UniformSampleCone, z

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 5

Speedup: 1.0×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. cancel-sign-sub 99.8%

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

   2. distribute-lft-neg-out 99.8%

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

   3. distribute-rgt-neg-out 99.8%

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

   4. sub-neg 99.8%

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

   5. associate-+r- 99.8%

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

   6. +-commutative 99.8%

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

   7. neg-mul-1 99.8%

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

   8. *-commutative 99.8%

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

   9. distribute-rgt-out-- 99.8%

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

   10. fma-define 99.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(ux, -1 - \left(-maxCos\right), 1\right)} \]

   11. sub-neg 99.9%

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

   12. remove-double-neg 99.9%

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

   13. +-commutative 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return 1.0f + (maxCos * (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 + (maxcos * (ux - (ux / maxcos)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Add Preprocessing

3. Taylor expanded in maxCos around inf 97.7%

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

   1. sub-neg 97.7%

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

   2. +-commutative 97.7%

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

   3. mul-1-neg 97.7%

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

   4. associate-+l+ 97.8%

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

   5. +-commutative 97.8%

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

   6. distribute-lft-out 98.6%

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

   7. mul-1-neg 98.6%

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

   8. unsub-neg 98.6%

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

   9. rgt-mult-inverse 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 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

Derivation

1. Initial program 99.8%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 4: 98.1% 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 99.8%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Add Preprocessing

3. Taylor expanded in maxCos around inf 97.7%

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

   1. sub-neg 97.7%

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

   2. +-commutative 97.7%

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

   3. mul-1-neg 97.7%

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

   4. associate-+l+ 97.8%

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

   5. +-commutative 97.8%

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

   6. distribute-lft-out 98.6%

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

   7. mul-1-neg 98.6%

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

   8. unsub-neg 98.6%

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

   9. rgt-mult-inverse 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in maxCos around 0 98.6%

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

   1. mul-1-neg 98.6%

      \[\leadsto 1 + \color{blue}{\left(-ux\right)} \]

   2. sub-neg 98.6%

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

8. Simplified 98.6%

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

Alternative 5: 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 99.8%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Add Preprocessing

3. Taylor expanded in maxCos around inf 97.7%

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

   1. sub-neg 97.7%

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

   2. +-commutative 97.7%

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

   3. mul-1-neg 97.7%

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

   4. associate-+l+ 97.8%

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

   5. +-commutative 97.8%

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

   6. distribute-lft-out 98.6%

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

   7. mul-1-neg 98.6%

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

   8. unsub-neg 98.6%

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

   9. rgt-mult-inverse 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in ux around 0 71.4%

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

Reproduce

herbie shell --seed 2024152 
(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)))