → 99.9%

Time: 3.9s

Alternatives: 4

Speedup: 1.0×

• could not determine a ground truth

  1. ux = 0.07536157965660095
  2. uy = 3.2476998512720456e-6
  3. maxCos = 2.5045538963405115e-24

Specification

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(1 - ux\right) + ux \cdot maxCos \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (+ (- 1.0 ux) (* ux maxCos)))

C

float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (1.0e0 - ux) + (ux * maxcos)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(1.0) - ux) + (ux * maxCos);
end

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - ux\right) + ux \cdot maxCos \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (+ (- 1.0 ux) (* ux maxCos)))

C

float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (1.0e0 - ux) + (ux * maxcos)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(1.0) - ux) + (ux * maxCos);
end

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\left(1 - maxCos \cdot maxCos\right) \cdot ux}{-1 - maxCos} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (+ 1.0 (/ (* (- 1.0 (* maxCos maxCos)) ux) (- -1.0 maxCos))))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f + (((1.0f - (maxCos * maxCos)) * ux) / (-1.0f - maxCos));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 + (((1.0e0 - (maxcos * maxcos)) * ux) / ((-1.0e0) - maxcos))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - Float32(maxCos * maxCos)) * ux) / Float32(Float32(-1.0) - maxCos)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) + (((single(1.0) - (maxCos * maxCos)) * ux) / (single(-1.0) - maxCos));
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) + \color{blue}{ux} \cdot maxCos \]

   2. associate-+l+ N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)} \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)}\right) \]

   4. neg-mul-1 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(-1 \cdot ux + \color{blue}{ux} \cdot maxCos\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(-1 \cdot ux + maxCos \cdot \color{blue}{ux}\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(ux \cdot \color{blue}{\left(-1 + maxCos\right)}\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \color{blue}{\left(-1 + maxCos\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \color{blue}{-1}\right)\right)\right) \]

   9. +-lowering-+.f32 100.0%

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 + ux \cdot \left(maxCos + -1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(\left(maxCos + -1\right) \cdot \color{blue}{ux}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(\left(-1 + maxCos\right) \cdot ux\right)\right) \]

   3. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(\frac{-1 \cdot -1 - maxCos \cdot maxCos}{-1 - maxCos} \cdot ux\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(\frac{\left(-1 \cdot -1 - maxCos \cdot maxCos\right) \cdot ux}{\color{blue}{-1 - maxCos}}\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(\left(-1 \cdot -1 - maxCos \cdot maxCos\right) \cdot ux\right), \color{blue}{\left(-1 - maxCos\right)}\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(-1 \cdot -1 - maxCos \cdot maxCos\right), ux\right), \left(\color{blue}{-1} - maxCos\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 - maxCos \cdot maxCos\right), ux\right), \left(-1 - maxCos\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \left(maxCos \cdot maxCos\right)\right), ux\right), \left(-1 - maxCos\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(maxCos, maxCos\right)\right), ux\right), \left(-1 - maxCos\right)\right)\right) \]

   10. --lowering--.f32 100.0%

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(maxCos, maxCos\right)\right), ux\right), \mathsf{\_.f32}\left(-1, \color{blue}{maxCos}\right)\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto 1 + \color{blue}{\frac{\left(1 - maxCos \cdot maxCos\right) \cdot ux}{-1 - maxCos}} \]
7. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + ux \cdot \left(maxCos + -1\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (+ 1.0 (* ux (+ maxCos -1.0))))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f + (ux * (maxCos + -1.0f));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 + (ux * (maxcos + (-1.0e0)))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) + (ux * (maxCos + single(-1.0)));
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) + \color{blue}{ux} \cdot maxCos \]

   2. associate-+l+ N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)} \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)}\right) \]

   4. neg-mul-1 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(-1 \cdot ux + \color{blue}{ux} \cdot maxCos\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(-1 \cdot ux + maxCos \cdot \color{blue}{ux}\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(ux \cdot \color{blue}{\left(-1 + maxCos\right)}\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \color{blue}{\left(-1 + maxCos\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \color{blue}{-1}\right)\right)\right) \]

   9. +-lowering-+.f32 100.0%

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 + ux \cdot \left(maxCos + -1\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 3: 98.2% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (ux uy maxCos) :precision binary32 (- 1.0 ux))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f - ux;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 - ux
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) - ux)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) - ux;
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) + \color{blue}{ux} \cdot maxCos \]

   2. associate-+l+ N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)} \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)}\right) \]

   4. neg-mul-1 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(-1 \cdot ux + \color{blue}{ux} \cdot maxCos\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(-1 \cdot ux + maxCos \cdot \color{blue}{ux}\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(ux \cdot \color{blue}{\left(-1 + maxCos\right)}\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \color{blue}{\left(-1 + maxCos\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \color{blue}{-1}\right)\right)\right) \]

   9. +-lowering-+.f32 100.0%

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 + ux \cdot \left(maxCos + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{1 + -1 \cdot ux} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto 1 + \left(\mathsf{neg}\left(ux\right)\right) \]

   2. unsub-neg N/A

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

   3. --lowering--.f32 97.9%

      \[\leadsto \mathsf{\_.f32}\left(1, \color{blue}{ux}\right) \]

7. Simplified 97.9%

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

Alternative 4: 71.8% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos) :precision binary32 1.0)

C

float code(float ux, float uy, float maxCos) {
	return 1.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(1.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) + \color{blue}{ux} \cdot maxCos \]

   2. associate-+l+ N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)} \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)}\right) \]

   4. neg-mul-1 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(-1 \cdot ux + \color{blue}{ux} \cdot maxCos\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(-1 \cdot ux + maxCos \cdot \color{blue}{ux}\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{+.f32}\left(1, \left(ux \cdot \color{blue}{\left(-1 + maxCos\right)}\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \color{blue}{\left(-1 + maxCos\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \color{blue}{-1}\right)\right)\right) \]

   9. +-lowering-+.f32 100.0%

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 + ux \cdot \left(maxCos + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{1} \]
6. Step-by-step derivation

7. Simplified 71.4%

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

Reproduce

herbie shell --seed 2024149 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, z"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (- 1.0 ux) (* ux maxCos)))